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+є)-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.