License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6812/
Kawase, Yasushi ;
Makino, Kazuhisa ;
Seimi, Kento
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 , ... , 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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6812},
URN = {urn:nbn:de:0030-drops-68126},
doi = {10.4230/LIPIcs.ISAAC.2016.42},
annote = {Keywords: function composition, time-dependent scheduling}
}
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Keywords: |
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function composition, time-dependent scheduling |
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Collection: |
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27th International Symposium on Algorithms and Computation (ISAAC 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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07.12.2016 |
