Approximating Acyclicity Parameters of Sparse Hypergraphs

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello (PODS'99, PODS'01) in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx in SODA'06, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. Computing each of these width parameters is known to be an NP-hard problem. Moreover, the (generalized) hypertree width of an n-vertex hypergraph cannot be approximated within a logarithmic factor unless P=NP. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class. This extends the results of Feige, Hajiaghayi, and Lee from STOC'05 on approximating treewidth of H-minor-free graphs.publishedVersio

Beyond Hypertree Width: Decomposition Methods Without Decompositions

The general intractability of the constraint satisfaction problem has motivated the study of restrictions on this problem that permit polynomial-time solvability. One major line of work has focused on structural restrictions, which arise from restricting the interaction among constraint scopes. In this paper, we engage in a mathematical investigation of generalized hypertree width, a structural measure that has up to recently eluded study. We obtain a number of computational results, including a simple proof of the tractability of CSP instances having bounded generalized hypertree width

An Analytical Study of Large SPARQL Query Logs

With the adoption of RDF as the data model for Linked Data and the Semantic Web, query specification from end- users has become more and more common in SPARQL end- points. In this paper, we conduct an in-depth analytical study of the queries formulated by end-users and harvested from large and up-to-date query logs from a wide variety of RDF data sources. As opposed to previous studies, ours is the first assessment on a voluminous query corpus, span- ning over several years and covering many representative SPARQL endpoints. Apart from the syntactical structure of the queries, that exhibits already interesting results on this generalized corpus, we drill deeper in the structural char- acteristics related to the graph- and hypergraph represen- tation of queries. We outline the most common shapes of queries when visually displayed as pseudographs, and char- acterize their (hyper-)tree width. Moreover, we analyze the evolution of queries over time, by introducing the novel con- cept of a streak, i.e., a sequence of queries that appear as subsequent modifications of a seed query. Our study offers several fresh insights on the already rich query features of real SPARQL queries formulated by real users, and brings us to draw a number of conclusions and pinpoint future di- rections for SPARQL query evaluation, query optimization, tuning, and benchmarking

Hypergraph Acyclicity and Propositional Model Counting

We show that the propositional model counting problem #SAT for CNF- formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore, we present a polynomial time algorithm that computes a disjoint branches decomposition of a given hypergraph if it exists and rejects otherwise. Finally, we show that some slight extensions of the class of hypergraphs with disjoint branches decompositions lead to intractable #SAT, leaving open how to generalize the counting result of this paper