Nondeterministic property testing László Lovász Katalin Vesztergombi

G(k,G): labeled subgraph of G induced by k random nodes. Definitions September 20122

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 for every graph G with d 1 (G,P) > ε and every k ≥ k 0 we have G(k,G) ∈ P′ with probability at most 1/3. P: graph property Testable graph properties September 20123

Example: No edge. Testable graph properties: examples Example: All degrees ≤10. Example: Contains a clique with ≥ n/2 nodes. Example: Bipartite. Example: Perfect. September 20124

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 Example: triangle-free Testable graph properties: examples September 20125

Example: disjoint union of two isomorphic graphs Testable graph properties: examples Not testable! September 20126

Every hereditary graph property is testable. Alon-Shapira inherited by induced subgraphs Testable graph properties September 20127

Nondeterministically testable graph properties Divine help: coloring the nodes, orienting and coloring the edges Q: property of directed, colored graphs shadow(Q)={shadow(G): G  Q}; G: directed, edge and node-colored graph shadow(G): forget orientation, delete edges with certain colors, forget coloring P nondeterministically testable: P= shadow(Q), where Q is a testable property of colored directed graphs. September 20128

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 Nondeterministically testable graph properties September 20129

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

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

G AGAG WGWG September From graphs to functions

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

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

A graph property P is testable iff for every sequence (G n ) of graphs with |V(G n )|  and   (G n,P)  0, we have d 1 (G n,P)  0. Cut distance and property testing September L-Szegedy

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

September G n  W :  F t(F,G n )  t(F,W) Limit graphon of a graph sequence Equivalently:

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

Let G n be a sequence of graphs, and let U be a graphon such that G n  U. Then the graphs G n can be labeled so that Convergence in norm (W n ): sequence of uniformly bounded kernels with  W n    0. Then  W n Z    0 for every integrable function Z: [0,1] 2   R. September Borgs-Chayes-L-Sós-V L-Szegedy

k-graphons k-graphon: W=(W 1,...,W k ), where W 1,...,W k  W 0 and W W k =1 fractional k-coloration September Sample G(r,W): random x 1,...,x r  [0,1], connect i to j with color c with probability W c (x i,x j )

L n : sequence of k-edge-colored graphs. L n convergent: distribution of G(r,L n ) is convergent. Convergence of k-graphons September L n convergent sequence of k-colored graphs   k-graphon W : G(r,L n )  G(r,W) in distribution. Equivalently: L-Szegedy

H 1, H 2,... in Q shadow(H n )=G n ... J 2, J 1 shadow(J n )=F n close to Q G 1, G 2,...  ... F 2, F 1 in P far from P  Main Theorem: Proof September

Let W=(W 1,...,W k ) be a k-graphon, and let. Let F n  U. Then there exist k-colored graphs J n on V(J n ) = V(F n ) such that shadow(J n ) = F n and J n  W. Main Lemma September

September  F Proof (k=3, m=2) W 1 W = H 1 H 2

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

September Proof (cont)

September Proof (cont)

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

September 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) Bounded degree graphs (≤D)