Testing subgraphs in directed graphs

摘要

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least εn2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(ε,H)nh copies of H. This is proved by establishing a directed version of Szemerédi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of ε only for testing the property PH of being H-free.As is common with applications of the undirected regularity lemma, here too the function 1/f(ε,H) is an extremely fast growing function in ε. We therefore further prove a precise characterization of all the digraphs H, for which f(ε,H) has a polynomial dependency on ε. This implies a characterization of all the digraphs H, for which the property of being H-free has a one-sided error property tester whose query complexity is polynomial in 1/ε. We further show that the same characterization also applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in (Alon, Proceedings of the 42nd IEEE FOCS, IEEE, New York, 2001, pp. 434–441). Interestingly, it turns out that if PH has a polynomial query complexity, then there is a two-sided ε-tester for PH that samples only O(1/ε) vertices, whereas any one-sided tester for PH makes at least (1/ε)d/2 queries, where d is the average degree of H. We also show that the complexity of deciding if for a given directed graph H, PH has a polynomial query complexity, is NP-complete, marking an interesting distinction from the case of undirected graphs.For some special cases of directed graphs H, we describe very efficient one-sided error property-testers for testing PH. As a consequence we conclude that when H is an undirected bipartite graph, we can give a one-sided error property tester with query complexity O((1/ε)h/2), improving the previously known upper bound of O((1/ε)h2). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.