NDLI: Tree-Related Widths of Graphs and Hypergraphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Adler, Isolde
Copyright Year	2008
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. 579627] and generalized hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775808] 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. 400422]. It was recently proved in [G. Gottlob, Z. Mikls, 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. 579627].) 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. 2233] and hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775808]. 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.
Starting Page	102
Ending Page	123
Page Count	22
File Format	PDF
ISSN	08954801
DOI	10.1137/050623395
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	22
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-02-15
Access Restriction	Subscribed
Subject Keyword	tree-width hypergraphs Structural characterization of families of graphs hypergraph sandwich problem Graph algorithms Hypergraphs infinte hypergraphs hypertree-width
Content Type	Text
Resource Type	Article
Subject	Mathematics

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