Constraint solving via fractional edge covers

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. [2002]. 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 [Marx 2010a], it follows that bounded fractional hypertree width is now the most general known structural property that guarantees polynomialtime solvability.

The limited blessing of low dimensionality

“…There is a large body of literature on how structural restrictions, that is, restrictions on the constraint graph influences the complexity of CSP [9,17,1,18,15,19,28,27,29,16]. Specifically, we need a general result of Marx [28] stating that, in a precise technical sense, treewidth of the constraint graph governs the complexity of the problem.…”

Skew strikes back

“…A major recent result from AGM [2,20] is the key to bridging this gap: AGM derived a tight bound on the output size of a join query as a function of individual input relation sizes and a much finer notion of "width". The AGM bound leads to the notion of fractional query number and eventually fractional hypertree width (fhw) which is strictly more general than all of the above width notions [28].…”

“…Atserias-Grohe-Marx [2] and Grohe-Marx [20] proved the following remarkable inequality, which shall be referred to as the AGM's inequality henceforth. Let x " px F q FPE be any point in the following polyhedron: 8 > > < > > :…”

“…More recently and more generally, tight estimates for the natural join problem were derived by Grohe-Marx [20] and Atserias-Grohe-Marx [2] (henceforth AGM). In fact, similar bounds can be traced back to the 1940s in geometry, where it was known as the famous Loomis-Whitney inequality [26].…”

H-Db

“…Yet, they are often not very intricate and, in fact, tend to exhibit some limited degree of cyclicity, which suffices to retain most of the nice properties of acyclic ones. Thus, several efforts have been subsequently spent to investigate invariants that are best suited to identify nearly-acyclic hypergraphs, leading to the definition of a number of so-called structural decomposition methods (such as the (generalized) hypertree [3], the fractional hypertree [5], and the component hypertree [4] decompositions). These methods aim at transforming a given cyclic hypergraph into an acyclic one, by organizing its edges (or its nodes) into a polynomial number of clusters, and by suitably arranging these clusters as a tree, called decomposition tree.…”