Complexity analysis and algorithms for the Program Download Problem - Peng, Chao; Zhou, Jie; Zhu, Binhai; Zhu, Hong (Journal of combinatorial optimization, 2015)
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 PDP ( n , 1 , d ) , we prove that PDP ( n , 1 , d ) is NP-complete by a reduction from 3-SAT(3). We can extend the NP-hardness proof to PDP ( 2 , 3 , d ) where there are only two channels but each program could appear in each channel at most 3 times, although PDP ( 2 , 1 , d ) and 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 MPDP ( 2 , p , d ) , p ≥ 3 , which aims to maximize the number of desired programs downloaded in two channels.