## Abstract

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=∑
_{i}
ℓ
_{i}
. 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 language | English (US) |
---|---|

Pages (from-to) | 368-405 |

Number of pages | 38 |

Journal | Algorithmica |

Volume | 70 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2014 |

## Keywords

- Active time
- Algorithms
- Matchings
- Scheduling

## ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics