Abstract
In this paper we examine generalized satisfiability problems with limited variable occurrences. First, we show that 3 occurrences per variable suffice to make these problems as hard as their unrestricted version. Then we focus on generalized satisfiability problems with at most 2 occurrences per variable. It is known that some NP -complete generalized satisfiability problems become polynomially solvable when only 2 occurrences per variable are allowed. We identify two new families of generalized satisfiability problems, called local and binary, that are polynomially solvable when only 2 occurrences per variable are allowed. We achieve this result by means of a reduction to the \(\triangle\)-matroid parity problem, which is another important theme of this work.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. U.S.A. 43, 842–844 (1957)
Bouchet, A., Cunningham, W.H.: Delta-matroids, jump systems, and bisubmodular polyhedra. SIAM J. Discrete Math. 8(1), 17–32 (1995)
Bouchet, A.: Representability of Δ-matroids. Combinatorics, pp. 167–182 (1988)
Bouchet, A.: Coverings and delta-coverings. Integer programming and combinatorial optimization, Copenhagen, pp. 228–243 (1995)
Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of Boolean constraint satisfaction problems. SIAM, Philadelphia (2001)
Edmonds, J.: Paths, trees, and flowers. Canad. J. Math. 17, 449–467 (1965)
Feder, T.: Fanout limitations on constraint systems. Th. Comp. Sci. 255(1-2), 281–293 (2001)
Garey, M.R., Johnson, D.S.: Computers and intractability (1979)
Geelen, J., Iwata, S., Murota, K.: The linear delta-matroid parity problem. Technical Report RIMS Preprint 1149, Kyoto University (1997)
Istrate, G.I.: Looking for a version of schaeer’s dichotomy theorem when each variable occurs at most twice. Technical Report TR652, The University of Rochester (March 1997)
Kratochvíl, J., Savický, P., Tuza, Z.: One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM J. Comput. 22(1), 203–210 (1993)
Ladner, R.E.: On the structure of polynomial time reducibility. J. Assoc. Comput. Mach. 22, 155–171 (1975)
Lovász, L., Plummer, M.D.: Matching Theory (1986)
Lovász, L.: The membership problem in jump systems. J. Comb. Th. (B) 70(1), 45–66 (1997)
Lovász, L.: The matroid matching problem. Algebraic methods in graph theory, Szeged, vol. I, II, pp. 495–517 (1978)
Schaefer, T.J.: The complexity of satisfiability problems. Theory of Computing (1978)
Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dalmau, V., Ford, D.K. (2003). Generalized Satisfiability with Limited Occurrences per Variable: A Study through Delta-Matroid Parity. In: Rovan, B., Vojtáš, P. (eds) Mathematical Foundations of Computer Science 2003. MFCS 2003. Lecture Notes in Computer Science, vol 2747. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45138-9_30
Download citation
DOI: https://doi.org/10.1007/978-3-540-45138-9_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40671-6
Online ISBN: 978-3-540-45138-9
eBook Packages: Springer Book Archive