Fast Divide-and-Conquer Algorithms for Preemptive Scheduling Problems with Controllable Processing Times --- A Polymatroid Optimization Approach

  • Authors:

  • Natalia V. Shakhlevich;Akiyoshi Shioura;Vitaly A. Strusevich

  • Affiliations:

  • School of Computing, University of Leeds, Leeds, U.K. LS2 9JT;Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8579;Department of Mathematical Sciences, University of Greenwich, London, U.K. SE10 9LS

  • Venue:
  • ESA '08 Proceedings of the 16th annual European symposium on Algorithms
  • Year:
  • 2008

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Abstract

We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in $ \O({\rm T}_{\rm feas}(n) \times\log n)$ time by using our divide-and-conquer technique, where nis the number of jobs and Tfeas(n) denotes the time complexity of the corresponding feasible scheduling problem with njobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper.