A class of spectral bounds for Max k-Cut

Abstract In this paper we introduce a new class of bounds for the maximum k -cut problem on undirected edge-weighted simple graphs. The bounds involve eigenvalues of the weighted adjacency matrix together with geometrical parameters. They generalize previous results on the maximum (2-)cut problem and we demonstrate that they can strictly improve over other eigenvalue bounds from the literature. We also report computational results illustrating the potential impact of the new bounds.

[1]  Sanjeev Khanna,et al.  On the Hardness of Approximating Max k-Cut and its Dual , 1997, Chic. J. Theor. Comput. Sci..

[2]  D. V. Pasechnik,et al.  On approximate graph colouring and MAX-k-CUT algorithms based on the theta-function , 2002 .

[3]  B. Mohar,et al.  Eigenvalues and the max-cut problem , 1990 .

[4]  Walid Ben-Ameur,et al.  Spectral bounds for the maximum cut problem , 2008, Networks.

[5]  M. R. Rao,et al.  The partition problem , 1993, Math. Program..

[6]  B. Borchers.  CSDP, A C library for semidefinite programming , 1999 .

[7]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[8]  Ali Ridha Mahjoub,et al.  The Maximum Cut Problem , 2014 .

[9]  David P. Williamson,et al.  Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming , 2001, STOC '01.

[10]  Miguel F. Anjos,et al.  Solving k -way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method , 2013 .

[11]  Walid Ben-Ameur,et al.  A polynomial-time recursive algorithm for some unconstrained quadratic optimization problems , 2011, Discret. Appl. Math..

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

[13]  R. Sotirov,et al.  New bounds for the max-k-cut and chromatic number of a graph , 2015, 1503.06595.

[14]  P. Erdös.  On an extremal problem in graph theory , 1970 .

[15]  Andreas Eisenblätter,et al.  The Semidefinite Relaxation of the k -Partition Polytope Is Strong , 2002, IPCO.

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

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

[18]  Franz Rendl,et al.  Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations , 2009, Math. Program..

[19]  C. Moore,et al.  MAX k-CUT and approximating the chromatic number of random graphs , 2006 .

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

[21]  Alan M. Frieze,et al.  Improved approximation algorithms for MAXk-CUT and MAX BISECTION , 1995, Algorithmica.

[22]  Sébastien Le Digabel,et al.  Computational study of valid inequalities for the maximum k-cut problem , 2016, Ann. Oper. Res..

[23]  Walid Ben-Ameur,et al.  Spectral bounds for unconstrained (-1, 1)-quadratic optimization problems , 2010, Eur. J. Oper. Res..

[24]  V. Nikiforov.  Max k-cut and the smallest eigenvalue , 2016 .

[25]  Martin Grötschel,et al.  An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design , 1988, Oper. Res..

[26]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[27]  M. R. Rao,et al.  Facets of the K-partition Polytope , 1995, Discret. Appl. Math..