Optimization in very large graphs

László Lovász

Eötvös Loránd University, Budapest

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2December 2012

How is the graph given?

- Graph is HUGE.

- Not known explicitly, not even the number of nodes.

Idealize: define minimum amount of info.

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Dense case: cn2 edges.

- We can sample a uniform random node a bounded number of times, and see edges

between sampled nodes.

Bounded degree (d)

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

How is the graph given?

„Property testing”: Goldreich-Goldwasser-Ron,

Arora-Karger-Karpinski, Rubinfeld-Sudan,

Alon-Fischer-Krivelevich-Szegedy, Fischer,

Frieze-Kannan, Alon-Shapira

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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,...

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The maximum cut problem

maximize

Applications: optimization, statistical mechanics…

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

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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 high probability.

Alon-Fernandez de la Vega-Kannan-Karpinski

A graph parameter f can be estimated from samples iff

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

(b) If V(Gn)=V(Hn)=[n] and then f(Gn)-f(Hn)0

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

Density of maximum cut is estimable.

Estimable graph parameters

, 2

1max ( , ) ( , ) 0S T nn HGe S T e S T

n- ®

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blow up every node

cut distance

Borgs-Chayes-L-Sós-Vesztergombi

Nondeterministic parameter estimation

Divine help: coloring nodes and edges, orienting edges

L: directed, (edge)-colored graph

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

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f’(G) = max {f(L): L’ = G} ; “shadow of f”

f: parameter defined on colored oriented graphs

If f is estimable, then f’ is estimable.

L.-Vesztergombi

Testable graph properties

P testable: for every ε > 0 there is a k0 ≥ 1 and a

there is a „test property” P’, such that

(a) for every graph G ∈ P and every k ≥ k0,

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

(b) 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

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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

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Testable graph properties

Every hereditary graph property is testable

Alon-Shapira

inherited by induced subgraphs

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Nondeterministically testable graph properties

Q: property of directed, colored graphs

Q’={G’: GQ}; “shadow of Q”

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

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Divine help: coloring nodes and edges, orienting edges

L: directed, (edge)-colored graph

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

Nondeterministically testable graph properties

Examples: maximum cut contains at least n2/100 edges

contains a spanning subgraph with a testable property P

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

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N=NP for denseproperty testing

Nondeterministically testable graph properties

Every nondeterministically testable graph property is testable.

L-Vesztergombi

Proof via graph limit theory:pure existence proof

of an algorithm...

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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?

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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 s, dsim(U,s)

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Every graph contains a representative set

with at most nodes. 2(1/ )2O e

Strong representative set U:

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

for every node, dsim(U,v)

Every graph contains a strong representative set

with at most nodes. 2(log(1/ / ))2O e e

Alon

Similarity distance of nodes

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Voronoi diagram= weak regularity

partition

Representative set and regularity partition

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Max cut in dense graphs What answer to expect?

- Cannot list all nodes on one side

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

(after some preprocessing)

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Construct representative set U

How to compute a (weak) regularity partition?

Each node is in same class as closest representative.

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- Construct representative set U

- Compute pij = density between Voronoi cells 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.

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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

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Bad examples

Gn: random 3-regular graph

Fn: random 3-regular bipartite graph

Hn: GnGn

Large girth graphs

(except for a small part)

Expandergraphs

(except for a small part)

23

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Algorithms for bounded degree graphs

Maximum cut cannot be estimated in this model

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

PNP in this model

(random d-regular graph vs. union of two random d-regular graphs)

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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

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Distributed computing model

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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

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Distributed computing model

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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

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Algorithms for bounded degree graphs

Suppose you tell the agents any information aboutthe graph.

In the presence of global random bits, this does not help! Csóka

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Open problems

● Can an (almost) maximum independent node set be

computed in a random d-regular graph in this model?

● Can an (almost) maximum cut be

computed in a random d-regular graph in this model?

(Gn)/n tends to a limit with probability 1

MaxCut(Gn)/n tends to a limit with probability 1

Bayati-Gamarnik-Tetali 2010

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Thanks, that’s all!