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 = {S1, S2, ..., Sm} where each set Si ∈ 2 has a positive requirement κ(Si) 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 Si is defined as the first index j in the ordering such that the first j elements in the ordering contain κ(Si) elements in Si. This problem was introduced in [1] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [2,16]. 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 κ(S) amount of elements in S are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming relaxation and analysis are completely different from [2,16]. 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. This work was partially supported by NSF grant CCF-1016684. Sungjin Im Department of Computer Science, University of Illinois, 201 N. Goodwin Ave., Urbana, IL 61801, USA. Tel.: +1-217-2446433 Fax: +1-217-2654035 E-mail: [email protected] Maxim Sviridenko Department of Computer Science, University of Warwick, Coventry CV4 7AL,UK. Tel.:+44-24-76573792 E-mail: [email protected] Ruben van der Zwaan Department of Quantitative Economics, Maastricht University, The Netherlands. E-mail: [email protected]