Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle

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