A PTAS for Minimizing Weighted Flow Time on a Single Machine

An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. The currently best known polynomial time algorithm for the problem is a (2+")-approximation by Rohwedder and Wiese [STOC 2021], which builds on the prior break-through result by Batra, Garg, and Kumar [FOCS 2018] who found the first pseudo-polynomial time constant factor approximation algorithm for the problem, and on the result by Feige, Kulkarni, and Li [SODA 2019] who turned the latter into a polynomial time algorithm. However, it remains open whether the problem admits a PTAS. We answer this question in the affirmative and present a polynomial time (1+")-approximation algorithm for weighted flow time on a single machine. We use a reduction of the problem to a geometric covering problem, which was introduced in the mentioned (2+")-approximation algorithm and which loses only a factor of 1+"in the approximation ratio. However, unlike that algorithm, we solve the resulting instances of the covering problem exactly, rather than losing a factor 2+". Key for this is to identify and exploit structural properties of instances of that problem covering problem which arise in the reduction from weighted flow time.