Happy Birthday Ravi Algorithms on large graphs Lszl

Happy Birthday Ravi! Algorithms on large graphs László Lovász Eötvös Loránd University, Budapest May 2013 1

The Weak Regularity Lemma Cut norm of matrix A nxn: Cut distance of two graphs with V(G) = V(G’): (extends to edge-weighted) May 2013 2

The Weak Regularity Lemma Avereged graph G ( partition of V(G)) 1/2 1 0 Template graph G/ 1/2 1 May 2013 2/5 1/2 1 0 1/5 3

The Weak Regularity Lemma For every graph G and every >0 there is a partition with and Frieze – Kannan 1999 May 2013 4

Algorithms for large graphs How is the graph given? - Graph is HUGE. - Not known explicitly, not even the number of nodes. Idealize: define minimum amount of info. May 2013 5

Algorithms for large graphs Dense case: cn 2 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 May 2013 6

Algorithms for 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 Computing a constantcut, regularity partition, . . . size encoding The partition (cut, . . . ) can be computed in polynomial time. For every node, we can determine in constant time which class it belongs to May 2013 7

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 2013 8

Similarity distance of nodes s w u t v This is a metric, computable in the sampling model May 2013 9

Representative set Strong representative set U: for any two nodes in s, t U, dsim(s, t) > for all nodes s, dsim(U, s) Average representative set U: for any two nodes s, t U, dsim(s, t) > for a random node s, Edsim(U, s) 2 May 2013 10

Representative sets and regularity partitions If P = {S 1, . . . , Sk} is a weak regularity partition with error , then we can select nodes vi Si such that S = {v 1, . . . , vk} is an average representative set with error < 4. If S V is an average representative set with error , then the Voronoi cells of S form a weak regularity partition with error < 8 . L-Szegedy May 2013 11

Representative sets and regularity partitions Voronoi diagram = weak regularity partition May 2013 12

Representative sets Every graph has an average representative set with at most nodes. Every graph has a strong representative set with at most nodes. Alon If S V(G) and dsim(u, v)> for all u, v S, then May 2013 13

Representative sets Example: every average representative set has nodes. dimension 1/ May 2013 angle 14

Representative sets and regularity partitions For every graph G and >0 there are ui, vi {0, 1}V(G) and ai such that Frieze-Kannan May 2013 15

How to compute a (weak) regularity partition? Construct weak representative set U Each node is in same class as closest representative. May 2013 16

How to compute a maximum cut? - Construct representative set - Compute weights in template graph (use sampling) - Compute max cut in template graph Each node is on same side as closest representative. (Different algorithm implicit by Frieze-Kannan. ) May 2013 17

How to compute a maximum matching? 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|. May 2013 18

Nondeterministically estimable parameters Divine help: coloring the nodes, orienting and coloring the edges G: directed, (edge)-colored graph G’: forget orientation, delete some colors, forget coloring; shadow of G g: parameter defined on directed, colored graphs g’(H)=max{g(G): G’=H}; shadow of g f nondeterministically estimable: f=g’, where g is an estimable parameter of colored directed graphs. May 2013 19

Nondeterministically estimable parameters Examples: density of maximum cut Goldreich-Goldwasser-Ron edit distance from a testable property Fischer- Newman the graph contains a subgraph G’ with all degrees cn and |E(G’)| an 2 May 2013 20

Nondeterministically estimable parameters Every nondeterministically estimable graph paratemeter is estimable. L-Vesztergombi Every nondeterministically estimable graph N=NP for dense property testing pproperty is testable. L-Vesztergombi Proof via graph limit theory: pure existence proof of an algorithm. . . May 2013 21

How to compute a maximum matching? More generally, how to compute a witness in non-deterministic property testing? May 2013 22

Happy Birthday Ravi! May 2013 23