hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs ryan o’donnell...
TRANSCRIPT
![Page 1: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/1.jpg)
Hardness of Robust Graph Isomorphism, Lasserre Gaps,and Asymmetry of Random Graphs
Ryan O’Donnell (CMU)John Wright (CMU)
Chenggang Wu (Tsinghua)Yuan Zhou (CMU)
![Page 2: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/2.jpg)
Hardness of Robust Graph Isomorphism, Lasserre Gaps,and Asymmetry of Random Graphs
Ryan O’Donnell (CMU)John Wright (CMU)
Chenggang Wu (Tsinghua)Yuan Zhou (CMU)
![Page 3: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/3.jpg)
Motivating ExampleYesterday’s
Facebook graphYesterday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 4: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/4.jpg)
Motivating ExampleYesterday’s
Facebook graphYesterday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 5: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/5.jpg)
Motivating ExampleYesterday’s
Facebook graphYesterday’s
Facebook graph
Graph Isomorphism algorithm A should:• output “YES, same graph”• unscramble graph #2
![Page 6: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/6.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 7: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/7.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 8: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/8.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 9: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/9.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
Graph Isomorphism algorithm A
![Page 10: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/10.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
Graph Isomorphism algorithm A will:• output “NO, not isomorphic”• terminate
![Page 11: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/11.jpg)
Motivating ExampleYesterday’s
Facebook graphToday’s
Facebook graph
But these graphs are almost isomorphic!• can we detect this?• can we unscramble graph #2?
![Page 12: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/12.jpg)
Robust Graph Isomorphism
Given two “almost isomorphic” graphs,find the “best almost-isomorphism” between them
(or something close to it)
![Page 13: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/13.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 14: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/14.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 15: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/15.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 16: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/16.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 17: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/17.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 18: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/18.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 19: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/19.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 20: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/20.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 21: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/21.jpg)
G = (V(G), E(G)) H = (V(H), E(H))
Isomorphisms
π
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 22: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/22.jpg)
Isomorphisms
G = (V(G), E(G)) H = (V(H), E(H))
A bijection π:V(G) → V(H) is an isomorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(H)
![Page 23: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/23.jpg)
Isomorphisms, eq.
Pr[(π(u), π(v)) ∈ E(H)] = 1(u, v) E(G)
G = (V(G), E(G)) H = (V(H), E(H))
A bijection π:V(G) → V(H) is an isomorphism if
~ (assuming |E(G)| = |E(H)|)
![Page 24: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/24.jpg)
Isomorphisms, eq.
Pr[(π(u), π(v)) ∈ E(H)] = 1(u, v) E(G)
G = (V(G), E(G)) H = (V(H), E(H))
A bijection π:V(G) → V(H) is an isomorphism if
~
(uniformly random)
(assuming |E(G)| = |E(H)|)
![Page 25: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/25.jpg)
Isomorphisms, eq.
Pr[(π(u), π(v)) ∈ E(H)] = 1(u, v) E(G)
G = (V(G), E(G)) H = (V(H), E(H))
A bijection π:V(G) → V(H) is an isomorphism if
~ (assuming |E(G)| = |E(H)|)
![Page 26: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/26.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~ (assuming |E(G)| = |E(H)|)
![Page 27: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/27.jpg)
π
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~ (assuming |E(G)| = |E(H)|)
![Page 28: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/28.jpg)
π
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
Fact:π is an isomorphism ⇔ π is a 1-isomorphism
(assuming |E(G)| = |E(H)|)
![Page 29: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/29.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~ (assuming |E(G)| = |E(H)|)
![Page 30: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/30.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 31: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/31.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 32: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/32.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 33: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/33.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 34: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/34.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 35: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/35.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 36: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/36.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 37: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/37.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 38: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/38.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 39: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/39.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 40: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/40.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 41: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/41.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 42: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/42.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
π
(assuming |E(G)| = |E(H)|)
![Page 43: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/43.jpg)
π
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
Fact:This π is a ½-isomorphism.
