introduction to kernel lower bounds daniel lokshtanov
TRANSCRIPT
![Page 1: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/1.jpg)
Introduction to Kernel Lower Bounds
Daniel Lokshtanov
![Page 2: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/2.jpg)
What?
• Kernelization is a mathematical framework to analyze the quality of polynomial time pre-processing
• Until recently: Many upper bounds known. No ”non-trivial” lower bounds.
• This talk: Survey of recent lower bounds.
![Page 3: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/3.jpg)
Part I
Introduction to Kernelization
![Page 4: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/4.jpg)
Parameterization
• Hard to analyze pre-processing for NP-hard problems within classical complexity. Reason: poly-time compression = poly-time solution.
• We consider parameterized problems. Each instance I comes with a parameter k ≤ |I| that is supposed to reflect how hard the instance is. Small k = easier instance.
![Page 5: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/5.jpg)
Parameterization: Example
Point Line Cover
IN: n points in the plane, integer k.PARAMETER: kQUESTION: Can the points be covered by k
straight lines?
Notice – easier to solve when k is small.
![Page 6: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/6.jpg)
Kernelization
A f(k)-kernel for a problem P is an algorithm that:
• Takes as input an instance (I,k)• Runs in time poly(|I|)• Outputs an equivalent instance (I’,k’) with– |I’| ≤ f(k)– k’ ≤ f(k)
![Page 7: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/7.jpg)
Point Line Cover
IN: n points in the plane, integer k.PARAMETER: kQUESTION: Can the points be covered by k
straight lines?
![Page 8: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/8.jpg)
Point Line Cover
TASK: Shoot the little devils, with only 3 shots.If some line covers 4 devils, must use it. Otherwise need 4 shots.
![Page 9: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/9.jpg)
k2 - kernel for Point Line Cover
• R1: If some line covers more than k points delete all points on the line and decrease k by 1.
• R2: If no line covers at least n/k points, answer ”NO”
• If neither R1 nor R2 can be applied n ≤ k2
![Page 10: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/10.jpg)
Edge Clique Cover
IN: Graph G, integer k.PARAMETER: kQUESTION: Can the edges of G be covered by k
cliques?
![Page 11: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/11.jpg)
4k - Kernel for Edge Clique Cover
R1: If u and v are adjacent and have same neighbours, delete v.
R2: If R1 can’t be applied and n > 2k, answer NO.
If R1, R2 can’t be applied, then n < 2k and m < 4k.
![Page 12: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/12.jpg)
Recap
A k2 kernel for Point Line Cover polynomial kernel
A 4k kernel for Edge Clique Cover exponential kernel
Which all parameterized problems have f(k)-kernels for some function f?
Which parameterized problems have poly(k)- kernels?
![Page 13: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/13.jpg)
Which problems have f(k) - kernels?
Theorem[Folklore]: A decidable parameterized problem P has an
f(k)-kernel for some f
P is fixed parameter tractable (FPT), i.e. solvable in time g(k)nO(1) for some g.
![Page 14: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/14.jpg)
Kernelization Complexity
Q1: Does P have an f(k) kernel?P is FPT YESP is W-hard NO, unless FPT=W[1]
Q2: Does P have a poly(k) kernel.poly(k) kernel YESHow to say NO?
![Page 15: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/15.jpg)
Part II
Framework for ruling out polynomial kernels
![Page 16: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/16.jpg)
Longest Path
IN: Graph G, integer kPARAMETER: kQUESTION: Does G have a path of length k?
Known: 2knc time algorithm [Williams 09]Does Longest Path have a polynomial kernel?
![Page 17: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/17.jpg)
Poly kernel for Longest Path?
Suppose Longest Path has a kc kernel.
Set t = kc + 1 and consider t instances with the same parameter k: (G1,k), (G2,k) ... (Gt,k)
The instance (G1 U G2 ... U Gt, k) is a yes instance iff some (Gi, k) is.
Kernelize this instance – the kernel has kc < t bits. Less than one bit per original instance, was at least one of the instances ”solved”?
![Page 18: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/18.jpg)
Poly Kernel for Longest Path?
G1,k G2,k Gt,k
...Disjoint union
G’,k’
... G,k
Polynomial kernel
![Page 19: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/19.jpg)
OR-Distillation Algorithms
Detour back to classical problems.
An OR-distillation algorithm for a problem L• Takes as input instances I1... It.• Runs in polynomial time• Outputs an instance O of L’ such that– |O| ≤ max poly(|Ii|)
– O is ”yes” some Ii is “yes”.
![Page 20: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/20.jpg)
OR-Distillation Algorithms
Intuition: A distillation algorithm looks at several problem instances and pics the one ”most likely” to be a yes instance.
Should not exist for NP-hard problems.
Theorem [FS08]: Unless coNP NP/poly⊆ , no NP-hard problem has an OR-distillation algorithm.
