Approximating acyclicity parameters of sparse hypergraphs

TL;DR

This paper investigates the approximability of hypergraph acyclicity parameters, establishing bounds and algorithms for sparse hypergraphs with incidence graphs in certain minor-free classes, enhancing understanding of hypergraph width measures.

Contribution

It proves that hypertree widths are bounded by the treewidth of incidence graphs in apex-minor-free classes and provides constant-factor approximation algorithms for these cases.

Findings

Hypertree width is closely related to incidence graph treewidth in certain classes.

A constant-factor approximation algorithm is developed for hypergraphs with minor-free incidence graphs.

The results justify hypertree width as a meaningful measure of hypergraph acyclicity.

Abstract

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class.