# A weighted linear matroid parity algorithm

@article{Iwata2017AWL, title={A weighted linear matroid parity algorithm}, author={Satoru Iwata and Yusuke Kobayashi}, journal={Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing}, year={2017} }

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Lovász (1980) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. In this paper, we present a combinatorial, deterministic, strongly polynomial…

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## 23 Citations

Weighted Linear Matroid Parity

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This talk presents a recently developed polynomial-time algorithm for the weighted linear matroid parity problem and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann (1986) for the unweighted problem.

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A reduction of the weighted version of Mader's problem to weighted linear matroid parity, which leads to an O(n^5)-time algorithm for the former problem, where n denotes the number of vertices in the input graph.

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A rather simple, deterministic, and strongly polynomial-time algorithm based on Dijkstra's algorithm for the unconstrained shortest path problem and Edmonds' blossom shrinking technique in matching algorithms is presented, which clarifies a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem.

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It is conjecture that every 4-wise intersecting clutter is non-ideal, and the proof is proved in the binary case using Jaeger's 8-flow theorem for graphs and Seymour's characterization of the binary matroids with the sums of circuits property.

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A reduction of the weighted version of Mader's problem to weighted linear matroid parity, which leads to an O(n^5)-time algorithm for the former problem, where n denotes the number of vertices in the input graph.

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The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle...