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 \(K = 2\).

在 MAXSPACE 中，给定一组广告\(\mathcal {A}\)，希望将子集\({\mathcal {A}'\subseteq \mathcal {A}}\)安排到K 个槽位\({B_1 , \dots , B_K}\)大小为L。每个广告 \({A_i \in \mathcal {A}}\)都有一个尺寸 \(s_i\)和一个频率 \(w_i\)。如果任何广告位中的广告总大小最多为L，并且每个广告\({A_i \in \mathcal {A}'}\)恰好出现在\(w_i\)个广告位中，并且每个广告位最多出现一次，则时间表是可行的投币口。目标是找到一个可行的调度，使所有槽占用的空间总和最大化。我们考虑一种称为 MAXSPACE-R 的概括，其中广告 \(A_i\)也有发布日期 \(r_i\) ，并且仅在以下情况下才出现在广告位 \(B_j\)中：\({j \ge r_i}\)。对于这个变体，我们给出了 1/9 近似算法。此外，我们考虑 MAXSPACE-RDV，其中广告 \(A_i\)也有截止日期 \(d_i\)（并且只能出现在 具有\(r_i \le j \le d_i\ ) 的广告位 \(B_j\ ) 中） ），以及值 \(v_i\) ，它是每个分配的\(A_i\)副本的增益 （可以与 \(s_i\)无关）。当K受常数限制时，我们提出了该问题的多项式时间近似方案。这是人们可以预期的最好因素，因为 MAXSPACE 是强 NP 困难的，即使\(K = 2\)。