![Page 21: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/21.jpg)
OR-Composition algorithms: Intuition
OR-Composition = ”formalization of disjoint union”
OR-Composition + Kernel = OR-Distillation
![Page 22: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/22.jpg)
OR-Composition Algorithms
Back to parameterized problems.
An OR-composition algorithm for a problem P• Takes as input instances I1 ... It with parameter k • Runs in polynomial time• Outputs an instance (O,k’) of P such that– k’ ≤ poly(k)– (O,k’) is ”yes” some (Ii,k) is “yes”.
![Page 23: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/23.jpg)
OR-Composition for Longest Path
G1,k G2,k Gt,k
...Disjoint union
... G,k
![Page 24: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/24.jpg)
Ruling Out Polynomial Kernels
Theorem [BDFH08]: If a parameterization P of an NP-hard* problem L has a composition algorithm, then P has no polynomial kernel unless coNP NP/poly⊆ .
Corollary [BDFH08]: Longest Path has no polynomial kernel unless coNP NP/poly⊆ .
* Originally proved only for NP-complete. New statement/proof by Holger Dell
![Page 25: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/25.jpg)
Proof of [BDFH08]-Theorem
Given OR-Composition + Kernel for P we give an OR-distillation for L into OR(L). By [FS08] this implies that coNP NP/poly⊆ .
![Page 26: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/26.jpg)
I1 I2 I3 Itt instances of size n...
I1,1 I2,1 I3,2 It,n...
Parameterization
Group by parameter
OR-Composition
O1,k1 O2,k2 On,kn...n instances instead of t. ki ≤ poly(n)
![Page 27: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/27.jpg)
O1,k1 O3,k2 On,kn...n instances instead of t. ki ≤ poly(n)
Kernelization
O’1,k’1 O’2,k’2 O’n,k’n...n instances of sizepoly(n) each.
Forget parameter
O’1 O’2 O’n...n instances of sizepoly(n) each. This is one instance to OR(L)of size poly(n)
![Page 28: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/28.jpg)
Recap II
NP-hard + OR-composition = no poly kernel.
Longest Path has no polynomial kernelLongest Cycle has no polynomial kernel...
![Page 29: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/29.jpg)
AND-Distillations / Compositions
• We can define AND-Distillation / Composition similarly to the OR case
• AND-Composition + Kernel = AND-Distillation
• Conjecture [BDFH08]: No NP-hard problem has an AND-Distillation.
![Page 30: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/30.jpg)
AND-Compositions
• Some interesting problems have AND-compositions;– treewidth – pathwidth – ...width – vertex ranking
• Under ”AND-Distillation Conjecture” they have no polynomial kernel.
![Page 31: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/31.jpg)
Open Problem
Relate the ”AND-Distillation” conjecture to a reasonable assumption in classical / parameterized complexity
![Page 32: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/32.jpg)
Part III
Kernel lower bounds for more problems
![Page 33: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/33.jpg)
Next
Polynomial Parameter Transformations: Reductions to show kernel lower bouds
”Non-trivial” OR-Composition algorithms
![Page 34: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/34.jpg)
k-k-Paths
IN: Graph G, integer kPARAMETER: kQUESTION: Does G contain k vertex-disjoint k-
paths?
Disjoint union doesnt work as OR-composition. Other way to show no poly kernel?
![Page 35: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/35.jpg)
Polynomial Parameter Transformations
A Polynomial Parameter Transformation (PPT) from A to B is an algorithm that:
• Takes as input an instance (I,k) of A• Runs in polynomial time• Outputs an instance (O,k’) of B such that– k’ ≤ poly(k)– (O,k’) is ”yes” for B (I,k) is “yes” for A.
![Page 36: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/36.jpg)
Reduction between problems
Theorem [BTY09]:If there is a PPT from A to B, and a P-time reduction from B to A* then:B has a poly(k) kernel A has a poly(k) kernel
*If B is NP and A is NP-hard, a trivial p-time reduction exists.
![Page 37: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/37.jpg)
Proof of Theorem [BTY09]:
I,k I’,k’
O’,k’O*,k*
PPT
Kernel
P-timereduction
A
A
B
B
![Page 38: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/38.jpg)
Back to k-k-Paths
Theorem [L09]:to k-k-Paths have no polynomial kernel unless
coNP NP/poly⊆
G,k
k-Path
G,k
k-k-Paths
k-1 paths of length k
NP-completenessgives reductionback.
![Page 39: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/39.jpg)
Non-trivial Compositions?
• Next, excluding polynomial kernels for:– Bounded Universe Set Cover– Connected Vertex Cover (2-approximable!)– Steiner Tree
![Page 40: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/40.jpg)
Bounded Universe Set Cover
IN: Set family F={S1...Sm} over a universe U of size k, integer t
PARAMETER: kQUESTION: Is there a subfamily F’ F ⊆ of size ≤ t
such that F’ covers U?
Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP NP/poly⊆ .
![Page 41: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/41.jpg)
Steiner Tree
IN: Graph G=(V,E), subset S V⊆ of size k, integer tPARAMETER: tQUESTION: Is there a subtree T on ≤ t vertices of G,
containing S?
