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}
}
  • S. Iwata, Yusuke Kobayashi
  • Published 19 June 2017
  • Mathematics, Computer Science
  • Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
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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References

SHOWING 1-10 OF 58 REFERENCES
An augmenting path algorithm for linear matroid parity
TLDR
This paper presents an algorithm that uses timeO(mn3), wherem is the number of elements andn is the rank, which is based on the method of augmenting paths used in the algorithms for all subcases of the problem.
A Fast, Simpler Algorithm for the Matroid Parity Problem
  • J. Orlin
  • Mathematics, Computer Science
    IPCO
  • 2008
TLDR
This work shows how to solve the linear matroid parity problem as a sequence of matroid intersection problems, and shows the 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].
Matroid matching and some applications
Solving the linear matroid parity problem as a sequence of matroid intersection problems
TLDR
This paper presents an O(r4n) algorithm for the linear matroid parity problem, and identifies an important subclass of non-simple parity problems called ‘easy’ parity problems which can be solved as matroid intersection problems.
Matroid matching in pseudomodular lattices
TLDR
A further step of generalization is presented, showing that a good characterization can also be obtained for the class of socalled pseudomodular matroids, introduced by Björner and Lovász.
Shortest Disjoint S-Paths Via Weighted Linear Matroid Parity
TLDR
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.
Matroid Matching: The Power of Local Search
TLDR
A polynomial-time approximation scheme for unweighted matroid matching for general matroids is presented, and it is shown that natural linear-programming relaxations that have been studied have an integrality gap, and, moreover, $\Omega(n)$ rounds of the Sherali--Adams hierarchy are necessary to bring the gap down to a constant.
Algebraic Algorithms for Matching and Matroid Problems
TLDR
New algebraic approaches for two well-known combinatorial problems: nonbipartite matching and matroid intersection are presented and new randomized algorithms that exceed or match the efficiency of existing algorithms are yielded.
A Weighted Linear Matroid Parity Algorithm
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...
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2
3
4
5
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