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

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 rj , weight wj and size pj , 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.