Nondeterministic property testing Lszl Lovsz Katalin Vesztergombi Definitions

Nondeterministic property testing László Lovász Katalin Vesztergombi

Definitions G(k, G): labeled subgraph of G induced by k random nodes. September 2012 2

Testable graph properties P: graph property P testable: there is a test property P’, such that (a) for every graph G ∈ P and every k ≥ 1, G(k, G) ∈ P′ with probability at least 2/3, and (b) for every ε > 0 there is a k 0 ≥ 1 such that (a) for every graph G with d 1(G, P) > ε (b) and every k ≥ k 0 we have G(k, G) ∈ P′ (c) with probability at most 1/3. September 2012 3

Testable graph properties: examples Example: No edge. Example: All degrees ≤ 10. Example: Contains a clique with ≥ n/2 nodes. Example: Bipartite. Example: Perfect. September 2012 4

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 n 2 edges to get a triangle-free graph. Ruzsa - Szemerédi September 2012 5

Testable graph properties: examples Example: disjoint union of two isomorphic graphs Not testable! September 2012 6

Testable graph properties Every hereditary graph property is testable. Alon-Shapira inherited by induced subgraphs September 2012 7

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. September 2012 8

Nondeterministically testable graph properties Examples: maximum cut contains ≥n 2/100 edges contains a clique with ≥ n/2 nodes contains a spanning subgraph with a testable property P we can delete ≤n 2/100 edges to get a perfect graph September 2012 9

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 September 2012 10

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. ) September 2012 11

From graphs to functions G 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 AG WG September 2012 12

Kernels and graphons W = {W: [0, 1]2 R, symmetric, bounded, measurable} kernel W 0 = {W: [0, 1]2 [0, 1], symmetric, measurable} graphon graph G graphon WG September 2012 13

Cut distance cut norm on L ([0, 1]2) There is a finite definition. September 2012 cut distance 14

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 have d 1(Gn, P) 0. L-Szegedy September 2012 15

Convergence of a graph sequence Probability that random map V(F) V(G) preserves edges (G 1, G 2, …) convergent: F t(F, Gn) is convergent distribution of k-samples is convergent for all k (G 1, G 2, …) convergent Cauchy in the cut distance Borgs-Chayes-L-Sós-V September 2012 16

Limit graphon of a graph sequence Gn W : F t(F, Gn) t(F, W) Equivalently: September 2012 17

Limit graphon: existence and uniqueness For every convergent graph sequence (Gn) there is a W W 0 such that Gn W. Conversely, W (Gn) such that Gn W. L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs-Chayes-L September 2012 18

Convergence in norm Let Gn be a sequence of graphs, and let U be a graphon such that Gn U. Then the graphs Gn can be labeled so that Borgs-Chayes-L-Sós-V (Wn ): sequence of uniformly bounded kernels with Wn � 0. Then Wn. Z � 0 for every integra function Z: [0, 1]2 R. L-Szegedy September 2012 19

k-graphons k-graphon: W=(W 1, . . . , Wk), where W 1, . . . , Wk W 0 and W 1+. . . +Wk=1 fractional k-coloration Sample G(r, W): random x 1, . . . , xr [0, 1], connect i to j with color c with probability Wc(xi, xj) September 2012 20

Convergence of k-graphons Ln: sequence of k-edge-colored graphs. Ln convergent: distribution of G(r, Ln) is convergent. Ln convergent sequence of k-colored graphs k-graphon W : G(r, Ln) G(r, W) in distribution. Equivalently: L-Szegedy September 2012 21

Main Theorem: Proof in Q H 1, H 2, . . . close to Q shadow(Hn)=Gn G 1, G 2, . . . in P September 2012 . . . J 2, J 1 shadow(Jn)=Fn . . . F 2, F 1 far from P 22

Main Lemma Let W=(W 1, . . . , Wk) be a k-graphon, and let . Let Fn U. Then there exist k-colored graphs Jn on V(Jn) = V(Fn) such that shadow(Jn) = Fn and Jn W. September 2012 23

Proof (k=3, m=2) + W 1 F W 2 + = September 2012 H 1 24 H 2 24

Proof (cont) (H 1, H 2) fractional edge-2 -coloring (J 1, J 2) edge-2 -coloring by randomization Two things to prove: are small (Chernoff bound) September 2012 25

Proof (cont) September 2012 26

Proof (cont) September 2012 27

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. September 2012 28

Bounded degree graphs (≤D) Maximum cut cannot be estimated in this model (random D-regular graph vs. random bipartite D-regular graph) P NP in this model (random D-regular graph vs. union of two random D-regular graphs) September 2012 29