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 showas a new result -that computing hypertree decompositions is fixed-parameter intractable. Hypertree Decompositions: Definition and BasicsThis 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 are 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 caracterizes 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 polynopmial 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 of can be found in [14,17,2,1,16,18], most of which are available at the Hypertree Decompositions Homepage at http://si.deis.unical.it/ frank/Hypertrees. 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 hypergraphThe primal graph of a hypergraph H is the graph H = (V (H), {{v, w} | v = w, there exists an e ∈ E(H) such that {v, w} ⊆ e}).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 hypergraph H is a tuple (T, χ), where T = (V (T ), E(T )) is a tree and χ : V (T ) −→ 2 V (H) is a function associating a set of vertices χ(t) ⊆ V (H) to each vertex t of the

We generalize the notion of backdoor sets from propositional formulas to quantified Boolean formulas (QBF). This allows us to obtain hierarchies of tractable classes of quantified Boolean formulas with the classes of quantified Horn and quantified 2CNF formulas, respectively, at their first level, thus gradually generalizing these two important tractable classes. In contrast to known tractable classes based on bounded treewidth, the number of quantifier alternations of our classes is unbounded. As a side product of our considerations we develop a theory of variable dependency which is of independent interest.

No abstract

We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worst-case time and space requirements.

We consider the constraint satisfaction problem (CSP) parameterized by the treewidth of primal, dual, and incidence graphs, combined with several other basic parameters such as domain size and arity. We determine all combinations of the considered parameters that admit fixed-parameter tractability.