1. Graph limit theory: Algorithms László Lovász Eötvös Loránd University, Budapest May 20121

2. 2 General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Local vs. global extrema

3. May 20123 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,...

4. May 20094 The maximum cut problem maximize Applications: optimization, statistical mechanics… NP-hard, even with 6% error Hastad Polynomial-time computable with  13% error Goemans-Williamson

5. May 20095 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 graph with large probability. Alon-Fernandez de la Vega -Kannan-Karpinski

6. For a graph parameter f, the following are equivalent: (i) f can be estimated from samples (ii) (G n ) convergent  f(G n ) convergent (iii) f is unifomly continuous w.r.t. Estimable graph parameters

7. A graph parameter f can be estimated from samples iff (a) f(G(k)) is convergent as k , and (b)If V(G n )=V(H n ), d  (G n,H n )  0, then f(G n )-f(H n )  0, and (c) If |V(G n )| , v  V(G n ), then f(G n )-f(G n \v)  0 Density of maximum cut is estimable. Estimable graph parameters

8. Testable graph properties P testable: there is a test property P ’, such that (a)for every graph G ∈ P and every k ≥ 1,  (k,G) ∈ P ′ with probability at least 2/3, and (b) for every ε > 0 there is a k 0 ≥ 1 such that for every graph G with d 1 (G, P ) > ε and every k ≥ k 0 we have  (k,G) ∈ P ′ with probability at most 1/3. P : graph property

9. Testable graph properties Example: triangle-free Removal Lemma:   ’ if t( ,G)<  ’, then we can delete  n 2 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

10. Testable graph properties Every hereditary graph property is testable Alon-Shapira inherited by induced subgraphs

11. Nondeterministically testable graph properties Divine help: coloring the nodes, orienting and coloring the edges Q : property of directed, colored graphs Q ’={G’: G  Q }; “shadow of Q ” G: directed, (edge)-colored graph G’: forget orientation, delete some colors, forget coloring; “shadow of G” P nondeterministically testable: P = Q ’, where Q is a testable property of colored directed graphs.

12. Nondeterministically testable graph properties Examples: maximum cut contains at least n 2 /100 edges constains a spanning subgraph with a testable property P we can delete n 2 /100 edges to get a graph with a testable property P

13. Nondeterministically testable graph properties Every nondeterministically testable graph property is testable. L-Vesztergombi H 1, H 2,... in Q H n ’=G n ... J 2, J 1 J n ’=F n close to Q G 1, G 2,... ... F 2, F 1 in P far from P  Pure existence proof of an algorithm...

14. May 201214 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?

15. May 201215 This is a metric, computable in the sampling model Similarity distance of nodes s t v w u Representative set U: for any two nodes in U, d sim (s,t) >  for most nodes, d sim (U,v)  

16. May 201216 Voronoi diagram = weak regularity partition Representative set and regularity partition

17. May 201217 Representative set and regularity partition If P = {S 1,..., S k } is a partition of V(G) such that d  (G,G P ) = , then we can select nodes v i  S i such that the average similarity distance from S = {v 1,..., v k } 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,G P )  8 . L-Szegedy

18. May 200918 Max cut in dense graphs What answer to expect? - Cannot list for all nodes For any given node, we want to tell on which side of the cut it lies (after some preprocessing)

19. May 201219 Construct representative set U How to compute a (weak) regularity partition? Each node is in same class as closest representative.

20. May 201220 - Construct representative set U - Compute p ij = density between classes V i and V j (use sampling) - Compute max cut (U 1,U 2 ) in complete graph on U with edge-weights p ij How to compute the maximum cut? (Different algorithm implicit by Frieze-Kannan.) Each node is on same side as closest representative.

21. May 201221 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 Maximum cut cannot be estimated in this model (random d-regular graph vs. random bipartite d-regular graph)

22. May 201222 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

23. May 201223 Distributed computing model

24. May 201224 Algorithms for bounded degree graphs (Almost) maximum matching can be computed in bounded time. Nguyen-Onak (Almost) maximum flow and (almost) minimum cut can be computed in bounded time. Csoka

25. May 201225 Distributed computing model

26. May 201226 Algorithms for bounded degree graphs (Almost) maximum matching can be computed in 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

27. May 201227