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(n1)} 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(k1)} 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 timedependent 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 twovalued freeorder secretary problem can be solved in polynomial time. We finally prove that there exists no constantfactor 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 = {42:142:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770262},
ISSN = {18688969},
year = {2016},
volume = {64},
editor = {SeokHee Hong},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6812},
URN = {urn:nbn:de:0030drops68126},
doi = {10.4230/LIPIcs.ISAAC.2016.42},
annote = {Keywords: function composition, timedependent scheduling}
}
Keywords: 

function composition, timedependent scheduling 
Collection: 

27th International Symposium on Algorithms and Computation (ISAAC 2016) 
Issue Date: 

2016 
Date of publication: 

07.12.2016 