Approximation Algorithms for the MAXSPACE Advertisement Problem

In MAXSPACE, given a set of ads \(\mathcal {A}\), one wants to schedule a subset \({\mathcal {A}'\subseteq \mathcal {A}}\) into K slots \({B_1, \dots , B_K}\) of size L. Each ad \({A_i \in \mathcal {A}}\) has a size \(s_i\) and a frequency \(w_i\). A schedule is feasible if the total size of ads in any slot is at most L, and each ad \({A_i \in \mathcal {A}'}\) appears in exactly \(w_i\) slots and at most once per slot. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We consider a generalization called MAXSPACE-R for which an ad \(A_i\) also has a release date \(r_i\) and may only appear in a slot \(B_j\) if \({j \ge r_i}\). For this variant, we give a 1/9-approximation algorithm. Furthermore, we consider MAXSPACE-RDV for which an ad \(A_i\) also has a deadline \(d_i\) (and may only appear in a slot \(B_j\) with \(r_i \le j \le d_i\)), and a value \(v_i\) that is the gain of each assigned copy of \(A_i\) (which can be unrelated to \(s_i\)). We present a polynomial-time approximation scheme for this problem when K is bounded by a constant. This is the best factor one can expect since MAXSPACE is strongly NP-hard, even if \(K = 2\).