Matroids and Multicommodity Flows

The max-flow min-cut theorem and the two-commodity flow theorem may both be interpreted as equalities between the maximum feasible packing of certain circuits of a graph and the minimum capacity of certain cocircuits, and thus may both be expressed in matroid terms. We study the matroids in which a similar “ k -commodity flow theorem” holds. (Thus for k = 1 and 2 it holds for polygon matroids of graphs.) We find for example that such a theorem holds for bond matroids of graphs for all k ; and that, for any matroid, if it holds for k = 4, then it holds for all k . We obtain excluded minor characterizations for every k ⩾ 2, and also study the k = 1 case which is still unsolved.

[1]  P. Seymour.  On Odd Cuts and Plane Multicommodity Flows , 1981 .

[2]  Alfred Lehman,et al.  On the width-length inequality , 1979, Math. Program..

[3]  B. Rothschild,et al.  On Two Commodity Network Flows , 1966, Oper. Res..

[4]  Paul D. Seymour,et al.  Decomposition of regular matroids , 1980, J. Comb. Theory, Ser. B.

[5]  T. C. Hu.  Multi-Commodity Network Flows , 1963 .

[6]  P. D. Seymour,et al.  On Multi‐Colourings of Cubic Graphs, and Conjectures of Fulkerson and Tutte , 1979 .

[7]  B. Rothschild,et al.  MULTICOMMODITY NETWORK FLOWS. , 1969 .

[8]  Robert E. Bixby,et al.  ℓ-matrices and a Characterization of Binary Matroids , 1974, Discret. Math..

[9]  Jack Edmonds,et al.  Matching, Euler tours and the Chinese postman , 1973, Math. Program..

[10]  K. Wagner.  Über eine Eigenschaft der ebenen Komplexe , 1937 .

[11]  Paul D. Seymour,et al.  The matroids with the max-flow min-cut property , 1977, J. Comb. Theory B.

[12]  Paul D. Seymour,et al.  A two-commodity cut theorem , 1978, Discret. Math..

[13]  W. T. Tutte.  A homotopy theorem for matroids. II , 1958 .

[14]  W. T. Tutte.  Matroids and graphs , 1959 .

[15]  Paul D. Seymour,et al.  A note on the production of matroid minors , 1977, J. Comb. Theory, Ser. B.

[16]  G. Dirac.  A Property of 4-Chromatic Graphs and some Remarks on Critical Graphs , 1952 .

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

[18]  T. Gallai.  Über reguläre Kettengruppen , 1959 .

[19]  L. M. Kelly,et al.  A characterization of sets of n points which determine n hyperplanes , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.