# Optimal Composition Ordering Problems for Piecewise Linear Functions

## Abstract

In this paper, we introduce maximum composition ordering problems. The input is n real functions $$f_1,\dots ,f_n:\mathbb {R}\rightarrow \mathbb {R}$$ and a constant $$c\in \mathbb {R}$$. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation $$\sigma :[n]\rightarrow [n]$$ which maximizes $$f_{\sigma (n)}\circ f_{\sigma (n-1)}\circ \dots \circ f_{\sigma (1)}(c)$$, where $$[n]=\{1,\dots ,n\}$$. The maximum partial composition ordering problem is to compute a permutation $$\sigma :[n]\rightarrow [n]$$ and a nonnegative integer $$k~(0\le k\le n)$$ which maximize $$f_{\sigma (k)}\circ f_{\sigma (k-1)}\circ \dots \circ f_{\sigma (1)}(c)$$. We propose $$\mathrm {O}(n\log n)$$ time algorithms for the maximum total and partial composition ordering problems for monotone linear functions $$f_i$$, which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum total composition ordering problem can be solved in polynomial time if $$f_i$$ is of the form $$\max \{a_ix+b_i,d_i,x\}$$ for some constants $$a_i\,(\ge 0)$$, $$b_i$$ and $$d_i$$. As a corollary, we show that the two-valued free-order secretary problem can be solved in polynomial time. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if $$f_i$$’s are monotone, piecewise linear functions with at most two pieces, unless P $$=$$ NP.

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## Acknowledgements

The first author is supported by JSPS KAKENHI Grant Numbers 26887014 and JP16K16005. The second author is supported by supported by JSPS KAKENHI Grant Numbers JP24106002, JP25280004, JP26280001, and JST CREST Grant Number JPMJCR1402, Japan.

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Correspondence to Yasushi Kawase.

An extended abstract of this paper appears in ISAAC 2016 [16].

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Kawase, Y., Makino, K. & Seimi, K. Optimal Composition Ordering Problems for Piecewise Linear Functions. Algorithmica 80, 2134–2159 (2018). https://doi.org/10.1007/s00453-017-0397-y