16 research outputs found
Faster counting and sampling algorithms using colorful decision oracle
In this work, we consider -{\sc Hyperedge Estimation} and -{\sc Hyperedge Sample} problem in a hypergraph in the query complexity framework, where denotes the set of vertices and denotes the set of hyperedges. The oracle access to the hypergraph is called {\sc Colorful Independence Oracle} ({\sc CID}), which takes (non-empty) pairwise disjoint subsets of vertices as input, and answers whether there exists a hyperedge in having (exactly) one vertex in each . The problem of -{\sc Hyperedge Estimation} and -{\sc Hyperedge Sample} with {\sc CID} oracle access is important in its own right as a combinatorial problem. Also, Dell {\it{et al.}}~[SODA '20] established that {\em decision} vs {\em counting} complexities of a number of combinatorial optimization problems can be abstracted out as -{\sc Hyperedge Estimation} problems with a {\sc CID} oracle access.
The main technical contribution of the paper is an algorithm that estimates with such that { by using at most many {\sc CID} queries, where denotes the number of vertices in the hypergraph and is a constant that depends only on }. Our result coupled with the framework of Dell {\it{et al.}}~[SODA '21] implies improved bounds for a number of fundamental problems