Let M=(V,r) be a matroid, and let G=(V,E) be a graph. The matroid matching problem is to find a matching [math]F \subseteq E[/math] of maximum cardinality for which V(F) is independent in M. If M is the free matroid, then we obtain the maximum matching problem. Other special cases include the minimum pinning set problem and Mader's disjoint S-paths problem.
It can be assumed w.l.o.g. that G is a collection of independent edges, since the structure of G can be encoded in the matroid. This version is usually called the matroid parity problem.
In general, the matroid matching problem is hard: it may require an exponential number of oracle calls . Lovász  gave a min-max formula and a polynomial algorithm for matroid matching in linearly represented matroids. More efficient deterministic algorithms have been found by Gabow and Stallmann  and Orlin and Vande Vate , while Cheung, Lau and Leung  developed fast randomized algorithms.
The Lovász formula and the Gabow-Stallmann algorithm were extended to the linear delta-matroid parity problem by Geelen, Iwata, and Murota . Additional results on matroid matching problems involving combinatorially defined matroids can be found in the thesis of Makai .
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- L. Lovász, Selecting independent lines from a family of lines in a space, Acta Sci. Math. 42 (1980), 121–131.
- L. Lovász, The matroid matching problem, in: Algebraic methods in graph theory (Szeged, 1978), Colloq. Math. Soc. János Bolyai 25 (1981), 495–517.
- L. Lovász, Matroid matching and some applications, DOI link
- H.N. Gabow, N. Stallmann, An augmenting path algorithm for linear matroid parity, DOI link
- J.B. Orlin, J.H. Vande Vate, Solving the linear matroid parity problem as a sequence of matroid intersection problems, DOI link
- H.Y. Cheung, L.C. Lau, K.M. Leung, Algebraic algorithms for linear matroid parity problems, DOI link, author link
- J.F. Geelen, S. Iwata, K. Murota, The linear delta-matroid parity problem, DOI link, author link
- J.F. Geelen, S. Iwata, Matroid matching via mixed skew-symmetric matrices, DOI link, author link
- M. Makai, Parity problems of combinatorial polymatroids, Ph.D. Thesis, ELTE, 2009, PDF link