Complexity analysis and algorithms for the Program Download Problem

Chao Peng, Jie Zhou, Binhai Zhu, Hong Zhu

Abstract: In this paper, we consider the Program Download Problem (PDP) which is to download a set of desired programs from multiple channels. When the problem is to decide whether the download can be done by a given deadline \(d\) and each program appears in each of the \(n\) channels at most once, denoted as \(\textit{PDP}(n,1,d)\), we prove that \(\textit{PDP}(n,1,d)\) is NP-complete by a reduction from 3-SAT(3). We can extend the NP-hardness proof to \(\textit{PDP}(2,3,d)\) where there are only two channels but each program could appear in each channel at most 3 times, although \(\textit{PDP}(2,1,d)\) and \(\textit{PDP}(2,2,d)\) are both in P. We show that the aligned version of the problem (APDP) is polynomially solvable by reducing it to a maximum flow problem. For a different version of the problem, MPDP, where the objective is to maximize the number of program downloaded before a given deadline \(d\), we prove that it is fixed-parameter tractable. Finally, we devise an approximation algorithm for \(\textit{MPDP}(2,p,d),\,p\ge 3\), which aims to maximize the number of desired programs downloaded in two channels.

External IDs:dblp:journals/jco/PengZZZ15