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.

Example

Let G = (U,V,E) be a bipartite graph. One may define a matroid M_U on the ground set E, in which a set of edges is independent if no two of the edges have the same endpoint in U. Similarly one may define a matroid M_V in which a set of edges is independent if no two of the edges have the same endpoint in V. Any set of edges that is independent in both M_U and M_V has the property that no two of its edges share an endpoint; that is, it is a matching. Thus, the largest common independent set of M_U and M_V is a maximum matching in G.

Extension

The matroid intersection problem becomes NP-hard when three matroids are involved, instead of only two.

References

• Brezovec, Carl; Cornuéjols, Gérard; Glover, Fred (1986), "Two algorithms for weighted matroid intersection", Mathematical Programming 36 (1): 39–53, doi:10.1007/BF02591988 .
• Aigner, Martin; Dowling, Thomas (1971), "Matching theory for combinatorial geometries", Transactions of the American Mathematical Society 158 (1): 231–245, doi:10.1090/S0002-9947-1971-0286689-5 .
• Edmonds, Jack (1979), "Matroid intersection", Discrete Optimization I, Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium, Annals of Discrete Mathematics, 4, pp. 39–49, doi:10.1016/S0167-5060(08)70817-3 .
• Frank, András (1981), "A weighted matroid intersection algorithm", Journal of Algorithms 2 (4): 328–336, doi:10.1016/0196-6774(81)90032-8 .
• Frederickson, Greg N.; Srinivas, Mandayam A. (1989), "Algorithms and data structures for an expanded family of matroid intersection problems", SIAM Journal on Computing 18 (1): 112–138, doi:10.1137/0218008 .
• Gabow, Harold N.; Tarjan, Robert E. (1984), "Efficient algorithms for a family of matroid intersection problems", Journal of Algorithms 5 (1): 80–131, doi:10.1016/0196-6774(84)90042-7 .
• Lawler, Eugene L. (1975), "Matroid intersection algorithms", Mathematical Programming 9 (1): 31–56, doi:10.1007/BF01681329 .