When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.42
URN: urn:nbn:de:0030-drops-68126
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6812/
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### Optimal Composition Ordering Problems for Piecewise Linear Functions

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### Abstract

In this paper, we introduce maximum composition ordering problems. The input is n real functions f_1 , ... , f_n : R to R and a constant c in R. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation sigma : [n] to [n] which maximizes f_{sigma(n)} circ f_{sigma(n-1)} circ ... circ f_{sigma(1)}(c), where [n] = {1, ... , n}. The maximum partial composition ordering problem is to compute a permutation sigma : [n] to [n] and a nonnegative integer k (0 le k le n) which maximize f_{sigma(k)} circ f_{sigma(k-1)} circ ... circ f_{sigma(1)}(c). We propose 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 partial composition ordering problem can be solved in polynomial time if f i is of the form max{a_i x + b_i , c_i } for some constants a_i (ge 0), b_i and c_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.

### BibTeX - Entry

```@InProceedings{kawase_et_al:LIPIcs:2016:6812,
author =	{Yasushi Kawase and Kazuhisa Makino and Kento Seimi},
title =	{{Optimal Composition Ordering Problems for Piecewise Linear Functions}},
booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages =	{42:1--42:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-026-2},
ISSN =	{1868-8969},
year =	{2016},
volume =	{64},
editor =	{Seok-Hee Hong},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},