Faster Counting and Sampling Algorithms using Colorful Decision Oracle

Abstract

In this work, we consider d - Hyperedge Estimation and d - Hyperedge Sample problems that deal with estimation and uniform sampling of hyperedges in a hypergraph ℋ( U (ℋ), ℱ(ℋ) in the query complexity framework, where U (ℋ) denotes the set of vertices and ℱ(ℋ) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle ( CID ), which takes d (non-empty) pairwise disjoint subsets of vertices A 1 ,..., A d ⊆ U (ℋ) as input and answers whether there exists a hyperedge in ℋ having exactly one vertex in each A i for all i ∈ {1, 2, ..., d }. Apart from the fact that d - Hyperedge Estimation and d - Hyperedge Sample problems with CID oracle access seem to be nice combinatorial problems, Dell et al. [SODA’20 & SICOMP’22] established that decision vs. counting complexities of a number of combinatorial optimization problems can be abstracted out as d - Hyperedge Estimation problem with a CID oracle access. The main technical contribution of this article is an algorithm that estimates m = |ℱ(ℋ)| with \(\widehat{m}\) such that \(\begin{equation*} \frac{1}{C_{d}\log ^{d-1} n} \;\le \; \frac{\widehat{m}}{m} \;\le \; C_{d} \log ^{d-1} n \end{equation*}\) by using at most C d log d +2 n CID queries, where n denotes the number of vertices in the hypergraph ℋ and C d is a constant that depends only on d . Our result, when coupled with the framework proposed by Dell et al. (SODA’20 & SICOMP’22), leads to implies improved bounds for (1 ± ε)-approximation (where ε ∈ (0,1)) for the following fundamental problems: Edge Estimation using the Bipartite Independent Set ( BIS ) query. We improve the bound obtained by Beame et al. (ITCS’18 & TALG’20). Triangle Estimation using the Tripartite Independent Set ( TIS ) query. Currently, Dell et al.’s result gives the best bound for the case of triangle estimation in general graphs (SODA’20 & SICOMP’22). The previous best bound for the case of graphs with low co-degree (co-degree of a graph is the maximum number of triangles incident over any edge of the graph) was due to Bhattacharya et al. (ISAAC’19 & TOCS’21). We improve both of these bounds. Hyperedge Estimation & Sampling using Colorful Independence Oracle ( CID ). We give an improvement over the bounds obtained by Dell et al. (SODA’20 & SICOMP’22).