In the MAXSPACE problem, given a set of ads A, one wants to schedule a subset A' of A into K slots B_1, ..., B_K of size L. Each ad A_i in 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 A' appears in exactly w_i slots. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We introduce a generalization called MAXSPACE-R in which each ad A_i also has a release date r_i >= 1, and may only appear in a slot B_j with j >= r_i. We also introduce a generalization of MAXSPACE-R called MAXSPACE-RD in which each ad A_i also has a deadline d_i <= K, and may only appear in a slot B_j with r_i <= j <= d_i. 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 1/9-approximation algorithm for MAXSPACE-R and a polynomial-time approximation scheme for MAXSPACE-RD when K is bounded by a constant. This is the best factor one can expect, since MAXSPACE is NP-hard, even if K = 2.

中文翻译：

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MAXSPACE 广告问题的近似算法

在 MAXSPACE 问题中，给定一组广告 A，我们希望将 A 的子集 A' 安排到 K 个大小为 L 的槽 B_1, ..., B_K 中。 A 中的每个广告 A_i 都有一个大小 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 A' 中的每个广告 A_i 恰好出现在 w_i 个广告位中，则时间表是可行的。目标是找到一个可行的时间表，使所有插槽占用的空间总和最大化。我们引入了一个称为 MAXSPACE-R 的泛化，其中每个广告 A_i 也有一个发布日期 r_i >= 1，并且可能只出现在 j >= r_i 的插槽 B_j 中。我们还介绍了 MAXSPACE-R 的泛化，称为 MAXSPACE-RD，其中每个广告 A_i 也有一个截止日期 d_i <= K，并且可能只出现在 r_i <= j <= d_i 的插槽 B_j 中。这些参数模拟广告子集对应于具有公告日期的商业活动的情况，该公告日期可能在某个定义的时间段之后到期。我们提出了 MAXSPACE-R 的 1/9 近似算法和 MAXSPACE-RD 的多项式时间近似方案，当 K 受常数限制时。这是可以预期的最佳因素，因为 MAXSPACE 是 NP 难的，即使 K = 2。