Abstract
Consider a matrix with m rows and n pairs of columns. The linear matroid parity problem (LMPP) is to determine a maximum number of pairs of columns that are linearly independent. We show how to solve the linear matroid parity problem as a sequence of matroid intersection problems. The algorithm runs in \(\mathcal{O}(m^3n)\). Our algorithm is comparable to the best running time for the LMPP, and is far simpler and faster than the algorithm of Orlin and Vande Vate [10], who also solved the LMPP as a sequence of matroid intersection problems. In addition, the algorithm may be viewed naturally as an extension of the blossom algorithm for nonbipartite matchings.
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References
Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)
Edmonds, J.: Submodular functions, matroids, and certain polyhedral. In: Guy, R., Hanani, H., Sauer, N., Schónheim, J. (eds.) Combinatorial Structures and Their Applications (proceedings Calgary International Conference on Combinatorial Structures and Their Applications, Calgary, Alberta, 1969), pp. 69–87. Gordon and Breach, New York (1970)
Gabow, H.N., Stallmann, M.: An augmenting path algorithm for linear matroid parity. Combinatorica 6, 123–150 (1986)
Jensen, P.M., Korte, B.: Complexity of matroid property algorithms. SIAM Journal on Computing 11, 184–190 (1982)
Lawler, E.L.: Optimal matroid intersections. In: Guy, R., Hanani, H., Sauer, N., Schónheim, J. (eds.) Combinatorial Structures and Their Applications (proceedings Calgary International Conference on Combinatorial Structures and Their Applications, Calgary, Alberta, 1969), pp. 233–234. Gordon and Breach, New York (1970)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Hold, Rinehart and Winston, New York (1976)
Lovász, L.: Matroid matching and some applications. Journal of Combinatorial Theory Series B 28, 208–236 (1980)
Lovász, L.: The matroid matching problem. In: Lovász, L., Sós, V.T. (eds.) Algebraic Methods in Graph Theory, Vol. II (Colloquium Szeged, 1978). Colloquia, Mathematica Societatis János Bolyai, vol. 25, pp. 495–517. North-Holland, Amsterdam (1981)
Lovász, L., Plummer, M.D.: Matching Theory. Akadémiai Kiadó, Budapest (1986) (also North-Holland Mathematics Studies vol. 121. North-Holland, Amsterdam)
Orlin, J.B., Vande Vate, J.H.: Solving the matroid parity problem as a sequence of matroid intersection problems. Mathematical Programming 47, 81–106 (1990)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 2. Springer, New York (2003)
Stallmann, M., Gabow, H.N.: An augmenting path algorithm for the parity problem on linear matroids. In: 25th Annual Symposium on Foundations of Computer Science, New York, pp. 217–227. IEEE, Los Alamitos (1984)
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Orlin, J.B. (2008). A Fast, Simpler Algorithm for the Matroid Parity Problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2008. Lecture Notes in Computer Science, vol 5035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68891-4_17
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DOI: https://doi.org/10.1007/978-3-540-68891-4_17
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