Constrained Multilinear Detection and Generalized Graph Motifs

TL;DR

This paper presents a novel algebraic sieving technique for detecting constrained multilinear monomials, leading to efficient algorithms for graph motif problems and related optimization variants.

Contribution

It introduces a new algebraic method for detecting constrained multilinear monomials and applies it to develop faster algorithms for Graph Motif and its optimization variants.

Findings

Developed an $O^*(2^k)$-time algorithm for the Graph Motif problem.

Extended the approach to solve the Closest Graph Motif variant.

Provided evidence that faster algorithms could imply breakthroughs for Set Cover.

Abstract

We introduce a new algebraic sieving technique to detect constrained multilinear monomials in multivariate polynomial generating functions given by an evaluation oracle. As applications of the technique, we show an $O^*(2^k)$-time polynomial space algorithm for the $k$-sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute Graph Motif, and Min-Add Graph Motif. Finally, we provide a piece of evidence that our result might be essentially tight: the existence of an $O^*((2-\epsilon)^k)$-time algorithm for the Graph Motif problem implies an $O((2-\epsilon')^n)$-time algorithm for Set Cover.