A model for minimizing active processor time

Jessica Chang*, Harold N. Gabow, Samir Khuller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job J i has an integer length ℓ i as well as a set T i of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t so long as it is "active" at t. The goal is to schedule all the jobs in the fewest number of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is non-zero). In other words, subject to ℓ i units of each job i being scheduled in its feasible region and at each slot at most B jobs being scheduled, we are interested in minimizing the total time during which the machine is active. We present a linear time algorithm for the case where jobs are unit length and each T i is a single interval, assuming that jobs are given in sorted order. For general T i, we show that the problem is NP-complete even for B=3. However when B=2, we show that it can be efficiently solved. In addition, we consider a version of the problem where jobs have arbitrary lengths and can be preempted at any point in time. For general B, the problem can be solved by linear programming. For B=2, the problem amounts to finding a triangle-free 2-matching on a special graph. We extend the algorithm of Babenko et al. (Proceedings of the 16th Annual International Conference on Computing and Combinatorics, pp. 120-129, 2010) to handle our variant, and also to handle non-unit length jobs. This yields an O(√Lm) time algorithm to solve the preemptive scheduling problem for B=2, where L=∑ ii . We also show that for B=2 and unit length jobs, the optimal non-preemptive schedule has active time at most 4/3 times that of the optimal preemptive schedule; this bound extends to several versions of the problem when jobs have arbitrary length.

Original languageEnglish (US)
Pages (from-to)368-405
Number of pages38
Issue number3
StatePublished - Nov 2014


  • Active time
  • Algorithms
  • Matchings
  • Scheduling

ASJC Scopus subject areas

  • General Computer Science
  • Computer Science Applications
  • Applied Mathematics


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