A combinatorial algorithm for computing the rank of a generic partitioned matrix with $$2 \times 2$$ submatrices

Hirai, Hiroshi; Iwamasa, Yuni

doi:10.1007/s10107-021-01676-5

Cited by 7 publications

(13 citation statements)

References 14 publications

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“…In this section, we consider an algebraic combinatorial optimization problem for a 2 ×2partitioned matrix (1.3). As an application of the cost-scaling Deg-Det algorithm, we extend the combinatorial rank computation in [15] to the deg-det computation.…”

Section: Proof Of the Sensitivity Theoremmentioning

confidence: 99%

“…Hirai and Iwamasa [15] showed that the rank computation of a 2×2-partitioned matrix can be formulated as the cardinality maximization problem of certain algebraically constraint 2-matchings in a bipartite graph. Based on this formulation and partly inspired by the Wong sequence method [9,10], they gave a combinatorial augmenting-path type O(n 4 )-time algorithm to obtain a maximum matching and an optimal solution S, T in (R 2×2 ).…”

Section: Proof Of the Sensitivity Theoremmentioning

confidence: 99%

“…We see Lemma 4.4 below for an essence of the proof. In [15], a stronger notion of a (2-)matching is used, and it is shown that |M| = rank(A M ) is checked in O(n 2 )-time (by assigning a valid labeling (VL)). An A-consistent 2-matching is called maximum if it has the maximum cardinality over all A-consistent 2-matchings.…”

Section: Proposition 42 ([15]mentioning

confidence: 99%

“…Although rank A can be computed by the above-mentioned nc-rank algorithms, the problem has a more intriguing combinatorial nature. Hirai and Iwamasa [15] showed that rank A is equal to the maximum size of a certain algebraically constrained 2-matching (A-consistent 2-matching) on a bipartite graph, and they developed an augmenting-path type polynomial time algorithm to obtain a maximum A-consistent 2-matching. We apply our cost-scaling framework for a 2 × 2-partitioned matrix A with x ij replaced by [2] speeded up this algorithm, and also obtained a randomized FPTAS by using cost scaling.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Proof Of the Sensitivity Theoremmentioning

confidence: 99%

Section: Proof Of the Sensitivity Theoremmentioning

confidence: 99%

Section: Proposition 42 ([15]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A cost-scaling algorithm for computing the degree of determinants

Hirai¹,

Ikeda²

2020

Preprint

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“…This indicates a new direction in combinatorial optimization, since Edmonds' problem generalizes several important combinatorial optimization problems. Inspired by their algorithm, [27] developed a combinatorial polynomial time algorithm for a certain algebraically constraint 2-matching problem in a bipartite graph, which corresponds to the (commutative) Edmonds' problem for a linear symbolic matrix in [32]. Also, a noncommutative algebraic formulation that captures weighted versions of combinatorial optimization problems was studied in [25,26,40].…”

Section: Introductionmentioning

confidence: 99%

Computing the nc-rank via discrete convex optimization on CAT(0) spaces

Hamada

Hirai

2020

Preprint

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In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrixwhere each A i is an n × n matrix over a field K, and x i (i = 1, 2, . . . , m) are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for K = Q, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field K. We present a significantly different polynomial time algorithm that works on an arbitrary field K. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.

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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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A combinatorial algorithm for computing the rank of a generic partitioned matrix with $$2 \times 2$$ submatrices

Cited by 7 publications

References 14 publications

A cost-scaling algorithm for computing the degree of determinants

A cost-scaling algorithm for computing the degree of determinants

Computing the nc-rank via discrete convex optimization on CAT(0) spaces

A cost-scaling algorithm for computing the degree of determinants

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