DeepAI AI Chat
Log In Sign Up

Approximately counting and sampling small witnesses using a colourful decision oracle

by   Holger Dell, et al.
University of Oxford
University of Glasgow
IT University of Copenhagen

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 fine-grained 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).


page 1

page 2

page 3

page 4


Faster Counting and Sampling Algorithms using Colorful Decision Oracle

In this work, we consider d-Hyperedge Estimation and d-Hyperedge Sample ...

Fine-grained reductions from approximate counting to decision

The main problems in fine-grained complexity are CNF-SAT, the Orthogonal...

Triangle Estimation using Polylogarithmic Queries

Estimating the number of triangles in a graph is one of the most fundame...

Nearly optimal independence oracle algorithms for edge estimation in hypergraphs

We study a query model of computation in which an n-vertex k-hypergraph ...

Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time

We present a simple sublinear-time algorithm for sampling an arbitrary s...

Fast algorithms for general spin systems on bipartite expanders

A spin system is a framework in which the vertices of a graph are assign...

Distribution-Aware Sampling and Weighted Model Counting for SAT

Given a CNF formula and a weight for each assignment of values to variab...