Performance bounds of algorithms for scheduling advertisements on a web page

Consider a set of n advertisements (hereafter called "ads") A = {A(1),...,A(n)} competing to be placed in a planning horizon which is divided into N time intervals called slots. An ad A(i) is specified by its size si and frequency w(i). The size s(i) represents the amount of space the ad occupies in a slot. Ad Ai is said to be scheduled if exactly w(i) copies of A(i) are placed in the slots subject to the restriction that a slot contains at most one copy of an ad. In this paper, we consider two problems. The MINSPACE problem minimizes the maximum fullness among all slots in a feasible schedule where the fullness of a slot is the sum of the sizes of ads assigned to the slot. For the MAXSPACE problem, in addition, we are given a common maximum fullness S for all slots. The total size of the ads placed in a slot cannot exceed S. The objective is to find a feasible schedule A(1) subset of or equal to A of ads such that the total occupied slot space SigmaA(iis an element ofA), w(i)s(i) is maximized. We examine the complexity status of both problems and provide heuristics with performance guarantees.