Composition Orderings for Linear Functions and Matrix Multiplication Orderings

TL;DR

The paper develops a unified, angle-based framework for ordering compositions of linear functions and extends it to matrix multiplication orderings. It fully characterizes optimal orders for monotone linear functions, showing when local optimality implies global optimality and revealing a counterclockwise unimodal structure. It then extends to nondecreasing and general linear functions, delivering an FPT algorithm with respect to the number of decreasing functions and a dynamic-programming approach that achieves $O(2^k k n^6)$. For matrices, it proves efficient orderings for 2×2 cases under simultaneous triangularizability and nonnegative determinants, connects to max-plus algebra, and establishes NP-hardness for several natural generalizations, including higher dimensions and target-approximation versions. These results unify scheduling-inspired composition problems with algebraic and combinatorial techniques, and delineate the boundary between tractable and intractable instances.

Abstract

We consider composition orderings for linear functions of one variable. Given $n$ linear functions $f_1,\dots,f_n$ and a constant $c$, the objective is to find a permutation $σ$ that minimizes/maximizes $f_{σ(n)}\circ\dots\circ f_{σ(1)}(c)$. 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 several interesting properties on optimal composition orderings such as the equivalence between local and global optimality. Furthermore, by using the characterization above, we provide a fixed-parameter tractable (FPT) algorithm for the composition ordering problem for general linear functions, with respect to the number of decreasing linear functions. We next deal with matrix multiplication orderings as a generalization of composition of linear functions. Given $n$ matrices $M_1,\dots,M_n\in\mathbb{R}^{m\times m}$ and two vectors $w,y\in\mathbb{R}^m$, where $m$ denotes a positive integer, the objective is to find a permutation $σ$ that minimizes/maximizes $w^\top M_{σ(n)}\dots M_{σ(1)} y$. The problem is also viewed as a generalization of flow shop scheduling through a limit. By this extension, we show that the multiplication ordering problem for $2\times 2$ 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 finally prove that three possible natural generalizations are NP-hard: 1) when $m=2$, 2) when $m\geq 3$, and 3) the target version of the problem.

Figures (4)

• Figure 1: Vector representations of $f(x)=-x$ and $g(x)=2x-\sqrt{3}$.
• Figure 2: $Permutation \sigma_{l,m,r}$ obtained from $\sigma$ by swapping two adjacent intervals.
• Figure 3: The vector representation for $f_1, \ldots, f_5$.
• Figure 4: The vector representation for $f_1, ..., f_7$.

Theorems & Definitions (96)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8