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• W2091522522 abstract "A hypergraph pair is a pair $(G,H)$ where G and H are hypergraphs on the same set of vertices. We extend the definitions of hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 64 (2002), pp. 579–627] and generalized hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775–808] from hypergraphs to hypergraph pairs. We show that for constant k the problem of deciding whether a hypergraph pair has generalized hypertree-width $leq k$, is equivalent to the hypergraph sandwich problem (HSP) [A. Lustig and O. Shmueli, J. Algorithms, 30 (1999), pp. 400–422]. It was recently proved in [G. Gottlob, Z. Miklós, and Th. Schwentick, Proceedings of the Symposium on Principles of Database Systems $(PODS/07)$] that the HSP is NP-complete. For constant k there is a polynomial time algorithm that decides whether a given hypergraph pair has hypertree-width $leq k$. (For hypertree-width of hypergraphs, this was shown in [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 64 (2002), pp. 579–627].) It follows that the HSP is solvable in polynomial time for inputs $(G,H)$ satisfying: $operatorname{ghw}(G,H)leq 1$ if and only if $operatorname{hw}(G,H)leq 1$. Besides this practical interest, hypergraph pairs serve as a tool for giving a common proof for the game theoretic characterizations of tree-width [P. D. Seymour and R. Thomas, J. Combin. Theory Ser. B, 58 (1993), pp. 22–33] and hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775–808]. Furthermore, they enable us to show a compactness property of generalized hypertree-width for a large class of hypergraphs, the hypergraphs with finite character. Finally, we present two examples showing that neither hypertree-width of hypergraph pairs nor hypertree-width of hypergraphs has the compactness property." @default.
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• W2091522522 date "2008-01-01" @default.
• W2091522522 modified "2023-09-23" @default.
• W2091522522 title "Tree-Related Widths of Graphs and Hypergraphs" @default.
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• W2091522522 doi "https://doi.org/10.1137/050623395" @default.
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