research-article

Algebraic Algorithms for Linear Matroid Parity Problems

Authors:

Kai Man LeungAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 10, Issue 3

Article No.: 10, Pages 1 - 26

https://doi.org/10.1145/2601066

Published: 01 May 2014 Publication History

Abstract

We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mr^ω-1), where m and r are the number of columns and the number of rows, respectively, and ω ≈ 2.3727 is the matrix multiplication exponent. This improves the O(mr^ω)-time algorithm by Gabow and Stallmann and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr²), which does not need fast matrix multiplication.

We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint S-path problem, we present an O(n^ω)-time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n⁴)-time randomized algorithm where n is the number of vertices, and an O(n³)-time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω (n⁴) and Ω (n³), respectively.

The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint S-path are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.

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Iwata SYokoi Y(2023)Finding Maximum Edge-Disjoint Paths Between Multiple TerminalsSIAM Journal on Computing10.1137/22M149480452:5(1230-1268)Online publication date: 17-Oct-2023
https://doi.org/10.1137/22M1494804
Matoya KOki T(2022)Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity ProblemsSIAM Journal on Discrete Mathematics10.1137/21M142175136:3(2121-2158)Online publication date: 6-Sep-2022
https://doi.org/10.1137/21M1421751
Hirai HIkeda M(2022)A cost-scaling algorithm for computing the degree of determinantscomputational complexity10.1007/s00037-022-00227-431:2Online publication date: 23-Jul-2022
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Index Terms

Algebraic Algorithms for Linear Matroid Parity Problems
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 10, Issue 3

June 2014

176 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2620785

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Copyright © 2014 ACM.

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Publication History

Published: 01 May 2014

Accepted: 01 December 2013

Revised: 01 July 2013

Received: 01 March 2011

Published in TALG Volume 10, Issue 3

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Cited By

Iwata SYokoi Y(2023)Finding Maximum Edge-Disjoint Paths Between Multiple TerminalsSIAM Journal on Computing10.1137/22M149480452:5(1230-1268)Online publication date: 17-Oct-2023
https://doi.org/10.1137/22M1494804
Matoya KOki T(2022)Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity ProblemsSIAM Journal on Discrete Mathematics10.1137/21M142175136:3(2121-2158)Online publication date: 6-Sep-2022
https://doi.org/10.1137/21M1421751
Hirai HIkeda M(2022)A cost-scaling algorithm for computing the degree of determinantscomputational complexity10.1007/s00037-022-00227-431:2Online publication date: 23-Jul-2022
https://doi.org/10.1007/s00037-022-00227-4
Iwata SKobayashi Y(2021)A Weighted Linear Matroid Parity AlgorithmSIAM Journal on Computing10.1137/17M114170951:2(STOC17-238-STOC17-280)Online publication date: 7-Jan-2021
https://doi.org/10.1137/17M1141709
Kim EKwon O(2021)A Polynomial Kernel for Distance-Hereditary Vertex DeletionAlgorithmica10.1007/s00453-021-00820-z83:7(2096-2141)Online publication date: 1-Jul-2021
https://dl.acm.org/doi/10.1007/s00453-021-00820-z
Matoya KOki T(2021)Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity ProblemsInteger Programming and Combinatorial Optimization10.1007/978-3-030-73879-2_16(223-237)Online publication date: 5-May-2021
https://doi.org/10.1007/978-3-030-73879-2_16
Ebrahimi JStraszak DVishnoi N(2020)Subdeterminant Maximization via Nonconvex Relaxations and Anti-ConcentrationSIAM Journal on Computing10.1137/19M130952349:6(1249-1270)Online publication date: 14-Dec-2020
https://doi.org/10.1137/19M1309523
Chakrabarty DTat Lee YSidford ASingla SChiu-wai Wong S(2019)Faster Matroid Intersection2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2019.00072(1146-1168)Online publication date: Nov-2019
https://doi.org/10.1109/FOCS.2019.00072
Huang CKakimura NKamiyama N(2019)Exact and approximation algorithms for weighted matroid intersectionMathematical Programming: Series A and B10.1007/s10107-018-1260-x177:1-2(85-112)Online publication date: 1-Sep-2019
https://dl.acm.org/doi/10.1007/s10107-018-1260-x
Iwata SKobayashi YHatami HMcKenzie PKing V(2017)A weighted linear matroid parity algorithmProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055436(264-276)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055436
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