Faster Counting and Sampling Algorithms using Colorful Decision Oracle
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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 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, 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 m such that 1/C_dlog^d-1 n ≤ m/m ≤ C_dlog ^d-1 n . by using at most C_dlog ^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 a number of fundamental problems.
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