New Bounds for Time-Dependent Scheduling with Uniform Deterioration

TL;DR

This paper introduces new approximation bounds for time-dependent scheduling with uniform deterioration, providing algorithms with guarantees for minimizing makespan and total completion time.

Contribution

It develops the first approximation algorithms with bounds depending on deterioration rate for uniform deterioration scheduling problems.

Findings

O(1+1/β)-approximation for makespan problem

O(1+1/β^2)-approximation for total completion time

Greedy algorithms effective for extreme deterioration rates

Abstract

Time-dependent scheduling with linear deterioration involves determining when to execute jobs whose processing times degrade as their beginning is delayed. Each job i is associated with a release time r_i and a processing time function p_i(s_i)=alpha_i + beta_i*s_i, where alpha_i, beta_i>0$ are constants and s_i is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. Here, we take a step forward by developing new bounds and approximation results for the interesting special case of the problems with uniform deterioration, i.e.\ beta_i=beta, for each i. The key contribution is a O(1+1/beta)-approximation algorithm for the makespan problem and a O(1+1/beta^2)-approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with beta=O(1/n) and beta=Omega(n), where n is the number of jobs. Our analysis is based on a new approach for comparing computed and optimal schedules via bounding pseudomatchings.