(assuming |E(G)| = |E(H)|)
![Page 44: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/44.jpg)
Approximate Isomorphisms
Pr[(π(u), π(v)) ∈ E(H)] = α(u, v) E(G)
A bijection π:V(G) → V(H) is an α-isomorphism if
~
G and H are α-isomorphicif they have a β-isomorphism,
and β ≥ α.
(assuming |E(G)| = |E(H)|)
![Page 45: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/45.jpg)
Approximate GISO
(c, s)-approximate GISO
Given G and H, output:
• YES if G and H are c-isomorphic • NO if G and H are not s-isomorphic
Fact:(1, s)-approximate GISO is no harder than GISO.
Not so clear for (1-ε, s)-approximate GISO…
c > s,c “close to 1”s “far from 1”
![Page 46: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/46.jpg)
Robust GISO
Given G and H which are (1-ε)-isomorphic,output a (1-r(ε))-isomorphism.
(r(ε)→0 as ε→0+)
• Robust algorithms previously studied for CSPs• a characterization conjectured by [Guruswami and Zhou 2011]• confirmed by [Barto and Kozik 2012]
• Robust GISO introduced in [WYZV 2013]• gives a robust GISO algorithm if G and H are trees• which other classes of graphs have robust GISO algorithms?
![Page 47: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/47.jpg)
Approximate GISO, a brief history
• [Arora et al. 2002] give a PTAS for this problem in the case of dense graphs– Our graphs will be sparse, i.e. m = O(n)
• [Arvind et al. 2012] have shown hardness of approximation results for variants of our problem– e.g., GISO with colored graphs
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GISO Hardness
• Famously not known to be in P or NP-complete
• Evidence that it’s not NP-complete
• What about robust GISO?
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Our result
Assume Feige’s Random 3XOR Hypothesis.Then there is no poly-time algorithm for Robust GISO.
Thm:
There exists a constant ε0 such that:For all ε > 0, no poly-time algorithm can distinguish between:• (1-ε)-isomorphic graphs G and H• not (1-ε0)-isomorphic graphs G and H
In other words, no poly-time algorithm solves (1-ε, 1-ε0)-approximate GISO.
(constantly far apart)
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Our (newer) result
Assume RP ≠ NP.Then there is no poly-time algorithm for Robust GISO.
Thm:
There exists a constant ε0 such that:For all ε > 0, no poly-time algorithm can distinguish between:• (1-ε)-isomorphic graphs G and H• not (1-ε0)-isomorphic graphs G and H
In other words, no poly-time algorithm solves (1-ε, 1-ε0)-approximate GISO.
(constantly far apart)
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Algorithms for GISO
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GISO Algorithms
Algorithm Runtime
Brute force
Weisfeiler-Lehman (WL) algorithm
[Babai Luks 83]
O(n!) ≈ O(2n log n)
exp(O(n log n))
exp(O(n log n)1/2)
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WLk algorithm
• Standard heuristic for GISO• Larger k, more powerful. Runs in time nk + O(1).• By [Atserias and Maneva 2013], equivalent to
something familiar:
WLk
Level-(k+1) Sherali-Adams LP for GISO
Level-k Sherali-Adams LP for GISOWLk-1
Level-(k-1) Sherali-Adams LP for GISO
…
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WLk/Level-k Sherali-Adams LP
• “Super LP”• Once speculated that WLk solves GISO with
k= O(log n)• Some graphs require k = Ω(n) rounds
[Cai, Fürer, Immerman 1992]
• How do SDPs do? What about the Lasserre/SOS “Super-Duper SDP”?
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Cai, Fürer, Immerman Instance
• Some graphs require k = Ω(n) rounds [Cai, Fürer,
Immerman 1992]• Basically encoded a 3XOR instance as a pair of
graphs.• Our main theorem is similar – we reduce from
random 3XOR.• Known that random 3XOR is as hard as
possible for Lasserre SDP [Schoenebeck 2008]
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Our result
There exists a constant ε0 such thatΩ(n) levels of the Lasserre/SOS hierarchy
are needed to distinguish:
Thm:
• YES: G and H are isomorphic • NO: G and H are (1-ε0)-isomorphic
(constantly far apart)
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Our proof
Assume Feige’s Random 3XOR Hypothesis.Then there is no algorithm for Robust GISO.
