# Preemptive and non-preemptive Generalized Min Sum Set Cover

## Abstract

In the (non-preemptive) Generalized Min Sum Set Cover Problem, we are given n ground elements and a collection of sets S = {S1, S 2,⋯, Sm} where each set Si ε 2 |n| has a positive requirement k(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 k(Si) elements in Si. 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. © Sungjin Im, Maxim Sviridenko and Ruben van der Zwaan.