Approximately counting and sampling small witnesses using a colourful decision oracle

Abstract

In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the finegrained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691).

Counting Small Induced Subgraphs with Hereditary Properties

“…Finally, we show that, whenever Φ is not meagre, the bipartite property Ψ Φ will satisfy the strong hardness condition in Theorem 5, yielding not only #W [1]-hardness, but also the conditional lower bound under ETH. 10 In what follows, we will describe both steps in more detail separately.…”

“…For more general classes of properties, there are only partial results, and to the best of our knowledge the complexity of approximating #IndSub(Φ) is still open for edge-monotone properties. However, there are strong recent meta-theorems such as the k-Hypergraph framework due to Dell, Lapinskas, and Meeks [10] that yield efficient approximation algorithms for #IndSub(Φ) by reduction to vertex-coloured decision problems.…”

Faster Counting and Sampling Algorithms using Colorful Decision Oracle

“…We do the improvement in approximation guarantee as well as query complexity in Rough Estimation algorithm (as stated in Theorem 1.3), as compared to Coarse algorithm of Dell et al [DLM20] (as stated in Lemma 5.1), by a careful analysis of the intersection pattern of the hypergraphs and setting the sampling probability parameters in Verify Estimate (Algorithm 1) algorithm in a nontrivial way, which is evident from the description of Algorithm 1 and its analysis.…”

“…* In [BGK + 18], the oracle is named as Generalized Partite Independent Set oracle. Here, we follow the same suit as Dell et al[DLM20] with respect to the name of the oracle.…”

On Triangle Estimation Using Tripartite Independent Set Queries

“…Note that the Triangle-Estimation can also be thought of as Hyperedge Estimation problem in a 3-uniform hypergrah. Very recently, Dell et al [15] and Bhattacharya et al [8], independently, showed that the bound on ∆ E is not necessary to solve Triangle-Estimation by using polylogarithmic many TIS queries. Also, both Dell et al [15] and Bhattacharya et al [8], independently, generalized our result to c-uniform hypergraphs, where c ∈ N is a constant.…”