# Faster Counting and Sampling Algorithms Using Colorful Decision Oracle

## Document Type

Conference Article

## Publication Title

Leibniz International Proceedings in Informatics, LIPIcs

## Abstract

In this work, we consider d-Hyperedge Estimation and d-Hyperedge Sample problem 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, i ∈ {1, 2, . . ., d}. The problem of d-Hyperedge Estimation and d-Hyperedge Sample with CID oracle access is important in its own right as a combinatorial problem. Also, Dell et al. [SODA’20] established that decision vs counting complexities of a number of combinatorial optimization problems can be abstracted out as d-Hyperedge Estimation problems with a CID oracle access. The main technical contribution of the paper is an algorithm that estimates m = |F(H)| with mb such that 1 ≤ mmb ≤ Cd logd-1 n. Cd logd-1 n by using at most Cd logd+2 n many 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 coupled with the framework of Dell et al. [SODA’21] implies improved bounds for the following fundamental problems: Edge Estimation using the Bipartite Independent Set (BIS). We improve the bound obtained by Beame et al. [ITCS’18, TALG’20]. Triangle Estimation using the Tripartite Independent Set (TIS). The previous best bound for the case of graphs with low co-degree (Co-degree for an edge in the graph is the number of triangles incident to that edge in the graph) was due to Bhattacharya et al. [ISAAC’19, TOCS’21], and Dell et al.’s result gives the best bound for the case of general graphs [SODA’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’21].

## DOI

10.4230/LIPIcs.STACS.2022.10

## Publication Date

3-1-2022

## Recommended Citation

Bhattacharya, Anup; Bishnu, Arijit; Ghosh, Arijit; and Mishra, Gopinath, "Faster Counting and Sampling Algorithms Using Colorful Decision Oracle" (2022). *Conference Articles*. 390.

https://digitalcommons.isical.ac.in/conf-articles/390