Nondeterministic property testing

1. Nondeterministic property testing László Lovász Katalin Vesztergombi

2. Definitions G(k,G): labeled subgraph of G induced by k random nodes.

3. Testable graph properties P: graph property • Ptestable:there is a test property P’, such that • for every graph G ∈ Pand every k ≥ 1, • G(k,G) ∈P′with probability at least 2/3, and • (b) for every ε > 0 there is a k0≥ 1 such that • for every graph Gwith d1(G,P) > ε • and every k ≥ k0we have G(k,G) ∈P′ • with probability at most 1/3.

4. Testable graph properties: examples Example: No edge. Example: All degrees ≤10. Example: Contains a clique with ≥ n/2 nodes. Example: Bipartite. Example: Perfect.

5. Testable graph properties: examples Example: triangle-free G’: sampled induced subgraph G’ not triangle-free  G not triangle free G’ triangle-free  with high probability, G has few triangles Removal Lemma:  ’ if t(,G)<’, then we can delete n2 edges to get a triangle-free graph. Ruzsa - Szemerédi

6. Testable graph properties: examples Example: disjoint union of two isomorphic graphs Not testable!

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

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

9. Nondeterministically testable graph properties Examples: maximum cut contains ≥n2/100 edges contains a clique with ≥ n/2 nodes contains a spanning subgraph with a testable property P we can delete ≤n2/100 edges to get a perfect graph

10. Main Theorem Every nondeterministically testable graph property is testable. L-V „P=NP” for property testing in dense graphs Pure existence proof of an algorithm

11. Restrictions and extensions Node-coloring can be encoded into the edge-coloring. We will not consider orientation of edges. Equivalent: Certificate is given by unary and binary relations. Ternary etc? Theorem is false if functions are allowed besides relations. (Example: union of two isomorphic graphs.)

12. From graphs to functions 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 G AG WG

13. Kernels and graphons W= {W: [0,1]2R, symmetric, bounded, measurable} kernel W0 = {W: [0,1]2[0,1], symmetric, measurable} graphon graph G graphon WG

14. Cut distance cut norm on L([0,1]2) cut distance There is a finite definition.

15. Cut distance and property testing A graph property P is testable iff for every sequence (Gn) of graphs with |V(Gn)| and d(Gn,P)0, we haved1(Gn,P)0. L-Szegedy

16. Convergence of a graph sequence distribution of k-samples is convergent for all k Probability that random map V(F)V(G) preserves edges (G1,G2,…) convergent:Ft(F,Gn) is convergent (G1,G2,…) convergent Cauchy in the cut distance Borgs-Chayes-L-Sós-V

17. Limit graphon of a graph sequence GnW : Ft(F,Gn)  t(F,W) Equivalently:

18. Limit graphon: existence and uniqueness For every convergent graph sequence(Gn) there is a WW0 such thatGnW. Conversely, W(Gn) such thatGnW. L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs-Chayes-L

19. Convergence in norm Let Gn be a sequence ofgraphs, and let U be a graphon such thatGnU. Then the graphs Gn can be labeled so that Borgs-Chayes-L-Sós-V (Wn ):sequence of uniformly boundedkernels withWn0. Then WnZ0 for every integrable function Z:[0,1]2R. L-Szegedy

20. k-graphons k-graphon: W=(W1,...,Wk), where W1,...,WkW0 and W1+...+Wk=1 fractional k-coloration Sample G(r,W): random x1,...,xr[0,1], connect i to j with color c with probability Wc(xi,xj)

21. Convergence of k-graphons Ln: sequence of k-edge-coloredgraphs. Ln convergent:distribution of G(r,Ln) is convergent. • Ln convergent sequence of k-coloredgraphs • k-graphonW: G(r,Ln)G(r,W)in distribution. • Equivalently: L-Szegedy

22. Main Theorem: Proof close to Q  in Q  ... J2, J1 H1, H2, ... shadow(Hn)=Gn shadow(Jn)=Fn G1, G2, ...  ... F2, F1 in P far from P

23. Main Lemma Let W=(W1,...,Wk) be a k-graphon, and let . Let FnU.Then there existk-colored graphs Jn on V(Jn) = V(Fn) such that shadow(Jn) =Fn andJnW.

24. Proof (k=3, m=2) +  W 1 W 2 F + = H 1 H 2 24

25. Proof (cont) (H1, H2) fractional edge-2-coloring  (J1, J2) edge-2-coloring by randomization are small (Chernoff bound) are small Two things to prove:

26. Proof (cont)

27. Proof (cont)

28. Bounded degree graphs (≤D) Sampling method:We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth.

29. Bounded degree graphs (≤D) Maximum cut cannot be estimated in this model (random D-regular graph vs. randombipartite D-regular graph) PNP in this model (random D-regular graph vs. union of two random D-regular graphs)