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 a 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^-(1)). In particular, given constants d > c 1, we output a bipartite graph that has n left nodes and n^c right nodes with right-degree of d so that any right set of size at most n⁽¹⁾ expands by factor of (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 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]).