Matching, matroids, and extensions

Abstract.Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones—jump systems and path-matchings.

[1]  R. Rado.  A THEOREM ON INDEPENDENCE RELATIONS , 1942 .

[2]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[3]  R. Rado.  Note on Independence Functions , 1957 .

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

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

[6]  N. S. Barnett,et al.  Private communication , 1969 .

[7]  D. J. A. Welsh,et al.  A greedy algorithm for solving a certain class of linear programmes , 1973, Math. Program..

[8]  M. Koren.  Extreme degree sequences of simple graphs , 1973 .

[9]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[10]  W. Cunningham,et al.  A primal algorithm for optimum matching , 1978 .

[11]  László Lovász,et al.  Matroid matching and some applications , 1980, J. Comb. Theory, Ser. B.

[12]  Gérard Cornuéjols,et al.  A matching problem with side conditions , 1980, Discret. Math..

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

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

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

[16]  L. Lovász.  Matching Theory (North-Holland mathematics studies) , 1986 .

[17]  Timothy F. Havel,et al.  Some Combinatorial Properties of Discriminants in Metric Vector Spaces , 1986 .

[18]  André Bouchet,et al.  Greedy algorithm and symmetric matroids , 1987, Math. Program..

[19]  David G. Kirkpatrick,et al.  On Restricted Two-Factors , 1988, SIAM J. Discret. Math..

[20]  William H. Cunningham,et al.  Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra , 1995, SIAM J. Discret. Math..

[21]  R. Graham,et al.  Handbook of Combinatorics , 1995 .

[22]  András Frank,et al.  Minimal Edge-Coverings of Pairs of Sets , 1995, J. Comb. Theory B.

[23]  Kazuo Murota,et al.  Valuated Matroid Intersection II: Algorithms , 1996, SIAM J. Discret. Math..

[24]  The optimal path-matching problem , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[25]  W. Cunningham,et al.  Matroid optimization and algorithms , 1996 .

[26]  Satoru Fujishige,et al.  A Min-Max Theorem for Bisubmodular Polyhedra , 1997, SIAM J. Discret. Math..

[27]  László Lovász.  The Membership Problem in Jump Systems , 1997, J. Comb. Theory, Ser. B.

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

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

[30]  David Hartvigsen,et al.  The Square-Free 2-Factor Problem in Bipartite Graphs , 1999, IPCO.

[31]  James F. Geelen,et al.  An Algebraic Matching Algorithm , 2000, Comb..

[32]  C_4-Free 2-Factors in Bipartite Graphs , 2001 .

[33]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[34]  András Frank,et al.  Note on the path-matching formula , 2002, J. Graph Theory.

[35]  András Frank.  Restricted t-matchings in bipartite graphs , 2003, Discret. Appl. Math..