2026-07-26T01:27:46Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:6812 2024-03-06T10:38:56Z ddc:004 open_access

Optimal Composition Ordering Problems for Piecewise Linear Functions Kawase, Yasushi Makino, Kazuhisa Seimi, Kento function composition time-dependent scheduling 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Yasushi Kawase and Kazuhisa Makino and Kento Seimi 2016 Is Part Of LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2016.42 urn:nbn:de:0030-drops-68126 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.42 eng https://creativecommons.org/licenses/by/3.0/legalcode