The TV advertisements scheduling problem

Abstract

A TV channel has a single advertisement break of duration h and a convex continuous function $$f{:}\;[0,h] \rightarrow \mathbb {R}^+$$ representing the TV rating points within the advertisement break. Given n TV advertisements of different durations $$p_j$$ that sum up to h, and willingness to pay coefficients $$w_j$$ , the objective is to schedule them on the TV break in order to maximize the total revenue of the TV channel $$\sum _j w_j \int _{c_j-p_j}^{c_j} f(t) dt,$$ where $$[c_j-p_j,c_j)$$ is the broadcast time interval of TV advertisement j. We show that this problem is NP-hard and propose a fully polynomial time approximation scheme, using a special dominance property of an optimal schedule and the technique of K-approximation sets and functions introduced by Halman et al. (Math Oper Res 34:674–685, 2009).