Minor-matching hypertree width

Abstract

In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, μ-tw, for graphs and hypergraphs, such that bounding the width guarantees that set of maximal independent sets has a polynomially-sized restriction to each decomposition bag. The relaxed conditions of the decomposition allow a much wider class of graphs and hypergraphs to have bounded width compared to other tree decompositions. We show that, for fixed k, there are 2(11 k+o(1))(n 2) n-vertex graphs of minor-matching hypertree width at most k. A number of problems including Maximum Independence Set, k-Colouring, and Homomorphism of uniform hypergraphs permit polynomial-time solutions for hypergraphs with bounded minor-matching hypertree width and bounded rank. We show that for any given k and any graph G, it is possible to construct a decomposition of minor-matching hypertree width at most O(k3), or to prove that μ-tw(G) > k in time nO(k3). This is done by presenting a general algorithm for approximating the hypertree width of well-behaved measures, and reducing μ-tw to such measure. The result relating the restriction of the maximal independent sets to a set S with the set of induced matchings intersecting S in graphs, and minor matchings intersecting S in hypergraphs, might be of independent interest.