Thm:
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Our proof
• By a reduction from (a variant of) 3XOR
Instance I
x1 + x2 + x3 = 0 (mod 2)
x10 + x15 + x1 = 1 (mod 2)
x4 + x5 + x12 = 1 (mod 2)
…
x7 + x8 + x9 = 0 (mod 2)
xi ∈ {0, 1}
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Our proof
• By a reduction from (a variant of) 3XOR
Instance I
x1 + x2 + x3 = 0 (mod 2)
x10 + x15 + x1 = 1 (mod 2)
x4 + x5 + x12 = 1 (mod 2)
…
x7 + x8 + x9 = 0 (mod 2)
000 010 100 110001 011 101 111
xi ∈ {0, 1}
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Our proof
• By a reduction from (a variant of) 3XOR
Instance I
x1 + x2 + x3 = 0 (mod 2)
x10 + x15 + x1 = 1 (mod 2)
x4 + x5 + x12 = 1 (mod 2)
…
x7 + x8 + x9 = 0 (mod 2)
000 010 100 110001 011 101 111
xi ∈ {0, 1}
![Page 61: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/61.jpg)
Our proof
• By a reduction from (a variant of) 3XOR
Instance I
x1 + x2 + x3 = 0 (mod 2)
x10 + x15 + x1 = 1 (mod 2)
x4 + x5 + x12 = 1 (mod 2)
…
x7 + x8 + x9 = 0 (mod 2)
3XOR easy to solve on satisfiable instances (Gaussian elimination)
Thm: [Håstad 2001]
Given a 3XOR instance I, it is NP-hard to distinguish between:
• YES: I is (1 - ε)-satisfiable• NO: I is (½ + ε)-satisfiable
xi ∈ {0, 1}
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Our proof
• By a reduction from (a variant of) 3XOR
Instance I
x1 + x2 + x3 = 0 (mod 2)
x10 + x15 + x1 = 1 (mod 2)
x4 + x5 + x12 = 1 (mod 2)
…
x7 + x8 + x9 = 0 (mod 2)
3XOR easy to solve on satisfiable instances (Gaussian elimination)
Thm: [Håstad 2001]
Given a 3XOR instance I, it is NP-hard to distinguish between:
• YES: I is almost-satisfiable• NO: I is far-from-satisfiable
xi ∈ {0, 1}
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Our proof
• By a reduction from (a variant of) 3XOR
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Far-from-satisfiable3XOR instance I
Far-from-isomorphicgraphs (G, H)
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Our proof
• By a reduction from (a variant of) 3XOR
Almost-satisfiable3XOR instance I
Far-from-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Far-from-isomorphicgraphs (G, H)
(reduction)
✔
✗(only works for most far-from-satisfiable 3XOR instances)
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Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
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Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
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Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
x3 + x5 + x7 = ? (mod 2)
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Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
x3 + x5 + x7 = ? (mod 2)
…
![Page 69: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/69.jpg)
Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
x3 + x5 + x7 = ? (mod 2)
…
x9 + x10 + xn = ? (mod 2)
![Page 70: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/70.jpg)
Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
x3 + x5 + x7 = ? (mod 2)
…
x9 + x10 + xn = ? (mod 2)
![Page 71: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/71.jpg)
Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = ? (mod 2)
x3 + x5 + x7 = ? (mod 2)
…
x9 + x10 + xn = ? (mod 2)
![Page 72: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/72.jpg)
Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = 0 (mod 2)
x3 + x5 + x7 = 1 (mod 2)
…
x9 + x10 + xn = 1 (mod 2)
![Page 73: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/73.jpg)
Random 3XOR
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
• n variables• m equations (m = C * n)
Of thepossible sets of size 3, pick m of them
( )n3
x1 + x2 + x3 = 0 (mod 2)
x3 + x5 + x7 = 1 (mod 2)
…
x9 + x10 + xn = 1 (mod 2) For some C > 0,~50%-satisfiable whp.
