Minor-matching hypertree width

In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, mu-t omega, 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(1-1/k+0(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 hyper-tree 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(k(3)), or to prove that mu-t omega(G) > k in time n(O(k3)). This is done by presenting a general algorithm for approximating the hypertree width of well-behaved measures, and reducing mu-t omega 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.