Eötvös Loránd Tudományegyetem, Budapest The many facets of the Regularity Lemma Lovász László Eötvös Loránd Tudományegyetem, Budapest lovasz@cs.elte.hu Cim May 2012

The Lemma Original version Given  >0 The nodes of any graph can be partitioned into a small number of essentially equal parts so that most bipartite graphs between 2 parts are essentially random (with different densities). difference at most 1 with  k2 exceptions for subsets X,Y of parts Vi,Vj # of edges between X and Y is pij|X||Y| ±  (n/k)2 May 2012

The Lemma Weaker and Stronger Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon-Fisher-Krivelevich -M.Szegedy 2000 Tao 2006 L-B.Szegedy 2007 May 2012

The Lemma Weak version S pij: density between Vi and Vj May 2012

The Lemma Weak version replace edges between Vi and Vj by random edges with density pij to get G' May 2012

The Lemma Original version Vi X Y Vj May 2012

The Lemma Strong version May 2012

Regularity lemma in pictures Pixel pictures 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 AG G WG May 2012

Regularity lemma in pictures Pixel pictures May 2012

Regularity lemma in pictures Pixel pictures essentially random Nodes can be so ordered May 2012

Removal Lemma A great application >0 ’>0 # of triangles ’n3  we can delete n2 edges to get a triangle-free graph. Ruzsa - Szemerédi Implies: the r3(n) theorem,... Dependence of 1/' on 1/ is at most tower of height log(1/) Fox May 2012

Counting Lemma Subgraph densities t(F,G): Probability that random map V(F)V(G) preserves edges (Can be defined for weighted graphs G) |t(F,G) - t(F,H)|  |E(F)| □(G,H) If □(G,H) is small, then G and G’ are similar in many other respects... May 2012

Linear algebra version Low rank approximation Cut norm of nxn matrix A: : Frieze - Kannan May 2012

Analytic version Approximation in Hilbert space May 2012

Analytic version 2. Distance of graphs cut distance (a) V(G) = V(G') (b) |V(G)| = |V(G')| (c) |V(G)| =n, |V(G')|=m (blow up nodes, or fractional overlay) May 2012

Analytic version 2. Distance of graphs “Weak" Regularity Lemma (approximation form): May 2012

Analytic version Graphons W0 = {W: [0,1]2 [0,1], symmetric, measurable} "graphon" t(F,WG) = t(F,G): Probability that random map V(F)V(G) preserves edges May 2012

Analytic version 2. Distance of functions May 2012

Analytic version 2. Approximation by stepfunctions “Weak" Regularity Lemma for graphons: May 2012

Analytic version 2 Approximation by stepfunctions “Strong" Regularity Lemma: May 2012

Analytic version 3 Compactness Strongest (but non-effective) Regularity Lemma: 1-step Martingale Convergence Theorem k-step May 2012

Analytic version 2. Deriving the strong version 1-step 3-step 2-step -neighborhoods with radius εk L1-neigborhoods with radius ε0 May 2012

Geometric version Similarity metric w u This is a metric Representative set U: for any two nodes in U, dsim(s,t) >  for most nodes, dsim(U,v)   May 2012

= weak regularity partition Geometric version Representative set and regularity Voronoi diagram = weak regularity partition May 2012

Geometric version Representative set and regularity If P = {S1, . . . , Sk} is a partition of V(G) such that d(G,GP) = , then we can select nodes viSi such that the average similarity distance from S = {v1, . . . , vk} is at most 4. If SV 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 May 2012

Geometric version Regularity and dimension Every graph can be partitioned into sets with similarity diameter <. Alon Graph with similarity distance is almost finite dimensional May 2012

Algorithm What answer to expect? - Cannot list for all nodes! Input: We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. Output: For any given node, we want to tell in which class it belongs to (after some preprocessing) May 2009

Algorithm Computing a weak regularity partition Construct representative set U sim(s,t) is computable in the sampling model Each node is in same class as closest representative. May 2012

Congratulations Endre! And finally... Congratulations Endre! May 2012