Effective formulation reductions for the quadratic assignment problem

In this paper we study two formulation reductions for the quadratic assignment problem (QAP). In particular we apply these reductions to the well known Adams and Johnson [2] integer linear programming formulation of the QAP. We analyze two cases: In the first case, we study the effect of constraint reduction. In the second case, we study the effect of variable reduction in the case of a sparse cost matrix. Computational experiments with a set of 30 QAPLIB instances, which range from 12 to 32 locations, are presented. The proposed reductions turned out to be very effective.

[1]  T. Mautor,et al.  A parallel depth first search branch and bound algorithm for the quadratic assignment problem , 1995 .

[2]  Jeff T. Linderoth,et al.  Solving large quadratic assignment problems on computational grids , 2002, Math. Program..

[3]  Franz Rendl,et al.  A New Lower Bound Via Projection for the Quadratic Assignment Problem , 1992, Math. Oper. Res..

[4]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[5]  T. Koopmans,et al.  Assignment Problems and the Location of Economic Activities , 1957 .

[6]  Alfonsas Misevicius,et al.  A Modified Simulated Annealing Algorithm for the Quadratic Assignment Problem , 2003, Informatica.

[7]  Michael J. Brusco,et al.  Using Quadratic Assignment Methods to Generate Initial Permutations for Least-Squares Unidimensional Scaling of Symmetric Proximity Matrices , 2000, J. Classif..

[8]  Ashish Tiwari,et al.  A greedy genetic algorithm for the quadratic assignment problem , 2000, Comput. Oper. Res..

[9]  Jens Clausen,et al.  Solving Large Quadratic Assignment Problems in Parallel , 1997, Comput. Optim. Appl..

[10]  E. Lawler.  The Quadratic Assignment Problem , 1963 .

[11]  Ekrem Duman,et al.  The quadratic assignment problem in the context of the printed circuit board assembly process , 2007, Comput. Oper. Res..

[12]  Peter Hahn,et al.  A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method , 1998, Eur. J. Oper. Res..

[13]  Panos M. Pardalos,et al.  The Quadratic Assignment Problem: A Survey and Recent Developments , 1993, Quadratic Assignment and Related Problems.

[14]  Manfred Padberg,et al.  Location, Scheduling, Design and Integer Programming , 2011, J. Oper. Res. Soc..

[15]  Nair Maria Maia de Abreu,et al.  A survey for the quadratic assignment problem , 2007, Eur. J. Oper. Res..

[16]  David Connolly.  An improved annealing scheme for the QAP , 1990 .

[17]  Teodor Gabriel Crainic,et al.  Parallel Branch‐and‐Bound Algorithms , 2006 .

[18]  V. Kaibel.  Polyhedral Methods for the QAP , 2000 .

[19]  Clayton W. Commander,et al.  A Survey of the Quadratic Assignment Problem, with Applications , 2003 .

[20]  Monique Guignard-Spielberg,et al.  A level-2 reformulation-linearization technique bound for the quadratic assignment problem , 2007, Eur. J. Oper. Res..

[21]  Warren P. Adams,et al.  Improved Linear Programming-based Lower Bounds for the Quadratic Assignment Proglem , 1993, Quadratic Assignment and Related Problems.

[22]  Zvi Drezner,et al.  Extensive experiments with hybrid genetic algorithms for the solution of the quadratic assignment problem , 2008, Comput. Oper. Res..

[23]  Franz Rendl,et al.  Semidefinite Programming Relaxations for the Quadratic Assignment Problem , 1998, J. Comb. Optim..

[24]  P. Gilmore.  Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem , 1962 .

[25]  M Dorigo,et al.  Ant colonies for the quadratic assignment problem , 1999, J. Oper. Res. Soc..

[26]  P. Pardalos,et al.  State of the art in global optimization: computational methods and applications , 1996 .

[27]  Maurício Resende.  A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming , 1996 .

[28]  Arthur M. Geoffrion,et al.  Scheduling Parallel Production Lines with Changeover Costs: Practical Application of a Quadratic Assignment/LP Approach , 1976, Oper. Res..

[29]  Michael Jünger,et al.  On the SQAP-Polytope , 2000, SIAM J. Optim..

[30]  Michael Jünger,et al.  Box-inequalities for quadratic assignment polytopes , 2001, Math. Program..

[31]  Nicos Christofides,et al.  An Exact Algorithm for the Quadratic Assignment Problem on a Tree , 1989, Oper. Res..

[32]  Barbaros Ç. Tansel,et al.  A branch-and-cut algorithm for quadratic assignment problems based on linearizations , 2007, Comput. Oper. Res..

[33]  Panos M. Pardalos,et al.  Lower bounds for the quadratic assignment problem , 1994, Ann. Oper. Res..

[34]  L. Kaufman,et al.  An algorithm for the quadratic assignment problem using Bender's decomposition , 1978 .

[35]  Jens Clausen,et al.  Joining forces in solving large-scale quadratic assignment problems in parallel , 1997, Proceedings 11th International Parallel Processing Symposium.

[36]  V. Deineko,et al.  The Quadratic Assignment Problem: Theory and Algorithms , 1998 .

[37]  Yong Xia.  Improved Gilmore-Lawler Bound for Quadratic Assignment Problems , 2007 .

[38]  Panos M. Pardalos,et al.  A Greedy Randomized Adaptive Search Procedure for the Quadratic Assignment Problem , 1993, Quadratic Assignment and Related Problems.

[39]  Nathan W. Brixius,et al.  Solving quadratic assignment problems using convex quadratic programming relaxations , 2001 .

[40]  Franz Rendl,et al.  QAPLIB – A Quadratic Assignment Problem Library , 1997, J. Glob. Optim..

[41]  Michael Jünger,et al.  The QAP-polytope and the star transformation , 2001, Discret. Appl. Math..

[42]  Zvi Drezner,et al.  Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming , 1995, Oper. Res..

[43]  Kurt M. Anstreicher,et al.  Recent advances in the solution of quadratic assignment problems , 2003, Math. Program..