1. Algorithms on large graphs László Lovász Eötvös Loránd University, Budapest May 20131

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

3. May 20133 The Weak Regularity Lemma Avereged graph G   (  partition of V(G)) 11/2 Template graph G/  1 1/2 1 0 0 2/5 1/5

4. May 20134 The Weak Regularity Lemma For every graph G and every  >0 there is a partition  with and Frieze – Kannan 1999

5. May 20135 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?

6. May 20136 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 Algorithms for large graphs

7. Computing a structure: find a maximum cut, regularity partition,... Computing a structure: find a maximum cut, regularity partition,... May 20137 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 be computed in polynomial time. For every node, we can determine in constant time which class it belongs to

8. May 20138 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?

9. May 20139 This is a metric, computable in the sampling model Similarity distance of nodes s t v w u

10. May 201310 Representative set Strong representative set U: for any two nodes in s,t  U, d sim (s,t) >  for all nodes s, d sim (U,s)   Average representative set U: for any two nodes s,t  U, d sim (s,t) >  for a random node s,  d sim (U,s)  2 

11. May 201311 Representative sets and regularity partitions If P = {S 1,..., S k } is a weak regularity partition with error , then we can select nodes v i  S i such that S = {v 1,..., v k } 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

12. May 201312 Voronoi diagram = weak regularity partition Representative sets and regularity partitions

13. May 201313 Every graph has an average representative set with at most nodes. Representative sets If S  V(G) and d sim (u,v)>  for all u,v  S, then Every graph has a strong representative set with at most nodes. Alon

14. May 201314 Example: every average representative set has nodes. Representative sets angle  dimension 1/ 

15. May 201315 Representative sets and regularity partitions Frieze-Kannan For every graph G and  >0 there are u i, v i  {0,1} V(G) and a i   such that

16. May 201316 Construct weak representative set U How to compute a (weak) regularity partition? Each node is in same class as closest representative.

17. May 201317 - 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.

18. May 201318 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?

19. 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 201319

20. Examples: density of maximum cut May 201320 the graph contains a subgraph G’ with all degrees  cn and |E(G’)|  an 2 edit distance from a testable property Fischer- Newman Goldreich-Goldwasser-Ron Nondeterministically estimable parameters

21. Every nondeterministically estimable graph pproperty is testable. L-Vesztergombi N=NP for dense property testing Every nondeterministically estimable graph paratemeter is estimable. L-Vesztergombi Proof via graph limit theory: pure existence proof of an algorithm... May 201321 Nondeterministically estimable parameters

22. May 201322 More generally, how to compute a witness in non-deterministic property testing? How to compute a maximum matching?

23. May 201323