Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0–1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.

[1]  Thorsten Koch,et al.  Branching rules revisited , 2005, Oper. Res. Lett..

[2]  Franz Rendl,et al.  Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition , 2006, Math. Program..

[3]  Christoph Helmberg,et al.  Fixing Variables in Semidefinite Relaxations , 1997, ESA.

[4]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[5]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[6]  Yin Zhang,et al.  Rank-Two Relaxation Heuristics for MAX-CUT and Other Binary Quadratic Programs , 2002, SIAM J. Optim..

[7]  P. Pardalos,et al.  Parallel branch and bound algorithms for quadratic zero–one programs on the hypercube architecture , 1990 .

[8]  Endre Boros,et al.  A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO) , 2008, Discret. Optim..

[9]  Masakatsu Muramatsu,et al.  A NEW SECOND-ORDER CONE PROGRAMMING RELAXATION FOR MAX-CUT PROBLEMS , 2003 .

[10]  G. Reinelt,et al.  2 Computing Exact Ground Statesof Hard Ising Spin Glass Problemsby Branch-and-Cut , 2005 .

[11]  Catherine A. Schevon,et al.  Optimization by simulated annealing: An experimental evaluation , 1984 .

[12]  Michael Armbruster,et al.  Branch-and-Cut for a Semidefinite Relaxation of Large-scale Minimum Bisection Problems , 2007 .

[13]  Franz Rendl,et al.  Solving the Max-cut Problem Using Eigenvalues , 1995, Discret. Appl. Math..

[14]  John E. Beasley,et al.  OR-Library: Distributing Test Problems by Electronic Mail , 1990 .

[15]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[16]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[17]  Frauke Liers Contributions to Determining Exact Ground-States of Ising Spin-Glasses and to their Physics , 2004 .

[18]  C. Helmberg,et al.  Solving quadratic (0,1)-problems by semidefinite programs and cutting planes , 1998 .

[19]  F. Glover,et al.  Adaptive Memory Tabu Search for Binary Quadratic Programs , 1998 .

[20]  Franz Rendl,et al.  Nonpolyhedral Relaxations of Graph-Bisection Problems , 1995, SIAM J. Optim..

[21]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning , 1989, Oper. Res..

[22]  Charles Delorme,et al.  Laplacian eigenvalues and the maximum cut problem , 1993, Math. Program..

[23]  M. Kojima,et al.  Second order cone programming relaxation of nonconvex quadratic optimization problems , 2001 .

[24]  Giovanni Rinaldi,et al.  New approaches for optimizing over the semimetric polytope , 2005, Math. Program..

[25]  Sahar Karimi,et al.  Max-cut Problem , 2007 .

[26]  P. Hansen Methods of Nonlinear 0-1 Programming , 1979 .

[27]  F. Rendl Semidefinite programming and combinatorial optimization , 1999 .

[28]  Angelika Wiegele Nonlinear optimization techniques applied to combinatorial optimization problems / Angelika Wiegele , 2006 .

[29]  John E. Beasley,et al.  Heuristic algorithms for the unconstrained binary quadratic programming problem , 1998 .

[30]  E. Spencer From the Library , 1936, British Journal of Ophthalmology.

[31]  Panos M. Pardalos,et al.  Computational aspects of a branch and bound algorithm for quadratic zero-one programming , 1990, Computing.

[32]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[33]  Franz Rendl,et al.  A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations , 2007, IPCO.

[34]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[35]  G. Rinaldi,et al.  Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm , 1995 .

[36]  Michael Jünger,et al.  Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS , 2000, Computational Combinatorial Optimization.

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

[38]  Michael Jünger,et al.  Experiments in quadratic 0–1 programming , 1989, Math. Program..

[39]  Francisco Barahona,et al.  Branch and Cut based on the volume algorithm: Steiner trees in graphs and Max-cut , 2006, RAIRO Oper. Res..

[40]  Franz Rendl,et al.  Semidefinite Programming and Graph Equipartition , 1998 .

[41]  Alain Billionnet,et al.  Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem , 2007, Math. Program..

[42]  Christoph Helmberg A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations , 2004, The Sharpest Cut.