Hypertree Decompositions: Structure, Algorithms, and Applications

We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixed-parameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertree decomposition and the associated notion of hypertree-width. The latter is a cyclicity measure for hypergraphs, and constitutes a hypergraph invariant as it is preserved under hypergraph isomorphisms. Many interesting NP-hard problems are polynomially solvable for classes of instances associated with hypergraphs of bounded width. This is also true for other hypergraph invariants such as treewidth, cutset-width, and so on. However, the advantage of hypertree-width with respect to other known hypergraph invariants is that it is more general and covers larger classes of instances of bounded width. The main concepts of hypertree decomposition and hypertree-width are introduced in the present section. A normal form for hypertree decompositions is described in Section 2. Section 3 describes the Robbers and Marshals game which characterizes hypertree-width. In Section 4 we use this game to explain why the problem of checking whether the hypertree-width of a hypergraph is ≤ k is feasible in polynomial time for each constant k. However, in Section 5 we show that this problem is fixed-parameter intractable with respect to k. In Section 6 we compare hypertree-width to other relevant hypergraph invariants. In Section 7 we discuss heuristics for computing hypertree decompositions. In Section 8 we show how hypertree decompositions can be beneficially applied for solving constraint satisfaction problems (CSPs). Finally, in Section 9 we list some open problems left for future research. Due to space limitations this paper is rather short, and most proofs are missing. A more thorough treatment can be found in [13,16,2,1,15,17], most of which are available at the Hypertree Decomposition Homepage at http://si.deis.unical.it/∼frank/Hypertrees. ? This paper was supported by the Austrian Science Fund (FWF) project: Nr. P17222-N04, Complementary Approaches to Constraint Satisfaction. Correspondence to: Georg Gottlob, Institut für Informationssysteme, TU Wien, Favoritenstr. 9-11/184-2, A-1040 Wien, Austria, E-mail: [email protected]. A hypergraph is a pair H = (V (H), E(H)), consisting of a nonempty set V (H) of vertices, and a set E(H) of subsets of V (H), the hyperedges of H . We only consider finite hypergraphs. Graphs are hypergraphs in which all hyperedges have two elements. For a hypergraph H and a set X ⊆ V (H), the subhypergraph induced by X is the hypergraph H [X ] = (X, {e ∩ X | e ∈ E(H)}). We let H \ X := H [V (H) \ X ]. The primal graph of a hypergraph H is the graph H = (V (H), {{v, w} | v 6= w, there exists an e ∈ E(H) such that {v, w} ⊆ e}). A hypergraph H is connected if H is connected. A set C ⊆ V (H) is connected (in H) if the induced subhypergraph H [C] is connected, and a connected component of H is a maximal connected subset of V (H). A sequence of nodes of V (H) is a path of H if it is a path of H . A tree decomposition of a hypergraphH is a tuple (T, χ), where T = (V (T ), E(T )) is a tree and χ : V (T ) −→ 2 (H) is a function associating a set of vertices χ(t) ⊆ V (H) to each vertex t of the decomposition tree T , such that for each e ∈ E(H) there is a node t ∈ V (T ) such that e ⊆ χ(t), and for each v ∈ V (H) the set {t ∈ V (T ) | v ∈ χ(t)} is connected in T . We assume the tree T in a tree decomposition to be rooted. For every node t, Tt denotes the rooted subtree of T with root t. For each such subtree Tt, let χ(Tt) =