\input zb-basic
\input zb-ioport
\iteman{io-port 06027237}
\itemau{Fomin, Fedor V.; Golovach, Petr A.; Thilikos, Dimitrios M.}
\itemti{Approximating acyclicity parameters of sparse hypergraphs.}
\itemso{Albers, Susanne (ed.) et al., STACS 2009. 26th international symposium on theoretical aspects of computer science, Freiburg, Germany, February 26--28, 2009. Wadern: Schloss Dagstuhl -- Leibniz Zentrum f\"ur Informatik (ISBN 978-3-939897-09-5). LIPICS -- Leibniz International Proceedings in Informatics 3, 445-456, electronic only (2009).}
\itemab
Summary: The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by {\it M. Grohe} and {\it D. Marx} [in: Proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithms. New York, NY: Association for Computing Machinery (ACM); Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). 289--298 (2006; Zbl 1192.68642)], 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 {\it U. Feige}, {\it M. T. Hajiaghayi} and {\it J. R. Lee} [in: STOC'05: Proceedings of the 37th annual ACM symposium on theory of computing. New York, NY: Association for Computing Machinery (ACM). 563--572 (2005; Zbl 1192.68893)] on approximating treewidth of H-minor-free graphs.
\itemrv{~}
\itemcc{}
\itemut{graph; hypergraph; hypertree width; treewidth}
\itemli{doi:10.4230/LIPIcs.STACS.2009.1803 http://subs.emis.de/LIPIcs/frontdoor\_ce51.html}
\end