On a weighted linear matroid intersection algorithm by Deg-Det computation

Furue, Hiroki; Hirai, Hiroshi

doi:10.1007/s13160-020-00413-3

Cited by 4 publications

(2 citation statements)

References 16 publications

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“…Our polynomial-time algorithm is obtained by truncating such low-degree terms. Note that in the case of the weighted linear matroid intersection, i.e., each A k is rank-1, such a care is not needed; see Furue & Hirai (2020); Hirai (2019) for details. First, we present the cost-scaling Deg-Det algorithm in the form that it updates A k instead of P, Q as follows:…”

Section: Truncation Of Low-degree Termsmentioning

confidence: 99%

A cost-scaling algorithm for computing the degree of determinants

Hirai

Ikeda

2022

comput. complex.

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In this paper, we address computation of the degree $$\deg {\rm Det} A$$ deg Det A of Dieudonné determinant $${\rm Det} A$$ Det A of $$\begin{aligned} A = \sum_{k=1}^m A_k x_k t^{c_k}, \end{aligned}$$ A = ∑ k = 1 m A k x k t c k , where $$A_k$$ A k are $$n \times n$$ n × n matrices over a field $$\mathbb{K}$$ K , $$x_k$$ x k are noncommutative variables, t is a variable commuting with $$x_k$$ x k , $$c_k$$ c k are integers, and the degree is considered for t. This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that $$\deg {\rm Det} A$$ deg Det A is obtained by a discrete convex optimization on a Euclidean building (Hirai 2019). We extend this framework by incorporating a cost-scaling technique and show that $$\deg {\rm Det} A$$ deg Det A can be computed in time polynomial of $$n,m,\log_2 C$$ n , m , log 2 C , where $$C:= \max_k |c_k|$$ C : = max k | c k | . We give a polyhedral interpretation of $$\deg {\rm Det}$$ deg Det , which says that $$\deg {\rm Det}$$ deg Det A is given by linear optimization over an integral polytope with respect to objective vector $$c = (c_k)$$ c = ( c k ) . Based on it, we show that our algorithm becomes a strongly polynomial one. We also apply our result to an algebraic combinatorial optimization problem arising from a symbolic matrix having $$2 \times 2$$ 2 × 2 -submatrix structure.

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Section: Truncation Of Low-degree Termsmentioning

confidence: 99%

A cost-scaling algorithm for computing the degree of determinants

Hirai

Ikeda

2022

comput. complex.

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“…Our polynomial time algorithm is obtained by truncating such low degree terms. Note that in the case of the weighted linear matroid intersection, i.e., each A k is rank-1, such a care is not needed; see [6,12] for details.…”

Section: Truncation Of Low-degree Termsmentioning

confidence: 99%

A cost-scaling algorithm for computing the degree of determinants

Hirai¹,

Ikeda²

2020

Preprint

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In this paper, we address computation of the degree deg Det A of Dieudonné determinant Det A ofwhere A k are n × n matrices over a field K, x k are noncommutative variables, t is a variable commuting x k , c k are integers, and the degree is considered for t. This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that deg Det A is obtained by a discrete convex optimization on a Euclidean building. We extend this framework by incorporating a cost scaling technique, and show that deg Det A can be computed in time polynomial of n, m, log 2 C, where C := max k |c k |. We apply this result to an algebraic combinatorial optimization problem arising from a symbolic matrix having 2 × 2-submatrix structure.

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Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

Hirai,

Iwamasa,

Oki

et al. 2024

Math. Program.

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We address the computation of the degrees of minors of a noncommutative symbolic matrix of form $$ A[c] :=\sum _{k=1}^m A_k t^{c_k} x_k, $$ A [ c ] : = ∑ k = 1 m A k t c k x k , where $$A_k$$ A k are matrices over a field $$\mathbb {K}$$ K , $$x_k$$ x k are noncommutative variables, $$c_k$$ c k are integer weights, and t is a commuting variable specifying the degree. This problem extends noncommutative Edmonds’ problem (Ivanyos et al. in Comput Complex 26:717–763, 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of A[c] of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and for membership of rank-2 Brascamp–Lieb polytopes.

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On a weighted linear matroid intersection algorithm by Deg-Det computation

Cited by 4 publications

References 16 publications

A cost-scaling algorithm for computing the degree of determinants

A cost-scaling algorithm for computing the degree of determinants

A cost-scaling algorithm for computing the degree of determinants

Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

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