Randomized heuristics for the Max-Cut problem

Given an undirected graph with edge weights, the MAX-CUT problem consists in finding a partition of the nodes into two subsets, such that the sum of the weights of the edges having endpoints in different subsets is maximized. It is a well-known NP-hard problem with applications in several fields, including VLSI design and statistical physics. In this article, a greedy randomized adaptive search procedure (GRASP), a variable neighborhood search (VNS), and a path-relinking (PR) intensification heuristic for MAX-CUT are proposed and tested. New hybrid heuristics that combine GRASP, VNS, and PR are also proposed and tested. Computational results indicate that these randomized heuristics find near-optimal solutions. On a set of standard test problems, new best known solutions were produced for many of the instances.

[1]  Mauricio G. C. Resende,et al.  Grasp: An Annotated Bibliography , 2002 .

[2]  Franz Rendl,et al.  A Spectral Bundle Method for Semidefinite Programming , 1999, SIAM J. Optim..

[3]  Panos M. Pardalos,et al.  GRASP With Path Relinking For The Three-Index Assignment Problem , 2000 .

[4]  Celso C. Ribeiro,et al.  A Hybrid GRASP with Perturbations for the Steiner Problem in Graphs , 2002, INFORMS J. Comput..

[5]  Renata M. Aiex,et al.  Parallel GRASP with path-relinking for job shop scheduling , 2003, Parallel Comput..

[6]  Yoji Kajitani,et al.  A graph- theoretic via minimization algorithm for two layer printed circuit boards , 1983 .

[7]  S. Benson,et al.  Mixed linear and semidefinite programming for combinatorial and quadratic optimization , 1999 .

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

[9]  P. Hansen,et al.  Developments of Variable Neighborhood Search , 2002 .

[10]  Franz Rendl,et al.  Solving Graph Bisection Problems with Semidefinite Programming , 2000, INFORMS J. Comput..

[11]  Marcus Peinado,et al.  Design and Performance of Parallel and Distributed Approximation Algorithms for Maxcut , 1997, J. Parallel Distributed Comput..

[12]  R. Pinter.  Optimal layer assignment for interconnect , 1984 .

[13]  Fred Glover,et al.  Improved Constructive Multistart Strategies for the Quadratic Assignment Problem Using Adaptive Memory , 1999, INFORMS J. Comput..

[14]  R. Monteiro,et al.  A projected gradient algorithm for solving the maxcut SDP relaxation , 2001 .

[15]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[16]  Celso C. Ribeiro,et al.  A Parallel Grasp for the Steiner Tree Problem in Graphs Using a Hybrid Local Search Strategy , 2000, J. Glob. Optim..

[17]  Masakazu Kojima,et al.  Numerical Evaluation of SDPA (Semidefinite Programming Algorithm) , 2000 .

[18]  Fred Glover,et al.  Tabu Search and Adaptive Memory Programming — Advances, Applications and Challenges , 1997 .

[19]  Celso C. Ribeiro,et al.  Variable neighborhood search for the degree-constrained minimum spanning tree problem , 2002, Discret. Appl. Math..

[20]  Rafael Martí,et al.  GRASP and Path Relinking for 2-Layer Straight Line Crossing Minimization , 1999, INFORMS J. Comput..

[21]  Linus Schrage,et al.  A More Portable Fortran Random Number Generator , 1979, TOMS.

[22]  Pablo Moscato,et al.  Handbook of Applied Optimization , 2000 .

[23]  Celso C. Ribeiro,et al.  Probability Distribution of Solution Time in GRASP: An Experimental Investigation , 2002, J. Heuristics.

[24]  Masakazu Kojima,et al.  Exploiting sparsity in primal-dual interior-point methods for semidefinite programming , 1997, Math. Program..

[25]  José Luis González Velarde,et al.  Computing tools for modeling, optimization and simulation : interfaces in computer science and operations research , 2000 .

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

[27]  M. Resende,et al.  A GRASP WITH PATH-RELINKING FOR PERMANENT VIRTUAL CIRCUIT ROUTING , 2001 .

[28]  David Hung-Chang Du,et al.  Efficient Algorithms for Layer Assignment Problem , 1987, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[29]  Pierre Hansen,et al.  Variable Neighborhood Search , 2018, Handbook of Heuristics.

[30]  Franz Rendl,et al.  A recipe for semidefinite relaxation for (0,1)-quadratic programming , 1995, J. Glob. Optim..

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

[32]  Mauricio G. C. Resende,et al.  Greedy Randomized Adaptive Search Procedures , 1995, J. Glob. Optim..

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

[34]  M. Resende,et al.  A probabilistic heuristic for a computationally difficult set covering problem , 1989 .

[35]  Celso C. Ribeiro,et al.  Local search with perturbations for the prize‐collecting Steiner tree problem in graphs , 2001, Networks.

[36]  F. Glover,et al.  Fundamentals of Scatter Search and Path Relinking , 2000 .

[37]  Fred Glover,et al.  Multi-Start and Strategic Oscillation Methods - Principles to Exploit Adaptive Memory , 2000 .