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Composition Orderings for Linear Functions and Matrix Multiplication Orderings

Authors Susumu Kubo , Kazuhisa Makino, Souta Sakamoto

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Author Details

Susumu Kubo
  • General Education Center, Tottori University of Environmental Studies, Japan
Kazuhisa Makino
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Souta Sakamoto
  • Acompany Co., Ltd., Nagoya, Japan

Funding

This work was partially supported by the joint project of Kyoto University and Toyota Motor Corporation, titled "Advanced Mathematical Science for Mobility Society."
  • Kubo, Susumu: Partially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Leading Initiative for Excellent Young Researchers Grant Number JPMXS0320200347.
  • Makino, Kazuhisa: Partially supported by JSPS KAKENHI Grant Numbers JP20H05967 and JP19K22841.

Acknowledgements

The authors thank Kristóf Bérczi (Eötvös Lor{á}nd Univ.), Yasushi Kawase (Univ. of Tokyo), and Takeshi Tokuyama (Kwansei Gakuin Univ.) for their helpful comments on the first version of this paper.

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Susumu Kubo, Kazuhisa Makino, and Souta Sakamoto. Composition Orderings for Linear Functions and Matrix Multiplication Orderings. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.44

Abstract

We first consider composition orderings for linear functions of one variable. Given n linear functions f_1,… ,f_n: ℝ → ℝ and a constant c ∈ ℝ, the objective is to find a permutation σ:[n] → [n] that minimizes/maximizes f_σ(n)∘⋯∘f_σ(1)(c), where [n] = {1, … , n}. It was first studied in the area of time-dependent scheduling, and known to be solvable in O(n log n) time if all functions are nondecreasing. In this paper, we present a complete characterization of optimal composition orderings for this case, by regarding linear functions as two-dimensional vectors. We also show the equivalence between local and global optimality in optimal composition orderings. Furthermore, by using the characterization above, we provide a fixed-parameter tractable (FPT) algorithm for the composition ordering problem with general linear functions, with respect to the number of decreasing linear functions. 
We next deal with matrix multiplication as a generalization of composition of linear functions. Given n matrices M₁,… , M_n ∈ ℝ^{m×m} and two vectors w,y ∈ ℝ^m, where m is a positive integer, the objective is to find a permutation σ:[n] → [n] that minimizes/maximizes w^⊤ M_σ(n) ⋯ M_σ(1) y. The matrix multiplication ordering problem has been studied in the context of max-plus algebra, but despite being a natural problem, it has not been explored in the conventional algebra to date. By extending the results for composition orderings for linear functions, we show that the matrix multiplication ordering problem with 2× 2 matrices is solvable in O(n log n) time if all the matrices are simultaneously triangularizable and have nonnegative determinants, and FPT with respect to the number of matrices with negative determinants, if all the matrices are simultaneously triangularizable. As the negative side, we prove that three possible natural generalizations are NP-hard. In addition, we derive the existing result for the minimum matrix multiplication ordering problem with 2 × 2 upper triangular matrices in max-plus algebra, which is an extension of the well-known Johnson’s rule for the two-machine flow shop scheduling, as a corollary of our result in the conventional algebra.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
Keywords
  • function composition
  • matrix multiplication
  • ordering problem
  • scheduling

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