![Page 42: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/42.jpg)
Steiner Tree
Theorem [DLS09]: Steiner Tree has no poly(k) kernel unless coNP NP/poly⊆ .
Proof: PPT from Bounded Universe Set Cover
Universe Terminals
Sets Non-Terminals
![Page 43: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/43.jpg)
Connected Vertex Cover
IN: Graph G=(V,E) integer k.PARAMETER: kQUESTION: Is there a set S of at most k vertices
such that G[S] is connected and every edge if G has at least one endpoint in S.
![Page 44: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/44.jpg)
Connected Vertex Cover
Theorem [DLS09]: Connected Vertex Cover has no poly(k) kernel unless coNP NP/poly⊆ .
Proof: PPT from Steiner Tree
Terminals
Non-Terminals
![Page 45: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/45.jpg)
Bounded Universe Set Cover
Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP NP/poly⊆ .
Proof plan: – Composition for ”Colored Bounded Universe Set
Cover”– PPT from Colored Bounded Universe Set Cover to
Bounded Universe Set Cover.
![Page 46: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/46.jpg)
Colored Bounded Universe Set Cover
IN: t set families F1={A1...Aa}, F2={B1...Bb}, Ft={X1...Xc} over a universe U of size k, integer t
PARAMETER: kQUESTION: Is there a family F’ = {Ai,Bj, ... Xl} of
size t containing one set of each color, such that F’ covers U?
![Page 47: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/47.jpg)
Composition, recap
An OR-composition algorithm for a problem P• Takes as input instances I1 ... It with parameter k • Runs in polynomial time• Outputs an instance (O,k’) of P such that– k’ ≤ poly(k)– (O,k’) is ”yes” some (Ii,k) is “yes”.
![Page 48: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/48.jpg)
Composition for CBUSC
Task: Given t instances of CBUSC all of size ≤ n and parameter k, output in polynomial time one ”equivalent” CBUSC instance.
Theorem [FKW04]: CBUSC instances with |U|=k can be solved in time O(2k|F|).
Trick: If t ≥ 2k then t2k|F| is polynomial, so wlog t < 2k.
![Page 49: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/49.jpg)
Composition for CBUSC
Plan: Glue the instances together on the universe.
BA B C CA
Universe
Sets
BA B C CA
Universe
Sets
BA B CA
Universe
Sets
C
BA B CA C
![Page 50: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/50.jpg)
Composition for CBUSC
GOOD: If one input is YES YESBAD: Can have NO + NO YES
Need to make sure: A solution picks sets from the same instance.
![Page 51: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/51.jpg)
ID’s and boxes
ID’s: Every instance gets a unique identification number from 0 to 2k-1, written in binary (k bits!)
Identification Check: Will check that for every pair of colors, the two solution verties of these colors come from the same instance = have the same ID.
![Page 52: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/52.jpg)
Boxes and ID’s
A box is a gadget containing k elements.
RED-BLUE box
BLUE-RED box
101100 101100101000
The red-blue and blue-red boxes together make sure that the blue and red solution vertices come from the same instance
![Page 53: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/53.jpg)
Composition for CBUSC
Modified plan:Glue the instances together on the universe. Add two boxes for every pair of colors.Universe size increases to O(k3), still poly(k).
Theorem [DLS09]: Colored Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆NP/poly.
![Page 54: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/54.jpg)
No kernel for Bounded Universe Set Cover
Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP NP/poly⊆ .
PPT from CBUSC to BUSC
BA B C CA
Universe
Sets
Universe
Sets
More Universe
![Page 55: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/55.jpg)
Epilogue
Compositions and Polynomial Parameter Transformations are tools to show kernel lower bounds.
Longest Path and Connected Vertex Cover are FPT but have no polynomial kernel unless coNP NP/poly⊆ .
![Page 56: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/56.jpg)
List of FPT problems with no poly(k) kernels unless coNP NP/poly⊆ .
• [HN06+FS08] k-Variable CNF-SAT • [BDFH08] Longest Path, Longest Cycle• [BTY09] Vertex Disjoint Paths, Cycles• [DLS09] Bounded Universe Hitting Set, Bounded
Universe Set Cover, Connected Vertex Cover, Steiner Tree, Capacitated Vertex Cover
• [KW09] Windmill-free Edge-Deletion• [KW09’] Cases of MinOnesSat • [JLS??] Dogson Score
![Page 57: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/57.jpg)
List of FPT problems with no poly(k) kernels unless AND-Distillation fails.
• [BDFH08] Treewidth, Pathwidth, Cutwidth, your-favourite width, and all sorts of stuff parameterized by them.
• [Z09] Vertex Ranking
![Page 58: Introduction to Kernel Lower Bounds Daniel Lokshtanov](https://reader034.vdocument.in/reader034/viewer/2022051214/56649c785503460f9492d6d1/html5/thumbnails/58.jpg)
THANK YOU!