General and Fractional Hypertree Decompositions

Abstract

Hypertree decompositions, as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHD) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H ), its generalized hypertree width ghw(H ), and its fractional hypertree width fhw(H ), respectively. It is known that hw(H ) ≤ k can be checked in polynomial time for fixed k, while checking ghw(H ) ≤ k is NP-complete for k ≥ 3. The complexity of checking fhw(H ) ≤ k for a fixed k has been open for over a decade.We settle this open problem by showing that checking fhw(H ) ≤ k is NP-complete, even for k = 2. The same construction allows us to prove also the NP-completeness of checking ghw(H ) ≤ k for k = 2. After that, we identify meaningful restrictions which make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw. INTRODUCTION AND BACKGROUNDResearch Challenges Tackled. In this work we tackle computational problems on hypergraph decompositions, which play a prominent role for answering Conjunctive Queries (CQs) and solving Constraint Satisfaction Problems (CSPs), which we discuss below.Many NP-hard graph-based problems become tractable for instances whose corresponding graphs have bounded treewidth. There are, however, many problems for which the structure of an instance is better described by a hypergraph than by a graph, for example, the above mentioned CQs and CSPs. Given that treewidth does not generalize hypergraph acyclicity 1 , proper hypergraph decomposition methods have been developed, in particular, hypertree decompositions (HDs) [27], the more general generalized hypertree decompositions (GHDs) [27], and the yet more general fractional hypertree decompositions (FHDs) [31], and corresponding notions of width of a hypergraph H have been defined: the hypertree width hw(H ), the generalized hypertree width ghw(H ), and the fractional hypertree width fhw(H ), where for every hypergraph H , fhw(H ) ≤ ghw(H ) ≤ hw(H ) holds. Definitions are given in Section 2. A number of highly relevant hypergraph-based problems such as CQ-evaluation and CSP-solving become tractable for classes of instances of bounded hw, ghw, or, fhw. For each of the mentioned types of decompositions it would thus be useful to be able to recognize for each constant k whether a given hypergraph H has corresponding width at most k, and if so, to compute such a decomposition. More formally, for decomposition ∈ {HD, GHD, FHD} and k > 0, we consider the following family of problems: 1 We here refer to the standard notion of hypergraph acyclicity, as used in [50] and [22], where it is called α -acyclicity. This notion is more general than other types of acyclicity that have been introduced in the literature.exists and answer 'no' otherwise. As shown in [27], Check(HD, k) is in Ptime. However, little is known about ...

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