Approximating Maximum Stable Set and Minimum Graph Coloring Problems with the Positive Semidefinite Relaxation

We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. Prom the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and over 6000 edges.

[1]  David C. Wood,et al.  A technique for colouring a graph applicable to large scale timetabling problems , 1969, Computer/law journal.

[2]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[3]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[4]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[5]  Egon Balas,et al.  A node covering algorithm , 1977 .

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

[7]  M. Powell,et al.  On the Estimation of Sparse Hessian Matrices , 1979 .

[8]  室 章治郎.  Michael R.Garey/David S.Johnson 著, "COMPUTERS AND INTRACTABILITY A guide to the Theory of NP-Completeness", FREEMAN, A5判変形判, 338+xii, \5,217, 1979 , 1980 .

[9]  John Cocke,et al.  Register Allocation Via Coloring , 1981, Comput. Lang..

[10]  John Cocke,et al.  A methodology for the real world , 1981 .

[11]  B. Pittel.  On the probable behaviour of some algorithms for finding the stability number of a graph , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  D. de Werra,et al.  An introduction to timetabling , 1985 .

[13]  Marek Kubale,et al.  A generalized implicit enumeration algorithm for graph coloring , 1985, CACM.

[14]  A. Gamst,et al.  Some lower bounds for a class of frequency assignment problems , 1986, IEEE Transactions on Vehicular Technology.

[15]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[16]  Gottfried Tinhofer,et al.  A branch and bound algorithm for the maximum clique problem , 1990, ZOR Methods Model. Oper. Res..

[17]  G. Anandalingam,et al.  Optimization of resource location in hierarchical computer networks , 1990, Comput. Oper. Res..

[18]  Alain Hertz,et al.  Tabaris: An exact algorithm based on tabu search for finding a maximum independent set in a graph , 1990, Comput. Oper. Res..

[19]  P. Pardalos,et al.  An exact algorithm for the maximum clique problem , 1990 .

[20]  Craig A. Morgenstern,et al.  Coloration neighborhood structures for general graph coloring , 1990, SODA '90.

[21]  Egon Balas,et al.  Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1991, SIAM J. Comput..

[22]  G. Nemhauser,et al.  A Strong Cutting Plane/Branch-and-Bound Algorithm for Node Packing , 1992 .

[23]  Panos M. Pardalos,et al.  A branch and bound algorithm for the maximum clique problem , 1992, Comput. Oper. Res..

[24]  Egon Balas,et al.  Addendum: Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1992, SIAM J. Comput..

[25]  Carlo Mannino,et al.  Edge projection and the maximum cardinality stable set problem , 1993, Cliques, Coloring, and Satisfiability.

[26]  Alan M. Frieze,et al.  Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION , 1995, IPCO.

[27]  Michael A. Trick,et al.  A Column Generation Approach for Graph Coloring , 1996, INFORMS J. Comput..

[28]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[29]  Y. Nesterov.  Quality of semidefinite relaxation for nonconvex quadratic optimization , 1997 .

[30]  Henry Wolkowicz,et al.  Strong Duality for Semidefinite Programming , 1997, SIAM J. Optim..

[31]  Y. Ye,et al.  Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics , 1998 .

[32]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1998, JACM.

[33]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[34]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[35]  Jon M. Kleinberg,et al.  The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover , 1998, SIAM J. Discret. Math..

[36]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[37]  Tamon Stephen,et al.  On a Representation of the Matching Polytope Via Semidefinite Liftings , 1999, Math. Oper. Res..

[38]  Yinyu Ye,et al.  Approximating quadratic programming with bound and quadratic constraints , 1999, Math. Program..

[39]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[40]  Levent Tunçel,et al.  On the Slater condition for the SDP relaxations of nonconvex sets , 2001, Oper. Res. Lett..