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Feige’s R3XOR Hypothesis
No poly-time algorithm can distinguish between:• an almost-satisfiable 3XOR instance• a random 3XOR instance
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Feige’s R3XOR Hypothesis
No poly-time algorithm can distinguish between:• an almost-satisfiable 3XOR instance• a random 3XOR instance
• Well-believed and “standard” complexity assumption
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Feige’s R3XOR Hypothesis
No poly-time algorithm can distinguish between:• an almost-satisfiable 3XOR instance• a random 3XOR instance
• Well-believed and “standard” complexity assumption
• Variants of this hypothesis used as basis for cryptosystems, hardness of approximation results, etc.
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Feige’s R3XOR Hypothesis
No poly-time algorithm can distinguish between:• an almost-satisfiable 3XOR instance• a random 3XOR instance
• Well-known complexity assumption
• Variants of this hypothesis used as basis for cryptosystems, hardness of approximation results, etc.
• Solvable in time 2O(n/log(n)) [Blum, Kalai, Wasserman 2003]
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Our proof
• Assume Feige’s R3XOR Hypothesis
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Far-from-satisfiable3XOR instance I
Far-from-isomorphicgraphs (G, H)
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Our proof
• Assume Feige’s R3XOR Hypothesis
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)
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Our proof
• Assume Feige’s R3XOR Hypothesis
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)(w.h.p.)
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The reduction
• Assume graph:3XOR instances → graphs
![Page 82: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/82.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
![Page 83: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/83.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 0 (mod 2)
…
x1 + x3 + x8 = 0 (mod 2)
instance sat(I)
![Page 84: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/84.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 0 (mod 2)
…
x1 + x3 + x8 = 0 (mod 2)
instance sat(I)
Fact:sat(I) is satisfiable (just set xi’s to 0)
![Page 85: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/85.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 0 (mod 2)
…
x1 + x3 + x8 = 0 (mod 2)
instance sat(I)
![Page 86: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/86.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 0 (mod 2)
…
x1 + x3 + x8 = 0 (mod 2)
instance sat(I)
• G := graph(I)• H := graph(sat(I))Output (G, H)
![Page 87: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/87.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
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Equation 0-Gadgetx + y + z = 0 (mod 2) 000 010 100 110
001 011 101 111
good assignments
![Page 89: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/89.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
000 010 100 110001 011 101 111
good assignments
![Page 90: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/90.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 91: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/91.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 92: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/92.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 93: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/93.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 94: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/94.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 95: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/95.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 96: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/96.jpg)
Equation 0-Gadgetx + y + z = 0 (mod 2)
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
Equation vertices:
000 010 100 110001 011 101 111
good assignments
![Page 97: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/97.jpg)
Equation 0-gadget, zoomed out
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
x + y + z = 0 (mod 2)
![Page 98: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/98.jpg)
Equation 0-gadget, zoomed out
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 0
x → 0y → 1z → 1
x → 1y → 0z → 1
x → 1y → 1z → 0
x y z
=
x + y + z = 0 (mod 2)
![Page 99: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/99.jpg)
Equation 1-Gadgetx + y + z = 1 (mod 2) 000 010 100 110
001 011 101 111
good assignments
![Page 100: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/100.jpg)
Equation 1-Gadgetx + y + z = 1 (mod 2) 000 010 100 110
001 011 101 111
good assignments
Variable vertices:
x
0 1
y
0 1
z
0 1
![Page 101: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/101.jpg)
Equation 1-Gadgetx + y + z = 1 (mod 2) 000 010 100 110
001 011 101 111
good assignments
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 1
x → 0y → 1z → 0
x → 1y → 0z → 0
x → 1y → 1z → 1
Equation vertices:
![Page 102: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/102.jpg)
Equation 1-Gadgetx + y + z = 1 (mod 2) 000 010 100 110
001 011 101 111
good assignments
Variable vertices:
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 1
x → 0y → 1z → 0
x → 1y → 0z → 0
x → 1y → 1z → 1
Equation vertices:
![Page 103: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/103.jpg)
Equation 1-gadget, zoomed out
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 1
x → 0y → 1z → 0
x → 1y → 0z → 0
x → 1y → 1z → 1
x + y + z = 1 (mod 2)
![Page 104: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/104.jpg)
Equation 1-gadget, zoomed out
x
0 1
y
0 1
z
0 1
x → 0y → 0z → 1
x → 0y → 1z → 0
x → 1y → 0z → 0
x → 1y → 1z → 1
x + y + z = 1 (mod 2)
x y z
=
![Page 105: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/105.jpg)
graph: 3XOR instance → Graph
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
![Page 106: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/106.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
→
