On Laminar Matroids and b-Matchings

We prove that three matroid optimisation problems, namely, the matchoid, matroid parity and matroid matching problems, all reduce to the b-matching problem when the matroids concerned are laminar. We then use this equivalence to show that laminar matroid parity polytopes are anely congruent to

[1]  T. Magnanti,et al.  Network Flows , 2011 .

[2]  Jan Vondrák,et al.  Matroid matching: the power of local search , 2010, STOC '10.

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

[4]  Tadayoshi Kohno,et al.  A Network-Flow-Based Scheduler: Design, Performance History, and Experimental Analysis , 2001, JEAL.

[5]  J. V. Vate.  Fractional matroid matchings , 1992, J. Comb. Theory B.

[6]  Bernhard Korte,et al.  Complexity of Matroid Property Algorithms , 1982, SIAM J. Comput..

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

[8]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[9]  J. Edmonds.  Matroids and the greedy algorithm , 1971, Math. Program..

[10]  Nicole Fassbinder,et al.  Combinatorial Optimization Networks And Matroids , 2016 .

[11]  Ralph E. Gomory,et al.  Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem , 2010, 50 Years of Integer Programming.

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

[13]  Jack Edmonds,et al.  Matching: A Well-Solved Class of Integer Linear Programs , 2001, Combinatorial Optimization.

[14]  T. Magnanti,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[15]  James G. Oxley,et al.  Matroid theory , 1992 .

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

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

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

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

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