Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In

Exact exponential-time algorithms for NP-hard problems is an emerging field, and an increasing number of new results are being added continuously. Two important NP-hard problems that have been studied for decades are the treewidth and the minimum fill problems. Recently, an exact algorithm was presented by Fomin, Kratsch, and Todinca to solve both of these problems in time ${\mathcal O}^{*}$(1.9601n). Their algorithm uses the notion of potential maximal cliques, and is able to list these in time ${\mathcal O}^{*}$(1.9601n), which gives the running time for the above mentioned problems. We show that the number of potential maximal cliques for an arbitrary graph G on n vertices is ${\mathcal O}^{*}$(1.8135n), and that all potential maximal cliques can be listed in ${\mathcal O}^{*}$(1.8899n) time. As a consequence of this results, treewidth and minimum fill-in can be computed in ${\mathcal O}^{*}$(1.8899n) time.

[1]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[2]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[3]  Ioan Todinca,et al.  Listing all potential maximal cliques of a graph , 2000, Theor. Comput. Sci..

[4]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[5]  D. Rose.  A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS , 1972 .

[6]  Paul D. Seymour,et al.  Tour Merging via Branch-Decomposition , 2003, INFORMS J. Comput..

[7]  Fedor V. Fomin,et al.  Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In , 2004, ICALP.

[8]  Jon M. Kleinberg,et al.  A deterministic (2-2/(k+1))n algorithm for k-SAT based on local search , 2002, Theor. Comput. Sci..

[9]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[10]  Dieter Kratsch,et al.  On treewidth approximations , 2004, Discret. Appl. Math..

[11]  Ioan Todinca,et al.  Treewidth and Minimum Fill-in: Grouping the Minimal Separators , 2001, SIAM J. Comput..

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

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

[14]  David J. Spiegelhalter,et al.  Local computations with probabilities on graphical structures and their application to expert systems , 1990 .

[15]  Anne Berry,et al.  Generating All the Minimal Separators of a Graph , 1999, Int. J. Found. Comput. Sci..

[16]  Eyal Amir,et al.  Efficient Approximation for Triangulation of Minimum Treewidth , 2001, UAI.

[17]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.