![Page 107: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/107.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 108: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/108.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 109: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/109.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 110: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/110.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 111: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/111.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 112: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/112.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 113: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/113.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 114: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/114.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 115: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/115.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 116: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/116.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 117: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/117.jpg)
graph: 3XOR instance → Graph
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
3XOR instance
Eq 1: x1 + x2 + x3 = 0 (mod 2)
Eq 2: x1 + x4 + xn = 1 (mod 2)
…
…
Eq m: x7 + x11 + xn = 1 (mod 2)
Place the equation gadget over equation and variable blobs
→
![Page 118: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/118.jpg)
The reduction
• Assume graph:3XOR instances → graphs
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 1 (mod 2)
…
x1 + x3 + x8 = 1 (mod 2)
3XOR instance I
x1 + x2 + x3 = 0 (mod 2)
x2 + x4 + xn = 0 (mod 2)
…
x1 + x3 + x8 = 0 (mod 2)
instance sat(I)
• G := graph(I)• H := graph(sat(I))Output (G, H)
![Page 119: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/119.jpg)
Zoomed-out picture
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
H
![Page 120: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/120.jpg)
Need to show
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)(w.h.p.)
![Page 121: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/121.jpg)
Need to show
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)(w.h.p.)
Completeness:
![Page 122: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/122.jpg)
Completeness
• Almost-satisfiable 3XOR instance I
![Page 123: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/123.jpg)
Completeness
• Satisfiable 3XOR instance I
![Page 124: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/124.jpg)
Completeness
• Satisfiable 3XOR instance I• Let f:{xi} → {0, 1} be a satisfying assignment
![Page 125: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/125.jpg)
Completeness
• Satisfiable 3XOR instance I• Let f:{xi} → {0, 1} be a satisfying assignment• We’ll use f to construct an isomorphism π
between G and H
![Page 126: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/126.jpg)
Completeness
• Satisfiable 3XOR instance I• Let f:{xi} → {0, 1} be a satisfying assignment• We’ll use f to construct an isomorphism π
between G and H• What should π look like?
![Page 127: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/127.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
01
01
![Page 128: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/128.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
If f(x1) = 0, π maps 0 vertex to 0 and 1
vertex to 1
01
01
![Page 129: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/129.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
If f(x2) = 1, π swaps the 0and the 1 vertices
![Page 130: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/130.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
If f(x3) = 1, π swaps the 0and the 1 vertices
![Page 131: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/131.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
?
![Page 132: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/132.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
?Fact:
• For every good even assignment, the 0-gadget has an isomorphism with the 0-gadget• For every good odd assignment, the 1-gadget has an isomorphism with the 0-gadget
![Page 133: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/133.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 134: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/134.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 135: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/135.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 136: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/136.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 137: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/137.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 138: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/138.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 139: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/139.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 140: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/140.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 141: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/141.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 142: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/142.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 143: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/143.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 144: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/144.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 145: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/145.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 146: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/146.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 147: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/147.jpg)
Gadget Isomorphism
(0,0,1) (0,1,0) (1,0,0) (1,1,1) (0,0,0) (0,1,1) (1,0,1) (1,1,0)
x
0 1
y
0 1
z
0 1
x
0 1
y
0 1
z
0 1
π only swaps the z vertices
000 010 100 110001 011 101 111
good assignments
x+y+z = 1 (mod 2) x+y+z = 0 (mod 2)
![Page 148: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/148.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
• Define π to be the appropriate isomorphism for each equation.• This π is an isomorphism between G and H.
