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Matroid parity problem

An instance of the matroid parity problem on a graphic matroid: given a graph with colored edges, having exactly two edges per color, find as large a forest as possible that again has exactly two edges per color.

In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid.[1] The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection.[1][2] It is also known as polymatroid matching, or the matchoid problem.[3]

Matroid parity can be solved in polynomial time for linear matroids. However, it is NP-hard for certain compactly-represented matroids, and requires more than a polynomial number of steps in the matroid oracle model.[1][4]

Applications of matroid parity algorithms include finding large planar subgraphs[5] and finding graph embeddings of maximum genus.[6] Matroid parity algorithms can also be used to find connected dominating sets and feedback vertex sets in graphs of maximum degree three.[7]

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