Abstract
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 \(K = 2\).
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This project was supported by São Paulo Research Foundation (FAPESP) grants #2015/11937-9, #2016/23552-7, #2017/21297-2, and #2020/13162-2, and National Council for Scientific and Technological Development (CNPq) grants #425340/2016-3, #312186/2020-7, and #311039/2020-0.
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Pedrosa, L.L.C., da Silva, M.R.C. & Schouery, R.C.S. Approximation Algorithms for the MAXSPACE Advertisement Problem. Theory Comput Syst (2024). https://doi.org/10.1007/s00224-024-10170-2
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DOI: https://doi.org/10.1007/s00224-024-10170-2