![Page 149: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/149.jpg)
Constructing π
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
f:{xi} → {0, 1} a satisfying assignment
• Define π to be the appropriate isomorphism for each equation.• This π is an isomorphism between G and H. ✔
![Page 150: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/150.jpg)
Need to show
(reduction)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)(w.h.p.)
(take my word for this … almost)
✔Soundness:
![Page 151: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/151.jpg)
Need to show
Random3XOR instance I
Far-from-isomorphicgraphs (G, H)(w.h.p.)
Almost-satisfiable3XOR instance I
Almost-isomorphicgraphs (G, H)
G and H are almost-isomorphic a (1-⇒ ε)-isomorphism π
• What must be true about π?
(w.h.p.)
![Page 152: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/152.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
H
![Page 153: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/153.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
Hπ
![Page 154: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/154.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
Hπ
![Page 155: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/155.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
Hπ
![Page 156: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/156.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
Hπ
![Page 157: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/157.jpg)
A possible π?
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
HπTake my word for this:If π is a (1-ε)-isomorphism, then none of these can happen (often).
![Page 158: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/158.jpg)
Green blobs map to green blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
π
G H
![Page 159: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/159.jpg)
Green blobs map to green blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
π
G H
![Page 160: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/160.jpg)
Green blobs map to green blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
π
G H
![Page 161: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/161.jpg)
Blue blobs map to blue blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
π
G H
![Page 162: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/162.jpg)
Blue blobs map to blue blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xnπ
G H
![Page 163: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/163.jpg)
Blue blobs map to blue blobs
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
πG H
![Page 164: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/164.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G1
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G2
π
G1 G2
![Page 165: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/165.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G1
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G2
π
Trust me on this:If π looks like this, then the rest of the proof goes through.
G1 G2
![Page 166: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/166.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G1
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G2
π
Trust me on this:If π looks like this, then the rest of the proof goes through.
G1 G2When does this fail?
![Page 167: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/167.jpg)
When does this fail?
• Fails when equation graph exhibits a lot of symmetry.
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
![Page 168: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/168.jpg)
When does this fail?
• Fails when equation graph exhibits a lot of symmetry.
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
e.g.:
![Page 169: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/169.jpg)
When does this fail?
• Fails when equation graph exhibits a lot of symmetry.
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
e.g.:
What do G and H look like?
![Page 170: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/170.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
G H
![Page 171: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/171.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 172: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/172.jpg)
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
Too much symmetry
![Page 173: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/173.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 174: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/174.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 175: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/175.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 176: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/176.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 177: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/177.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 178: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/178.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 179: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/179.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 180: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/180.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 181: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/181.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 182: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/182.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 183: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/183.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 184: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/184.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
![Page 185: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/185.jpg)
Too much symmetry
Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…Eq1 Eq2 Eq3
x1 x2 x3 x4 x5 x6…
…
Hπ could just shift everything over by one
This is not what we wanted!
![Page 186: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/186.jpg)
Why could we do this?
• The equation graph had a lot of symmetry.
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
![Page 187: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/187.jpg)
When does this fail?
• The equation graph had a lot of symmetry.
• But this graph was chosen randomly!
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
![Page 188: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/188.jpg)
When does this fail?
• The equation graph had a lot of symmetry.
• But this graph was chosen randomly!
• Maybe random graphs usually have very little symmetry?
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
![Page 189: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/189.jpg)
When does this fail?
• The equation graph had a lot of symmetry.
• But this graph was chosen randomly!
• Maybe random graphs usually have very little symmetry?
