Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs

We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a nonbipartite graph. Using standard techniques, we finally extend the result to capacitated b-matchings.

[1]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[2]  W. T. Tutte.  A Short Proof of the Factor Theorem for Finite Graphs , 1954, Canadian Journal of Mathematics.

[3]  Anja Fischer,et al.  Complete description for the spanning tree problem with one linearised quadratic term , 2013, Oper. Res. Lett..

[4]  J. Keilwagen,et al.  Exact algorithms and heuristics for the Quadratic Traveling Salesman Problem with an application in bioinformatics , 2014, Discret. Appl. Math..

[5]  Richard M. Karp,et al.  On Linear Characterizations of Combinatorial Optimization Problems , 1982, SIAM J. Comput..

[6]  S. Thomas McCormick,et al.  Matroid optimisation problems with nested non-linear monomials in the objective function , 2018, Math. Program..

[7]  Christoph Buchheim,et al.  The spanning tree problem with one quadratic term , 2013, CTW.

[8]  Volker Kaibel.  Polyhedral Combinatorics of QAPs with Less Objects than Locations , 1998 .

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

[10]  Alexander Schrijver.  Short proofs on the matching polyhedron , 1983, J. Comb. Theory, Ser. B.

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

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

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

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

[15]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[16]  Abdel Nasser,et al.  A Survey of the Quadratic Assignment Problem , 2014 .

[17]  Francesco Maffioli,et al.  On Minimum Reload Cost Cycle Cover , 2014, Discret. Appl. Math..

[18]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

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

[20]  Frauke Liers,et al.  An exact solution method for quadratic matching: The one-quadratic-term technique and generalisations , 2015, Discret. Optim..

[21]  A. Fischer.  A Polyhedral Study of Quadratic Traveling Salesman Problems , 2013 .

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