Preemptive and Non-Preemptive Generalized Min Sum Set Cover

Sungjin Im , Maxim Sviridenko , Ruben van Der Zwaan

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.

Mots clés: Set Cover; Approximation; Preemption; Latency; Average cover time

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Sungjin Im, Maxim Sviridenko, Ruben van Der Zwaan. Preemptive and Non-Preemptive Generalized Min Sum Set Cover. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.465-476. ⟨hal-00678169⟩