• If so, then we usually get our dream scenario.
x1x2
x3
x4
x5
x6x7
x8
xn
x9
x10
…
![Page 190: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/190.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
π
G H
![Page 191: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/191.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G1
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G2
π
Trust me on this:If π looks like this, then the rest of the proof goes through.
![Page 192: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/192.jpg)
A dream scenario
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G1
x1…
Eq1 Eq2 Eqm
…
x2 x3 xn
G2
π
Trust me on this:If π looks like this, then the rest of the proof goes through.
✔
![Page 193: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/193.jpg)
Robust asymmetryof random graphs
![Page 194: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/194.jpg)
Symmetric graphs
A symmetric graphis one in which you can rearrange the vertices
and get back the same graph.
![Page 195: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/195.jpg)
Symmetric graphs
A symmetric graphis one in which you can rearrange the vertices
and get back the same graph.
e.g.
![Page 196: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/196.jpg)
Symmetric graphs
A symmetric graphis one in which you can rearrange the vertices
and get back the same graph.
e.g. An asymmetric graph:
![Page 197: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/197.jpg)
Symmetric graphs (formally)
A symmetric graphis one with a nontrivial automorphism.
![Page 198: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/198.jpg)
Symmetric graphs (formally)
A symmetric graphis one with a nontrivial automorphism.
A permutation π on V(G) is an automorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(G)
![Page 199: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/199.jpg)
Symmetric graphs (formally)
A symmetric graphis one with a nontrivial automorphism.
A permutation π on V(G) is an automorphism if
(u, v) ∈ E(G) ⇔ (π(u), π(v)) ∈ E(G)
Fact: The (trivial) identity permutation π(v) = vis always an automorphism.
![Page 200: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/200.jpg)
Random graphs are asymmetric
• G(n, p) is asymmetric with high probability when . [Erdős and Rényi 63]ln n
n≤ p ≤ 1 – ln n
n
![Page 201: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/201.jpg)
Random graphs are asymmetric
• G(n, p) is asymmetric with high probability when . [Erdős and Rényi 63]
• Random d-regular graphs are asymmetric w.h.p. when 3 ≤ d ≤ n – 4.
ln nn
≤ p ≤ 1 – ln nn
![Page 202: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/202.jpg)
Random graphs are asymmetric
• G(n, p) is asymmetric with high probability when . [Erdős and Rényi 63]
• Random d-regular graphs are asymmetric w.h.p. when 3 ≤ d ≤ n – 4.
• What about (hyper-)graphs with m edges?
ln nn
≤ p ≤ 1 – ln nn
![Page 203: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/203.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
![Page 204: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/204.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
Does G have a good α-automorphism?Try 1:
![Page 205: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/205.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
Does G have a good α-automorphism?every graph has a 1-automorphism (identity permutation)
Try 1:
![Page 206: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/206.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
Does G have a good α-automorphism?every graph has a 1-automorphism (identity permutation)
Ignoring the identity permutation, does G have a good α-automorphism?
Try 1:
Try 2:
✗
![Page 207: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/207.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
Does G have a good α-automorphism?every graph has a 1-automorphism (identity permutation)
Ignoring the identity permutation, does G have a good α-automorphism?
Try 1:
Try 2:
✗every graph has a .99999-automorphism:
π(1) = 2π(2) = 1π(3) = 3π(4) = 4π(5) = 5
π(6) = 6π(7) = 7π(8) = 8 …π(n) = n
![Page 208: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/208.jpg)
Approximate automorphisms
Pr[(π(u), π(v)) ∈ E(G)] = α(u, v) E(G)
A permutation π on V(G) is an α-automorphism if
~
Does G have a good α-automorphism?every graph has a 1-automorphism (identity permutation)
Ignoring the identity permutation, does G have a good α-automorphism?
Try 1:
Try 2:
✗every graph has a .99999-automorphism:
Does G have a good α-automorphism which is far from the identity??
