Faster Counting and Sampling Algorithms Using Colorful Decision Oracle

Article Type

Research Article

Publication Title

ACM Transactions on Computation Theory

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 A1, . . ., Ad ⊆ U (H) as input and answers whether there exists a hyperedge in H having exactly one vertex in each Ai 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 m̂ such that by using at most Cd logd+2 n CID queries, where n denotes the number of vertices in the hypergraph H and Cd 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).

DOI

10.1145/3657605

Publication Date

6-10-2024

Comments

Open Access; Bronze Open Access; Green Open Access

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