$\newcommand{\cala}{\mathcal{A}}$ In MAXSPACE, given a set of ads $\cala$, one wants to schedule a subset ${\cala'\subseteq\cala}$ into $K$ slots ${B_1, \dots, B_K}$ of size $L$. Each ad ${A_i \in \cala}$ 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 ${A_i \in \cala'}$ 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 $K = 2$.

翻译：$\newcommand{\cala}{\mathcal{A}}$ 在MAXSPACE问题中，给定一组广告$\cala$，需将子集${\cala'\subseteq\cala}$调度到大小为$L$的$K$个时隙${B_1, \dots, B_K}$中。每个广告${A_i \in \cala}$具有大小$s_i$和频率$w_i$。若任意时隙中广告的总大小不超过$L$，且每个广告${A_i \in \cala'}$恰好出现在$w_i$个时隙中（每个时隙最多出现一次），则调度可行。目标是找到最大化所有时隙占用空间总和的可行调度。我们考虑称为MAXSPACE-R的泛化变体，其中广告$A_i$还具有释放时间$r_i$，且仅当${j \ge r_i}$时才能出现在时隙$B_j$中。针对该变体，我们给出了一个$1/9$近似算法。进一步地，考虑MAXSPACE-RDV变体，其中广告$A_i$具有截止时间$d_i$（仅能出现在满足$r_i \le j \le d_i$的时隙$B_j$中），以及价值$v_i$（表示$A_i$每次分配的收益，可独立于$s_i$）。当$K$被常数界定时，我们为该问题提出了一个多项式时间近似方案。由于MAXSPACE在$K = 2$时即为强NP难问题，因此这是可达到的最佳近似因子。