Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Suppose that you wish to sample a random graph G over n vertices and m edges conditioned on the event that G does not contain a “small" t-size graph H (e.g., clique) as a subgraph. Assuming that most such graphs are H-free, the problem can be solved by a simple rejected-sampling algorithm (that tests for t-cliques) with an expected running time of n
^O(t)
. Is it possible to solve the problem in running time that does not grow polynomially with n
^t
? In this paper, we introduce the general problem of sampling a “random looking'' graph G with a given edge density that avoids some arbitrary predefined t-size subgraph H. As our main result, we show that the problem is solvable with respect to some specially crafted k-wise independent distribution over graphs. That is, we design a sampling algorithm for k-wise independent graphs that supports efficient testing for subgraph-freeness in time f(t) · n
^c
where f is a function of t and the constant c in the exponent is independent of t. Our solution extends to the case where both G and H are d-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 n (i.e., n
^-ω(1)
). In particular, given constants d>c, we output a bipartite graph that has n left nodes, n
^c
right nodes with right-degree of d so that any right set of size at most n
^Ω(1)
expands by factor of Ω(d). 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 LDPC 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 parallel-cryptography (cf., Applebaum-Ishai-Kushilevitz, FOCS 2004; and Ishai-Kushilevitz-Ostrovsky-Sahai, STOC 2008).