Weighted flow time does not admit O(1)-competitive algorithms

ABSTRACT

We consider the classic online scheduling problem of minimizing the total weighted flow time on a single machine with preemptions. Here, each job j has an arbitrary arrival time r_j, weight w_j and size p_j, and given a schedule its flow time is defined as the duration of time since its arrival until it completes its service requirement. The first non-trivial algorithms with poly-logarithmic competitive ratio for this problem were obtained relatively recently, and it was widely believed that the problem admits a constant factor competitive algorithm. In this paper, we show an ω(1) lower bound on the competitive ratio of any deterministic online algorithm. Our result is based on a gap amplification technique for online algorithms. Starting with a trivial lower bound of 1, we give a procedure to improve the lower bound sequentially, while ensuring at each step that the size of the instance increases relatively modestly.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Sequencing and scheduling

Additional Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES
      F.1.2 Modes of Computation
          Subjects: Online computation

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms