Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time

TL;DR

This paper presents the first pseudo-polynomial time constant approximation algorithm for the weighted flow-time scheduling problem on a single machine, using a novel reduction to a structured Demand Multi-Cut problem and dynamic programming techniques.

Contribution

It introduces a new approximation algorithm for weighted flow-time on a single machine with pseudo-polynomial running time, leveraging a reduction to Demand Multi-Cut and structured dynamic programming.

Findings

First pseudo-polynomial time constant approximation for weighted flow-time.

Reduction to Demand Multi-Cut problem with exploitable structure.

Dynamic programming approach with smoothness properties.

Abstract

In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement p_j, release date r_j and weight w_j. The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The running time of our algorithm is polynomial in n, the number of jobs, and P, which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multi-cut problem on trees, which we call the Demand Multi-Cut problem. Even though we do not give a constant factor approximation algorithm for the Demand Multi-Cut problem on trees, we show that the specific instances of Demand Multi-Cut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of DP table.