Matroid intersection

In combinatorial optimization, the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are assigned real weights, the weighted matroid intersection problem is to find a common independent set with the maximum possible weight. These problems generalize many problems in combinatorial optimization including finding maximum matchings and maximum weight matchings in bipartite graphs and finding arborescences in directed graphs.

The matroid intersection theorem, due to Jack Edmonds,^[1] says that there is always a simple upper bound certificate, consisting of a partitioning of the ground set amongst the two matroids, whose value (sum of respective ranks) equals the size of a maximum common independent set. Based on this theorem, the matroid intersection problem for two matroids can be solved in polynomial time using matroid partitioning algorithms.

^ Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", in R. Guy; H. Hanam; N. Sauer; J. Schonheim (eds.), Combinatorial structures and their applications (Proc. 1969 Calgary Conference), Gordon and Breach, New York, pp. 69–87. Reprinted in M. Jünger et al. (Eds.): Combinatorial Optimization (Edmonds Festschrift), LNCS 2570, pp. 1126, Springer-Verlag, 2003.

[:0-1] Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", in R. Guy; H. Hanam; N. Sauer; J. Schonheim (eds.), Combinatorial structures and their applications (Proc. 1969 Calgary Conference), Gordon and Breach, New York, pp. 69–87. Reprinted in M. Jünger et al. (Eds.): Combinatorial Optimization (Edmonds Festschrift), LNCS 2570, pp. 1126, Springer-Verlag, 2003.

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