Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Abstract

Suppose that you wish to sample a random graph over vertices and edges conditioned on the event that does not contain a “small” -size graph (e.g., clique) as a subgraph. Assuming that most such graphs are -free, the problem can be solved by a simple rejected-sampling algorithm (that tests for -cliques) with an expected running time of . Is it possible to solve the problem in a running time that does not grow polynomially with ? In this paper, we introduce the general problem of sampling a “random looking” graph with a given edge density that avoids some arbitrary predefined -size subgraph . As our main result, we show that the problem is solvable with respect to some specially crafted -wise independent distribution over graphs. That is, we design a sampling algorithm for -wise independent graphs that supports efficient testing for subgraph-freeness in time , where is a function of and the constant in the exponent is independent of . Our solution extends to the case where both and are -uniform hypergraphs. We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially unbalanced expander graphs whose failure probability is negligible in (i.e., ). In particular, given constants , we output a bipartite graph that has left nodes and right nodes with right-degree of so that any right set of size at most expands by factor of . This result is extended to the setting of unique expansion as well. We observe that such a negligible-error construction can be employed in many useful settings and present applications in coding theory (batch codes and low-density parity-check codes), pseudorandomness (low-bias generators and randomness extractors), and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally computable cryptographic pseudorandom generators based on Goldreich’s one-wayness assumption resolving a central open problem in the area of parallel-time cryptography (e.g., Applebaum, Ishai, and Kushilevitz [SIAM J. Comput., 36 (2006), pp. 845–888] and Ishai et al. [Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 433–442]).