On exact algorithms for treewidth

We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential-time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time O*(2n). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the Traveling Salesman problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. We also consider the problem of computing Treewidth under the restriction that the space used is only polynomial and give a simple O*(4n) algorithm that requires polynomial space. We also show that with a more complicated algorithm using balanced separators, Treewidth can be computed in O*(2.9512n) time and polynomial space.

[1]  Sandeep Koranne,et al.  Boost C++ Libraries , 2011 .

[2]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[3]  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..

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

[5]  Arie M. C. A. Koster,et al.  Degree-Based Treewidth Lower Bounds , 2005, WEA.

[6]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[7]  Yngve Villanger,et al.  Computing Pathwidth Faster Than 2n , 2009, IWPEC.

[8]  Jacques Carlier,et al.  Heuristic and metaheuristic methods for computing graph treewidth , 2004, RAIRO Oper. Res..

[9]  Yngve Villanger,et al.  Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In , 2006, LATIN.

[10]  Mikko Koivisto,et al.  A space-time tradeoff for permutation problems , 2010, SODA '10.

[11]  Fedor V. Fomin,et al.  Exact algorithms for treewidth and minimum fill-in ∗ † , 2006 .

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

[13]  Nancy G. Kinnersley,et al.  The Vertex Separation Number of a Graph equals its Path-Width , 1992, Inf. Process. Lett..

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

[15]  Hisao Tamaki,et al.  Computing Directed Pathwidth in O(1.89 n ) Time , 2012, IPEC.

[16]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

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

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

[19]  Saharon Shelah,et al.  Expected Computation Time for Hamiltonian Path Problem , 1987, SIAM J. Comput..

[20]  Arie M. C. A. Koster,et al.  Treewidth computations II. Lower bounds , 2011, Inf. Comput..

[21]  Arie M. C. A. Koster,et al.  PREPROCESSING RULES FOR TRIANGULATION OF PROBABILISTIC NETWORKS * , 2005, Comput. Intell..

[22]  Dimitrios M. Thilikos,et al.  Fugitive-Search Games on Graphs and Related Parameters , 1994, Theor. Comput. Sci..

[23]  Dimitrios M. Thilikos,et al.  A Note on Exact Algorithms for Vertex Ordering Problems on Graphs , 2012, Theory of Computing Systems.

[24]  JOSEP DÍAZ,et al.  A survey of graph layout problems , 2002, CSUR.

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

[26]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[27]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[28]  M. Golumbic.  Chapter 3 - Perfect graphs , 2004 .

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

[30]  Fedor V. Fomin,et al.  Finding Induced Subgraphs via Minimal Triangulations , 2009, STACS.

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

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

[33]  Fabrizio Grandoni,et al.  Some New Techniques in Design and Analysis of Exact (Exponential) Algorithms , 2005, Bull. EATCS.

[34]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

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

[36]  Dan Geiger,et al.  A Practical Algorithm for Finding Optimal Triangulations , 1997, AAAI/IAAI.

[37]  Andreas Björklund,et al.  Determinant Sums for Undirected Hamiltonicity , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

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

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

[40]  Hans L. Bodlaender,et al.  Treewidth: Characterizations, Applications, and Computations , 2006, WG.

[41]  Michal Pilipczuk,et al.  On Cutwidth Parameterized by Vertex Cover , 2011, IPEC.

[42]  Fedor V. Fomin,et al.  Treewidth computation and extremal combinatorics , 2008, Comb..