Branch-Cut-and-Propagate for the Maximum k-Colorable Subgraph Problem with Symmetry

Given an undirected graph and a positive integer k, themaximum k-colorable subgraph problem consists of selecting a k-colorable induced subgraph of maximum cardinality. The natural integer programming formulation for this problem exhibits two kinds of symmetry: arbitrarily permuting the color classes and/or applying a non-trivial graph automorphism gives equivalent solutions. It is well known that such symmetries have negative effects on the performance of constraint/integer programming solvers. We investigate the integration of a branch-and-cut algorithm for solving the maximum k-colorable subgraph problem with constraint propagation techniques to handle the symmetry arising from the graph. The latter symmetry is handled by (non-linear) lexicographic ordering constraints and linearizations thereof. In experiments, we evaluate the influence of several components of our algorithm on the performance, including the different symmetry handling methods. We show that several components are crucial for an efficient algorithm; in particular, the handling of graph symmetries yields a significant performance speed-up.

[1]  Pascal Van Hentenryck,et al.  Static and Dynamic Structural Symmetry Breaking , 2006, CP.

[2]  Jean-François Puget,et al.  On the Satisfiability of Symmetrical Constrained Satisfaction Problems , 1993, ISMIS.

[3]  Toby Walsh,et al.  Breaking Row and Column Symmetries in Matrix Models , 2002, CP.

[4]  M. Jünger,et al.  50 Years of Integer Programming 1958-2008 - From the Early Years to the State-of-the-Art , 2010 .

[5]  Tobias Achterberg,et al.  Conflict analysis in mixed integer programming , 2007, Discret. Optim..

[6]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[7]  Edward C. Sewell,et al.  An improved algorithm for exact graph coloring , 1993, Cliques, Coloring, and Satisfiability.

[8]  François Margot,et al.  Symmetry in Integer Linear Programming , 2010, 50 Years of Integer Programming.

[9]  Jeff T. Linderoth,et al.  Orbital branching , 2007, Math. Program..

[10]  Laurence A. Wolsey,et al.  Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 4th International Conference, CPAIOR 2007, Brussels, Belgium, May 23-26, 2007, Proceedings , 2007, CPAIOR.

[11]  M. Pfetsch,et al.  Detecting Orbitopal Symmetries , 2009 .

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

[13]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[14]  Marc E. Pfetsch,et al.  The maximum k-colorable subgraph problem and orbitopes , 2011, Discret. Optim..

[15]  Isabel Méndez-Díaz,et al.  A Polyhedral Approach for Graph Coloring1 , 2001, Electron. Notes Discret. Math..

[16]  François Margot,et al.  Small covering designs by branch-and-cut , 2003, Math. Program..

[17]  Ian P. Gent,et al.  Symmetry in Constraint Programming , 2006, Handbook of Constraint Programming.

[18]  Thorsten Koch,et al.  Constraint Integer Programming: A New Approach to Integrate CP and MIP , 2008, CPAIOR.

[19]  Jeff T. Linderoth,et al.  Orbital Branching , 2007, IPCO.

[20]  Timo Berthold,et al.  Hybrid Branching , 2009, CPAIOR.

[21]  Aziz Moukrim,et al.  Pre-processing and Linear-Decomposition Algorithm to Solve the k-Colorability Problem , 2004, WEA.

[22]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[23]  R. Frucht.  Herstellung von Graphen mit vorgegebener abstrakter Gruppe , 1939 .

[24]  Sarah Ruepp,et al.  Benchmarking RWA strategies for dynamically controlled optical networks , 2008, Networks 2008 - The 13th International Telecommunications Network Strategy and Planning Symposium.

[25]  Isabel Méndez-Díaz,et al.  A cutting plane algorithm for graph coloring , 2008, Discret. Appl. Math..

[26]  Nicolas Barnier,et al.  Solving the Kirkman's schoolgirl problem in a few seconds , 2002 .

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

[28]  James M. Crawford,et al.  Symmetry-Breaking Predicates for Search Problems , 1996, KR.

[29]  Peter Jeavons,et al.  Symmetry Definitions for Constraint Satisfaction Problems , 2005, Constraints.

[30]  David Cohen,et al.  Principles and Practice of Constraint Programming - CP 2010 - 16th International Conference, CP 2010, St. Andrews, Scotland, UK, September 6-10, 2010. Proceedings , 2010, CP.

[31]  François Margot,et al.  Symmetric ILP: Coloring and small integers , 2007, Discret. Optim..

[32]  Isabel Méndez-Díaz,et al.  A Branch-and-Cut algorithm for graph coloring , 2006, Discret. Appl. Math..

[33]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[34]  Zbigniew W. Ras,et al.  Methodologies for Intelligent Systems , 1991, Lecture Notes in Computer Science.

[35]  V. Kaibel,et al.  Packing and partitioning orbitopes , 2006, math/0603678.

[36]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[37]  Toby Walsh,et al.  Propagation algorithms for lexicographic ordering constraints , 2006, Artif. Intell..

[38]  Robert Klein,et al.  Operations Research Proceedings 2008 , 2009 .

[39]  François Margot,et al.  Pruning by isomorphism in branch-and-cut , 2001, Math. Program..

[40]  Toby Walsh,et al.  On the Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry , 2010, CP.

[41]  Fabrizio Grandoni,et al.  A measure & conquer approach for the analysis of exact algorithms , 2009, JACM.

[42]  Marc E. Pfetsch,et al.  Orbitopal fixing , 2011, Discret. Optim..

[43]  Ted K. Ralphs,et al.  Integer and Combinatorial Optimization , 2013 .

[44]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .