Treewidth and Hypertree Width

Abstract

The chapter covers methods for identifying islands of tractability for NP-hard combinatorial problems by exploiting suitable properties of their graphical structure. Acyclic structures are considered, as well as nearly acyclic ones identified by means of so-called structural decomposition methods. In particular, the chapter focuses on the tree decomposition method, which is the most powerful decomposition method for graphs, and on the hypertree decomposition method, which is its natural counterpart for hypergraphs. These problem-decomposition methods give rise to corresponding notions of width of an instance, namely, treewidth and hypertree width. It turns out that many NP-hard problems can be solved efficiently over classes of instances of bounded treewidth or hypertree width: deciding whether a solution exists, computing a solution, and even computing an optimal solution (if some cost function over solutions is specified) are all polynomial-time tasks. Example applications include problems from artificial intelligence, databases, game theory, and combinatorial auctions.

An analytical study of large SPARQL query logs

“…We therefore investigated the tree-and hypertreewidth the CQs, CQ F -and CQ OF queries. We do not formally define tree-or hypertreewidth in this paper but instead refer to an excellent introduc-tion [13]. In the terminology of Gottlob et al, we investigate the generalized hypertree width of the canonical hypergraphs of queries.…”

Evaluating Datalog via Tree Automata and Cycluits

“…Yannakakis's result was generalized in two main directions. One direction [37] has investigated more general CQ classes, in particular CQs of bounded treewidth [33], hypertreewidth [38], and fractional hypertreewidth [43]. Bounding these query parameters to some fixed k makes query evaluation run in time O((|I| · |Q|) f (k) ) for some function f , hence in PTIME; for treewidth, since the decomposition can be computed in FPT-linear time [20], this goes down to O(|I| k · |Q|).…”

Hypertree Decompositions

“…The notion of tree decomposition [123] represents a significant success story in Computer Science (see, e.g., [65]). The associated notion of treewidth was meant to provide a measure of the degree of cyclicity in graphs and hypergraphs.…”