New Algorithms for Minimizing the Weighted Number of Tardy Jobs On a Single Machine

TL;DR

This paper investigates the complexity of minimizing the total weight of tardy jobs on a single machine, showing fixed parameter tractability when combining certain parameters and polynomial solvability under specific conditions.

Contribution

It introduces new fixed parameter tractability results and polynomial-time algorithms for special cases of the classical scheduling problem.

Findings

Problem is FPT when combining any two parameters

Polynomial-time solvable when processing times and weights are constant

Extends understanding of problem complexity beyond NP-hardness

Abstract

In this paper we study the classical single machine scheduling problem where the objective is to minimize the total weight of tardy jobs. Our analysis focuses on the case where one or more of three natural parameters is either constant or is taken as a parameter in the sense of parameterized complexity. These three parameters are the number of different due dates, processing times, and weights in our set of input jobs. We show that the problem belongs to the class of fixed parameter tractable (FPT)problems when combining any two of these three parameters. We also show that the problem is polynomial-time solvable when the latter two parameters are constant, complementing Karp's result who showed that the problem is NP-hard already for a single due date.