Encoding Treewidth into SAT

One of the most important structural parameters of graphs is treewidth , a measure for the "tree-likeness" and thus in many cases an indicator for the hardness of problem instances. The smaller the treewidth, the closer the graph is to a tree and the more efficiently the underlying instance often can be solved. However, computing the treewidth of a graph is NP -hard in general. In this paper we propose an encoding of the decision problem whether the treewidth of a given graph is at most k into the propositional satisfiability problem. The resulting SAT instance can then be fed to a SAT solver. In this way we are able to improve the known bounds on the treewidth of several benchmark graphs from the literature.

[1]  Arie M. C. A. Koster,et al.  Contraction and Treewidth Lower Bounds , 2004, J. Graph Algorithms Appl..

[2]  Hans L. Bodlaender,et al.  Discovering Treewidth , 2005, SOFSEM.

[3]  Peter Vojtáš.  SOFSEM 2005: Theory and Practice of Computer Science, 31st Conference on Current Trends in Theory and Practice of Computer Science, Liptovský Ján, Slovakia, January 22-28, 2005, Proceedings , 2005, SOFSEM.

[4]  Carsten Sinz,et al.  Towards an Optimal CNF Encoding of Boolean Cardinality Constraints , 2005, CP.

[5]  Jacques Carlier,et al.  New Lower and Upper Bounds for Graph Treewidth , 2003, WEA.

[6]  Vibhav Gogate,et al.  A Complete Anytime Algorithm for Treewidth , 2004, UAI.

[7]  Toby Walsh,et al.  Handbook of Constraint Programming , 2006, Handbook of Constraint Programming.

[8]  Rina Dechter,et al.  Tractable Structures for Constraint Satisfaction Problems , 2006, Handbook of Constraint Programming.

[9]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[10]  Alexander Grigoriev,et al.  Treewidth Lower Bounds with Brambles , 2005, Algorithmica.

[11]  Arie M. C. A. Koster,et al.  Treewidth: Computational Experiments , 2001, Electron. Notes Discret. Math..

[12]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[13]  B. A. Reed,et al.  Algorithmic Aspects of Tree Width , 2003 .

[14]  Hans L. Bodlaender,et al.  New Upper Bound Heuristics for Treewidth , 2005, WEA.

[15]  Peter van Beek,et al.  Principles and Practice of Constraint Programming - CP 2005, 11th International Conference, CP 2005, Sitges, Spain, October 1-5, 2005, Proceedings , 2005, CP.

[16]  Hans L. Bodlaender,et al.  A Branch and Bound Algorithm for Exact, Upper, and Lower Bounds on Treewidth , 2006, AAIM.

[17]  Klaus Jansen,et al.  Experimental and Efficient Algorithms , 2003, Lecture Notes in Computer Science.

[18]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[19]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .