Constraint Solving via Fractional Edge Covers

TL;DR

This paper introduces the concept of fractional hypertree width, a new structural property that guarantees polynomial-time solvability of constraint satisfaction problems, expanding the known classes beyond hypertree width.

Contribution

It defines fractional hypertree width, combining hypertree width and fractional edge cover number, and proves its effectiveness in ensuring polynomial-time solutions for CSPs.

Findings

Bounded fractional hypertree width implies polynomial-time solvability.

Fractional hypertree width generalizes hypertree width.

Efficient algorithms exist for finding fractional hypertree decompositions.

Abstract

Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number. Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions by Marx, it follows that bounded fractional hypertree width is now the most general known structural property that guarantees polynomial-time solvability.