Complexity analysis and algorithms for the Program Download Problem

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.