Faster Counting and Sampling Algorithms Using Colorful Decision Oracle
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,…, A_d ⊆ U(ℋ) as input, and answers whether there exists a hyperedge in H having (exactly) one vertex in each A_i, 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 m̂ such that
1/(C_{d)log^{d-1} n) ≤ m̂/m ≤ C_{d} log ^{d-1} n.
by using at most C_{d}log ^{d+2} n many 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 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].
Query Complexity
Subset Query
Hyperedge Estimation
and Colorful Independent Set oracle
Theory of computation~Streaming, sublinear and near linear time algorithms
10:1-10:16
Regular Paper
https://arxiv.org/abs/2201.04975
Anup
Bhattacharya
Anup Bhattacharya
National Institute of Science Education and Research, Bhubaneswar, India
https://sites.google.com/site/anupbtcs
Arijit
Bishnu
Arijit Bishnu
Indian Statistical Institute, Kolkata, India
https://www.isical.ac.in/~arijit/
Arijit
Ghosh
Arijit Ghosh
Indian Statistical Institute, Kolkata, India
https://sites.google.com/site/homepagearijitghosh/
Gopinath
Mishra
Gopinath Mishra
University of Warwick, Coventry, UK
https://sites.google.com/view/gopinathmishra/
10.4230/LIPIcs.STACS.2022.10
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http://arxiv.org/abs/2201.04975
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Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra
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