Improved Approximation Algorithms for Minimum-Space Advertisement Scheduling

We study a scheduling problem involving the optimal placement of advertisement images in a shared space over time. The problem is a generalization of the classical scheduling problem P∥Cmax, and involves scheduling each job on a specified number of parallel machines (not necessarily simultaneously) with a goal of minimizing the makespan. In 1969 Graham showed that processing jobs in decreasing order of size, assigning each to the currently-least-loaded machine, yields a 4/3-approximation for P∥Cmax. Our main result is a proof that the natural generalization of Graham’s algorithm also yields a 4/3-approximationto the minimum-space advertisement scheduling problem. Previously, this algorithm was only known to give an approximation ratio of 2, and the best known approximation ratio for any algorithm for the minimum-space ad scheduling problem was 3/2. Our proof requires a number of new structural insights, which leads to a new lower bound for the problem and a non-trivial linear programming relaxation. We also provide a pseudo-polynomial approximation scheme for the problem (polynomial in the size of the problem and the number of machines).