Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Matoya, Kazuki; Oki, Taihei

doi:10.1007/978-3-030-73879-2_16

Cited by 1 publication

(1 citation statement)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is already faster than the known algorithm [CLV01a] that takes O(m 4 n ω ) time 1 . We further develop a much faster algorithm using the sparse matrix representation, which is another matrix representation of the linear matroid parity problem introduced by Geelen and Iwata [GI05] that extends a sparse representation of the linear matroid intersection problem [Har09;Mur00]; see also [MO22]. The current fastest algorithm for linear matroid intersection [Har09] and parity [CLL14] both perform the search-to-decision reduction with the sparse representations and the divide-and-conquer strategy.…”

Section: Matrix Representation Of Fractional Linear Matroid Parity Vi...mentioning

confidence: 99%

Algebraic Algorithms for Fractional Linear Matroid Parity via Non-commutative Rank

Oki¹,

Soma²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix representations using the concept of non-commutative rank (nc-rank), which has recently attracted attention in the research of Edmonds' problem. We reveal that the nc-rank of the matrix representation of linear matroid parity corresponds to the optimal value of fractional linear matroid parity: a half-integral relaxation of linear matroid parity. Based on our representation, we present an algebraic algorithm for the fractional linear matroid parity problem by building a new technique to incorporate the search-to-decision reduction into the half-integral problem represented via the nc-rank. We further present a faster divide-and-conquer algorithm for finding a maximum fractional matroid matching and an algebraic algorithm for finding a dual optimal solution. They together lead to an algebraic algorithm for the weighted fractional linear matroid parity problem. Our algorithms are significantly simpler and faster than the existing algorithms.

show abstract

Section: Matrix Representation Of Fractional Linear Matroid Parity Vi...mentioning

confidence: 99%

Algebraic Algorithms for Fractional Linear Matroid Parity via Non-commutative Rank

Oki¹,

Soma²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Cited by 1 publication

References 30 publications

Algebraic Algorithms for Fractional Linear Matroid Parity via Non-commutative Rank

Algebraic Algorithms for Fractional Linear Matroid Parity via Non-commutative Rank

Contact Info

Product

Resources

About