Graph limit theory:

Algorithms

László Lovász

Eötvös Loránd University, Budapest

May 2012 1

May 2012 2

General questions about extremal graphs

- Which inequalities between subgraph densities are valid?

- Which graphs are extremal?

- Can all valid inequalities be proved using just Cauchy-Schwarz?

- Local vs. global extrema

May 2012 3

Algorithms for very large graphs

Parameter estimation: edge density, triangle density, maximum cut

Property testing: is the graph bipartite? triangle-free? perfect?

Computing a structure: find a maximum cut, regularity partition,...

4May 2009

The maximum cut problem

maximize

Applications: optimization, statistical mechanics…

NP-hard, even with 6% error HastadPolynomial-time computable with 13% error Goemans-Williamson

5May 2009

Max cut in dense graphs How to estimate the density?cut with many

edges

Sampling O(4) nodes the density of max cut in the induced subgraph

is closer than to the density of the max cut in the whole

graphwith large probability.

Alon-Fernandez de la Vega-Kannan-Karpinski

For a graph parameter f, the following are equivalent:

(i) f can be estimated from samples

(ii) (Gn) convergent f(Gn) convergent

(iii) f is unifomly continuous w.r.t. dX

Estimable graph parameters

A graph parameter f can be estimated from samples iff

(a) f(G(k)) is convergent as k, and

(b) If V(Gn)=V(Hn), d(Gn,Hn)0, then f(Gn)-f(Hn)0, and

(c) If |V(Gn)|, vV(Gn), then f(Gn)-f(Gn\v)0

Density of maximum cut is estimable.

Estimable graph parameters

Testable graph properties

P testable: there is a test property P’, such that

(a) for every graph G ∈ P and every k ≥ 1,

G(k,G) ∈ P′ with probability at least 2/3, and

(b) for every ε > 0 there is a k0 ≥ 1 such that

for every graph G with d1(G,P) > ε

and every k ≥ k0 we have G(k,G) ∈ P′

with probability at most 1/3.

P: graph property

Testable graph properties

Example: triangle-free

Removal Lemma:

’ if t(,G)<’, then we can delete n2 edges

to get a triangle-free graph.

Ruzsa - Szemerédi

G’: sampled induced subgraph

G’ not triangle-free G not triangle free

G’ triangle-free with high probability, G has few triangles

Testable graph properties

Every hereditary graph property is testable

Alon-Shapira

inherited by induced subgraphs

Nondeterministically testable graph properties

Divine help: coloring the nodes, orienting and coloring the edges

Q: property of directed, colored graphs

Q’={G’: GQ}; “shadow of Q”

G: directed, (edge)-colored graph

G’: forget orientation, delete some colors, forget coloring; “shadow of G”

P nondeterministically testable: P=Q’, where Q is a testable property of colored directed graphs.

Nondeterministically testable graph properties

Examples: maximum cut contains at least n2/100 edges

constains a spanning subgraph with a testable property P

we can delete n2/100 edges to get a graph with a testable property P

Nondeterministically testable graph properties

Every nondeterministically testable graph property is testable.

L-Vesztergombi

H1, H2, ...

in Q

Hn’=Gn

... J2, J1

Jn’=Fn

close to Q

G1, G2, ... ... F2, F1

in P

far from P

Pure existence proof

of an algorithm...

May 2012 14

Computing a structure: representative set

Representative set of nodes: bounded size, (almost) every node is “similar” to one of the nodes in the set

When are two nodes similar? Neighbors? Same neighborhood?

May 2012 15

sim( , ) : E E ( ) E ( )v u su vu w wvtwa a ad t as = -

This is a metric, computable in the sampling model

Similarity distance of nodes

st

v

wu

Representative set U:

for any two nodes in U, dsim(s,t) >

for most nodes, dsim(U,v)

May 2012 16

Voronoi diagram= weak regularity

partition

Representative set and regularity partition

May 2012 17

Representative set and regularity partition

If P = {S1, . . . , Sk} is a partition of V(G) such that d(G,GP) = , then we can select nodes viSi such that the average similarity distance from S = {v1, . . . , vk} is at most 4.

If SV and the average similarity distance from S is ,then the Voronoi cells of S form a partition P such that d(G,GP) 8.

L-Szegedy

18May 2009

Max cut in dense graphs What answer to expect?

- Cannot list for all nodes

For any given node, we want to tell on which side of the cut it lies

(after some preprocessing)

May 2012 19

Construct representative set U

How to compute a (weak) regularity partition?

Each node is in same class as closest representative.

May 2012 20

- Construct representative set U

- Compute pij = density between classes Vi and Vj

(use sampling)

- Compute max cut (U1,U2) in complete graph on U

with edge-weights pij

How to compute the maximum cut?

(Different algorithm implicit by Frieze-Kannan.)

Each node is on same side as closest representative.

May 2012 21

Bounded degree (d)

- We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth.

Algorithms for bounded degree graphs

Maximum cut cannot be estimated in this model

(random d-regular graph vs. randombipartite d-regular graph)

May 2012 22

Algorithms for bounded degree graphs

Cellular automata (1970’s): network of finite automata

Distributed computing (1980’s): agent at every node, bounded time

Constant time algorithms (2000’s): bounded number of nodes sampled, explored at bounded depth

May 2012 23

Distributed computing model

May 2012 24

Algorithms for bounded degree graphs

(Almost) maximum matching can be computedin bounded time.

Nguyen-Onak

(Almost) maximum flow and (almost) minimum cut can be computed in bounded time.

Csoka

May 2012 25

Distributed computing model

May 2012 26

Algorithms for bounded degree graphs

(Almost) maximum matching can be computedin bounded time.

Nguyen-Onak

(Almost) maximum flow and (almost) minimum cut can be computed in bounded time.

Csóka

need local random bits

no random bits need global random

bits

May 2012 27

Thanks, that’s all!