A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.

[1]  Éva Tardos,et al.  Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[2]  M. O'Kelly,et al.  A quadratic integer program for the location of interacting hub facilities , 1987 .

[3]  Martin Skutella,et al.  Convex quadratic and semidefinite programming relaxations in scheduling , 2001, JACM.

[4]  Hiroo Saito.  The Symmetric Quadratic Semi-Assignment Polytope , 2006, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[5]  Hanif D. Sherali,et al.  A simultaneous lifting strategy for identifying new classes of facets for the Boolean quadric polytope , 1995, Oper. Res. Lett..

[6]  Martine Labbé,et al.  A branch and cut algorithm for hub location problems with single assignment , 2005, Math. Program..

[7]  Vangelis F. Magirou,et al.  An algorithm for the multiprocessor assignment problem , 1989 .

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

[9]  Manfred W. Padberg,et al.  The boolean quadric polytope: Some characteristics, facets and relatives , 1989, Math. Program..

[10]  Tomomi Matsui,et al.  Linear Relaxation for Hub Network Design Problems , 2001 .

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

[12]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[13]  Jinhyeon Sohn,et al.  The single allocation problem in the interacting three-hub network , 2000, Networks.

[14]  Morton E. O'Kelly,et al.  Hub‐and‐Spoke Networks in Air Transportation: An Analytical Review , 1999 .

[15]  Donald F. Towsley.  Allocating programs containing branches and loops within a multiple processor system , 1986, IEEE Transactions on Software Engineering.

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

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

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

[19]  Warren P. Adams,et al.  A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems , 1986 .

[20]  Arie M. C. A. Koster,et al.  The partial constraint satisfaction problem: Facets and lifting theorems , 1998, Oper. Res. Lett..

[21]  Federico Malucelli,et al.  A polynomially solvable class of quadratic semi-assignment problems , 1996 .

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

[23]  Martine Labbé,et al.  Projecting the flow variables for hub location problems , 2004, Networks.

[24]  Harold S. Stone,et al.  Multiprocessor Scheduling with the Aid of Network Flow Algorithms , 1977, IEEE Transactions on Software Engineering.

[25]  Giorgio Gallo,et al.  Lower bounds for the quadratic semi-assignment problem , 1986 .

[26]  Andreas T. Ernst,et al.  Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem , 1998 .

[27]  Federico Malucelli,et al.  Quadratic Semi-Assignment Problems on Structured Graphs , 1994 .

[28]  Horst W. Hamacher,et al.  Adapting polyhedral properties from facility to hub location problems , 2004, Discret. Appl. Math..

[29]  T. Fujie,et al.  The Quadratic Semi-Assignment Polytope , 2004 .

[30]  Michael W. Carter,et al.  The indefinite zero-one quadratic problem , 1984, Discret. Appl. Math..

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

[32]  Alain Billionnet,et al.  Best reduction of the quadratic semi-assignment problem , 2001, Discret. Appl. Math..