Approximation Algorithms for Fair Repetitive Scheduling

TL;DR

This paper develops approximation algorithms with provable guarantees for a fair repetitive scheduling problem, addressing NP-hardness through LP-based and batching techniques for different processing time scenarios.

Contribution

It introduces new approximation algorithms with performance guarantees for fair repetitive scheduling, including LP-based schemes and batching methods for various processing time settings.

Findings

LP-based 2-approximation for day-dependent times

Polynomial-time approximation scheme for constant days

Simple approximation for day-invariant processing times

Abstract

We consider a recently introduced fair repetitive scheduling problem involving a set of clients, each asking for their associated job to be daily scheduled on a single machine across a finite planning horizon. The goal is to determine a job processing permutation for each day, aiming to minimize the maximum total completion time experienced by any client. This problem is known to be NP-hard for quite restrictive settings, with previous work offering exact solution methods for highly-structured special cases. In this paper, we focus on the design of approximation algorithms with provable performance guarantees. Our main contributions can be briefly summarized as follows: (i) When job processing times are day-dependent, we devise a polynomial-time LP-based $2$-approximation, as well as a polynomial-time approximation scheme for a constant number of days. (ii) With day-invariant processing times, we obtain a surprisingly simple $(\frac{1+\sqrt{2}}{2}+\epsilon)$-approximation in polynomial time. This setting is also shown to admit a quasi-polynomial-time approximation scheme for an arbitrary number of days. The key technical component driving our approximation schemes is a novel batching technique, where jobs are conceptually grouped into batches, subsequently leading either to a low-dimensional dynamic program or to a compact configuration LP. Concurrently, while developing our constant-factor approximations, we propose a host of lower-bounding mechanisms that may be of broader interest.