Solving the linear matroid parity problem as a sequence of matroid intersection problems

In this paper, we present an O(r4n) algorithm for the linear matroid parity problem. Our solution technique is to introduce a modest generalization, the non-simple parity problem, and identify an important subclass of non-simple parity problems called ‘easy’ parity problems which can be solved as matroid intersection problems. We then show how to solve any linear matroid parity problem parametrically as a sequence of ‘easy’ parity problems.In contrast to other algorithmic work on this problem, we focus on general structural properties of dual solutions rather than on local primal structures. In a companion paper, we develop these ideas into a duality theory for the parity problem.

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