# Minor-matching hypertree width

## TL;DR

This paper introduces minor-matching hypertree width, a new graph and hypergraph width measure that enables polynomial-time solutions for several problems on graphs with bounded width, expanding the class of tractable instances.

## Contribution

It defines minor-matching hypertree width, shows its properties, and provides algorithms for approximation and recognition, broadening the applicability of width-based graph algorithms.

## Key findings

- Bounded minor-matching hypertree width implies polynomial algorithms for Maximal Independent Set, Coloring, and Homomorphism.
- There are exponentially many graphs with bounded minor-matching hypertree width for fixed k.
- An algorithm can approximate or determine the width within a factor of O(k^3) in n^{O(k^3)} time.

## Abstract

In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, $\mu\text{-}tw$, 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 of bounded width compared to other tree decompositions. We show that, for fixed $k$, there are $2^{(1 - \frac1k + o(1)){n \choose 2}}$ $n$-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 $O(k^3)$, or to prove that $\mu\text{-}tw(G) > k$ in time $n^{O(k^3)}$. This is done by presenting a general algorithm for approximating the hypertree width of well-behaved measures, and reducing $\mu\text{-}tw$ 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.