A Polynomial-time Approximation Scheme for the MAXSPACE Advertisement Problem

In the MAXSPACE problem, given a set of ads A , one wants to place a subset A ′ ⊆ A into K slots B1, ..., BK of size L. Each ad A i ∈ A has a size si and a frequency wi. A schedule is feasible if the total size of ads in any slot is at most L, and each ad A i ∈ A ′ appears in exactly wi slots. The goal is to find a feasible schedule which maximizes the sum of the space occupied by all slots. We introduce a generalization, called MAXSPACE-RD, in which each ad Ai also has a release date ri ≥ 1 and a deadline di ≤ K, and may only appear in a slot Bj with ri ≤ j ≤ di. These parameters model situations where a subset of ads corresponds to a commercial campaign with an announcement date that may expire after some defined period. We present a polynomial-time approximation scheme for MAXSPACE-RD when K is bounded by a constant, i.e., for any e > 0, we give a polynomial-time algorithm which returns a solution with value at least (1−e)Opt, where Opt is the optimal value. This is the best factor one can expect, since MAXSPACE is NP-hard, even if K = 2.