Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle

Abstract

In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly and can be accessed only through its colorful independence oracle: The colorful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colorful with respect to a given vertex-coloring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. [ACM Trans. Algorithms, 16 (2020), 52], Dell and Lapinskas [Proceedings of STOC, ACM, 2018, pp. 281--288], and Bhattacharya et al. [Proceedings of ISAAC, 2019]. Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV, and weighted $k$-Clique) and parameterized complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to constraint satisfaction problems).