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 is an element of 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 [1] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [2, 15].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 [2, 15]. 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.