Optimal Composition Ordering Problems for Piecewise Linear Functions

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.