Counting H-Colorings of Partial k-Trees
The problem of counting all H-colorings of a graph G of n vertices is considered. While the problem is, in general, #P-complete, we give linear time algorithms that solve the main variants of this problem when the input graph G is a k-tree or, in the case where G is directed, when the underlying graph of G is a k-tree. Our algorithms remain polynomial even in the case where k = O(log n) or in the case where the size of H is O(n). Our results are easy to implement and imply the existence of polynomial time algorithms for a series of problems on partial k-trees such as core checking and chromatic polynomial computation.
KeywordsPolynomial Time Algorithm Input Graph Tree Decomposition Linear Time Algorithm Counting Problem
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- [CDF]Colin Cooper, Martin Dyer, and Alan Frieze On Marcov chains for randomly H-coloring a graph. Journal of Algorithms (to appear).Google Scholar
- [CMR]B. Courcelle, J.A. Makowski, and U. Rotics On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics (to appear).Google Scholar
- [Cou90a]Bruno Courcelle Graph rewriting: an algebraic and logical approach. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 192–242, Amsterdam, 1990. North Holland Publ. Comp.Google Scholar
- [DFJ99]Martin Dyer, Alan Frieze, and Mark Jerrum On counting independent sets in sparse graphs. In 40th Annual Symposium on Foundations of Computer Science, pages 210–217, 1999.Google Scholar
- [DG00]M.E. Dyer and C.S. Greenhill The complexity of counting graph homomorphisms. In 11th ACM/SIAM Symposium on Discrete Algorithms, pages 246–255, 2000.Google Scholar
- [DNS00]Josep Díaz, Jaroslav Ne#x0161;etřil, and Maria Serna H-coloring of large degree graphs. Technical Report No. 2000-465, KAM-DIMATIA Series, Charles University, 2000.Google Scholar
- [Lev73]L. Levin Universal sequential search problems. Problems of Information Transmissions, 9:265–266, 1973.Google Scholar
- [Ree92]B. Reed Finding approximate separators and computing tree-width quickly. In 24th ACM Symposium on Theory of Computing, pages 221–228, 1992.Google Scholar
- [RS85]Neil Robertson and Paul D. Seymour Graph minors — a survey. In I. Anderson editor, Surveys in Combinatorics, pages 153–171. Cambridge Univ. Press, 1985.Google Scholar