Try 3:
✗π(1) = 2π(2) = 1π(3) = 3π(4) = 4π(5) = 5
π(6) = 6π(7) = 7π(8) = 8 …π(n) = n
![Page 209: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/209.jpg)
Random graphs arerobustly asymmetric
Let G be a random n-vertex m-edge graph.Then WHP, any (1-ε)-automorphism for G is
O(ε)-close to the identity, for any large enough ε.
Thm:
![Page 210: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/210.jpg)
Random graphs arerobustly asymmetric
Let G be a random n-vertex m-edge graph.Then WHP, any (1-ε)-automorphism for G is
O(ε)-close to the identity, for any large enough ε.
Some restrictions:
• C*n ≤ m ≤
• ε ≥ ε0
Cn2
Also works for hypergraphs.
Thm:
![Page 211: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/211.jpg)
Q.E.D.
![Page 212: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/212.jpg)
Open Problems
![Page 213: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/213.jpg)
Can we explicitly constructrobustly asymmetric graphs?
• Currently, we can only generate robustly asymmetric graphs randomly.
![Page 214: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/214.jpg)
Can we explicitly constructrobustly asymmetric graphs?
• Currently, we can only generate robustly asymmetric graphs randomly.
• An explicit construction would prove NP-hardness of robust Graph Isomorphism.
![Page 215: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/215.jpg)
Can we explicitly constructrobustly asymmetric graphs?
• Currently, we can only generate robustly asymmetric graphs randomly.
• An explicit construction would prove NP-hardness of robust Graph Isomorphism.
• We don’t really have any great candidates yet…
![Page 216: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/216.jpg)
Improving hardness for approximating GISO
![Page 217: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/217.jpg)
PCP Theorem for GISO
There exists a constant ε0 such that:For all ε > 0, no poly-time algorithm can distinguish between:• (1-ε)-isomorphic graphs G and H• (1-ε0)-isomorphic graphs G and H
Our theorem:
![Page 218: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/218.jpg)
PCP Theorem for GISO
For all ε > 0, no poly-time algorithm can distinguish between:• (1-ε)-isomorphic graphs G and H• .99999…9999-isomorphic graphs G and H
Our theorem:
For all ε > 0, it is NP-hard to distinguish between:• satisfiable 3Sat instance • .99999…9999-satisfiable 3Sat instance
PCP theorem:
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A historical parallel?
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• .999…99-satisfiable
Given two graphs, can’t tell if they’re:• nearly isomorphic• .999…99-isomorphic
![Page 220: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/220.jpg)
A historical parallel?
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• .999…99-satisfiable
Parallel repetition [Raz 1995]
Given two graphs, can’t tell if they’re:• nearly isomorphic• .999…99-isomorphic
![Page 221: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/221.jpg)
A historical parallel?
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• .999…99-satisfiable
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• ~7/8-satisfiable
Parallel repetition [Raz 1995]
Given two graphs, can’t tell if they’re:• nearly isomorphic• .999…99-isomorphic
![Page 222: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/222.jpg)
A historical parallel?
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• .999…99-satisfiable
Given 3Sat instance, can’t tell if it’s:• fully satisfiable• ~7/8-satisfiable
Parallel repetition & Long code reduction [Raz 1995, Håstad 2001]
Given two graphs, can’t tell if they’re:• nearly isomorphic• .999…99-isomorphic
Parallel repetition for graphs?
???
![Page 223: Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs Ryan O’Donnell (CMU) John Wright (CMU) Chenggang Wu (Tsinghua) Yuan](https://reader037.vdocument.in/reader037/viewer/2022110319/56649c795503460f9492e5e4/html5/thumbnails/223.jpg)
An attempt: tensor product
• Given G = (V, E), G⊗G is graph with vertex set V×V and edge set
• Known that if G and H are not isomorphic, then G⊗G and H⊗H are not isomorphic. (under some mild conditions)
• If G and H are not (1-ε) isomorphic, are G⊗G and H⊗H not (1-ε)2 isomorphic?
((u1, u2), (v1, v2)) ∈ E(G⊗G) ⇔ (u1, v1) ∈ E and (u2, v2) ∈ E
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thanks!