Preemptive and non-preemptive generalized min sum set cover

In the (non-preemptive) Generalized Min Sum Set Cover Problem, we are given n ground elements and a collection of sets S = {S_1, S_2, ..., S_m} where each set S_i in 2^{[n]} has a positive requirement k(S_i) that has to be fulfilled. We would like to order all elements to minimize the total (weighted) cover time of all sets. The cover time of a set S_i is defined as the first index j in the ordering such that the first j elements in the ordering contain k(S_i) elements in S_i. This problem was introduced by [Azar, Gamzu and Yin, 2009] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We study the version where preemption is allowed. The difference is that elements can be fractionally scheduled and a set S is covered in the moment when k(S) amount of elements in S are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming and analysis are completely different from [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We also show that any preemptive solution can be transformed into a non-preemptive one by losing a factor of 6.2 in the objective function. As a byproduct, we obtain an improved 12.4-approximation for the non-preemptive problem. Keywords: Set Cover, Approximation, Preemption, Latency, Average cover time