An exact solution method for quadratic matching: The one-quadratic-term technique and generalisations

The quadratic matching problem (QMP) asks for a matching in a graph that optimises a quadratic objective in the edge variables. The QMP generalises the quadratic assignment problem as the latter can be seen as a perfect matching on a bipartite graph. In our approach, we strengthen the linearised IP-formulation by cutting planes that are derived from facets of the corresponding matching problem where only one quadratic term is present in the objective function (QMP1). As the influence of these cutting planes on the root bound decreases for instances with more quadratic terms in the objective, we present different methods to strengthen these cutting planes. Adopting the idea of the reformulation linearisation technique (RLT) we derive valid inequalities for the general QMP from QMP1 facets. Following the approach in Fomeni et?al. (2015) we replace cubic terms that appear in a reformulated QMP1 inequality by a quadratic estimator. Based on these methods we design and implement an exact branch-and-cut approach. When compared to solving the standard linearised IP formulation strengthened by cutting planes that result from the application of the RLT to the degree inequalities, we show that root bounds and cpu times for solving instances to optimality can significantly be improved, when the new method is applied.

[1]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[2]  R. Weismantel,et al.  A Semidefinite Programming Approach to the Quadratic Knapsack Problem , 2000, J. Comb. Optim..

[3]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[4]  Fred W. Glover,et al.  Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs , 2004, Discret. Optim..

[5]  R. Fortet.  L’algebre de Boole et ses applications en recherche operationnelle , 1960 .

[6]  Alain Billionnet,et al.  Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: The QCR method , 2009, Discret. Appl. Math..

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

[8]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[9]  Christoph Buchheim,et al.  Combinatorial optimization with one quadratic term: Spanning trees and forests , 2014, Discret. Appl. Math..

[10]  V. Kaibel,et al.  Angewandte Mathematik Und Informatik Universit at Zu K Oln a Basic Study of the Qap-polytope , 1996 .

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

[12]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

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

[14]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

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

[16]  Alberto Caprara,et al.  Constrained 0-1 quadratic programming: Basic approaches and extensions , 2008, Eur. J. Oper. Res..

[17]  Michel X. Goemans,et al.  Minimizing submodular functions over families of sets , 1995, Comb..

[18]  A. M. Geoffrion.  Duality in Nonlinear Programming: A Simplified Applications-Oriented Development , 1971 .

[19]  Martin Grötschel,et al.  Corrigendum to our paper “the ellipsoid method and its consequences in combinatorial optimization” , 1984, Comb..

[20]  Vladimir Kolmogorov,et al.  Feature Correspondence Via Graph Matching: Models and Global Optimization , 2008, ECCV.

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

[22]  Christoph von der Malsburg,et al.  Pattern recognition by labeled graph matching , 1988, Neural Networks.

[23]  Christoph Helmberg,et al.  The symmetric quadratic traveling salesman problem , 2013, Math. Program..

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

[25]  Pierre Hansen,et al.  Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem , 2009, Discret. Appl. Math..

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

[27]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[28]  J. Edmonds,et al.  Facets of I-matching polyhedra , 1974 .

[29]  Laura Klein.  Combinatorial optimization with one quadratic term , 2014 .

[30]  A. Barvinok,et al.  Combinatorial complexity of orbits in representations of the symmetric group , 1992 .

[31]  Hanif D. Sherali,et al.  Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems , 1990, Oper. Res..

[32]  Fred W. Glover,et al.  Technical Note - Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program , 1974, Oper. Res..

[33]  Christoph Buchheim,et al.  On the Quadratic Shortest Path Problem , 2015, SEA.

[34]  Marcus Oswald,et al.  Local cuts revisited , 2008, Oper. Res. Lett..

[35]  Marshall L. Fisher,et al.  Optimal Solution of Scheduling Problems Using Lagrange Multipliers: Part I , 1973, Oper. Res..

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

[37]  Dirk Oliver Theis,et al.  Odd minimum cut sets and b-matchings revisited , 2006, SIAM J. Discret. Math..

[38]  R. Burkard.  Quadratic Assignment Problems , 1984 .

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

[40]  F. Glover.  IMPROVED LINEAR INTEGER PROGRAMMING FORMULATIONS OF NONLINEAR INTEGER PROBLEMS , 1975 .

[41]  Adam N. Letchford,et al.  Cutting planes for RLT relaxations of mixed 0–1 polynomial programs , 2015, Math. Program..

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

[43]  Marcus Oswald,et al.  Speeding up IP-based algorithms for constrained quadratic 0–1 optimization , 2010, Math. Program..