Algorithms on large graphs
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Transcript of Algorithms on large graphs
Algorithms on large graphs
László Lovász Eötvös Loránd University, Budapest
May 2013 1
Happy Birthday Ravi!
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[ ]2 ,
1 max ijS T n i S j T
A An Í Î Î
= å åW
Cut norm of matrix Anxn:
The Weak Regularity Lemma
'2 , ( )
1( , ') max | ( , ) ( , ) |G GS T V Gd G G e S T e S T
n Í= -W
Cut distance of two graphs with V(G) = V(G’):
(extends to edge-weighted)
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The Weak Regularity Lemma
Avereged graph GP (P partition of V(G)) 11/2
Template graph G/P11/2
1/210
0
2/5
2/5
1/5
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The Weak Regularity Lemma
For every graph G and every >0 there is
a partition with
and 2(1/ )| | 2O e=P ( , )d G G e<PW
Frieze – Kannan 1999
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Algorithms for large graphs
- Graph is HUGE.
- Not known explicitly, not even the number of nodes.
Idealize: define minimum amount of info.
How is the graph given?
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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.
„Property testing”, constant time algorithms: Arora-
Karger-Karpinski, Goldreich-Goldwasser-Ron,
Rubinfeld-Sudan, Alon-Fischer-Krivelevich-Szegedy,
Fischer, Frieze-Kannan, Alon-Shapira
Algorithms for large graphs
Computing a structure: find a maximum cut, regularity partition,...Computing a structure: find a maximum cut, regularity partition,...
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Algorithms for large graphs
Parameter estimation: edge density, triangle density, maximum cut
Property testing: is the graph bipartite? triangle-free? perfect?
Computing a constant size encoding
The partition (cut,...) can becomputed in polynomial time.
For every node, we can determine in constant time which class
it belongs to
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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
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Representative set
Strong representative set U:
for any two nodes in s,tU, dsim(s,t) >
for all nodes s, dsim(U,s)
Average representative set U:
for any two nodes s,tU, dsim(s,t) >
for a random node s, Edsim(U,s) 2
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Representative sets and regularity partitions
If P = {S1, . . . , Sk} is a weak regularity partition with error , then we can select nodes viSi
such that S = {v1, . . . , vk} is an average representative set with error < 4.If SV is an average representative set with error , then the Voronoi cells of S form a weak regularity partition with error < 8.
L-Szegedy
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Voronoi diagram= weak regularity
partition
Representative sets and regularity partitions
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Every graph has an average representative set
with at most nodes. 2(1/ )2O e
Representative sets
If S V(G) and dsim(u,v)> for all u,vS, then2(log(1/ / ))2OS e e=
Every graph has a strong representative set
with at most nodes. Alon
2(log(1/ / ))2O e e
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Example: every average representative set
has nodes. 2(1/ )2 eW
Representative sets
angle dimension 1/
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Representative sets and regularity partitions
Frieze-Kannan
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1,k
TG i i i
i
A a u v k Oe
e=
æ ö÷ç ÷ç ÷ç- < è ø=åW
For every graph G and >0 there are ui, vi {0,1}V(G) and ai such that
sim( , ) : E E ( ) E ( )v u su vu w wvtwa a ad t as = -
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Construct weak 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
- Compute weights in template graph (use sampling)
- Compute max cut in template graph
How to compute a maximum cut?
(Different algorithm implicit by Frieze-Kannan.)
Each node is on same side as closest representative.
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Given a bigraph with bipartition {U,W} (|U|=|W|=n)
and c[0,1], find a maximum subgraph with all degrees
at most c|U|.
How to compute a maximum matching?
Nondeterministically estimable parameters
Divine help: coloring the nodes, orienting and coloring the edges
g: parameter defined on directed, colored graphs
g’(H)=max{g(G): G’=H}; shadow of g
G: directed, (edge)-colored graph
G’: forget orientation, delete some colors, forget coloring; shadow of G
f nondeterministically estimable: f=g’,where g is an estimable parameter of colored directed graphs. May 2013 19
Examples: density of maximum cut
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the graph contains a subgraph G’ with all degrees cn and |E(G’)| an2
edit distance from a testable property
Fischer- Newman
Goldreich-Goldwasser-Ron
Nondeterministically estimable parameters
Every nondeterministically estimable graph
pproperty is testable.L-Vesztergombi
N=NP for denseproperty testing
Every nondeterministically estimable graph
paratemeter is estimable.L-Vesztergombi
Proof via graph limit theory:pure existence proof
of an algorithm...
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Nondeterministically estimable parameters
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More generally, how to compute a witness in
non-deterministic property testing?
How to compute a maximum matching?
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Happy Birthday Ravi!