A common approximation framework for early work, late work, and resource leveling problems

Abstract

We study the approximability of two related machine scheduling problems. In the late work minimization problem, there are identical parallel machines and the jobs have a common due date. The objective is to minimize the late work , defined as the sum of the portion of the jobs done after the due date. A related problem is the maximization of the early work , defined as the sum of the portion of the jobs done before the due date. We describe a polynomial time approximation scheme for the early work maximization problem, and we extended it to the late work minimization problem after shifting the objective function by a positive value that depends on the problem data. We also prove an inapproximability result for the latter problem if the objective function is shifted by a constant which does not depend on the input. These results remain valid even if the number of the jobs assigned to the same machine is bounded. This leads to an extension of our approximation scheme to some variants of the resource leveling problem, for which no approximation algorithms were known.

Improved approximation schemes for early work scheduling on identical parallel machines with common due date

“…As mentioned in (Györgyi & Kis, 2020), the jobs with processing time p j ≥ d will be scheduled on distinct machines, and can be deleted from the instance with the machines processing them.…”

“…When m is not fixed, Györgyi and Kis (2020) proposed a PTAS for a more general case. Chen et al (2016) studied the online early work scheduling problem on parallel machines, and proposed an optimal online algorithm for two identical machines.…”

“…In this paper, we propose an EPTAS with running time O(n) for the early work scheduling problem, which improves the result in (Györgyi & Kis, 2020), and a FPTAS with running time O(n) when the number of machines is a fixed number, which improves the result in (Chen et al, 2020b), where n is the number of jobs, and the hidden constant depends on the desired accuracy.…”

Online and semi-online scheduling on two hierarchical machines with a common due date to maximize the total early work

“…Then, for the more general case where the number of machine m is fixed (m ≥ 2), Chen et al [3] proposed an FPTAS based on a dynamic programming approach. Finally, for the cases with is an arbitrary number of machines, Györgyi and Kis [7] proposed a PTAS, while Li [9] proposed an EPTAS. The total early work maximizationn has been introduced into online scheduling by Chen et al [4], who considered the parallel machine model with a common due date.…”