Scheduling Two Agents on a Single Machine: A Parameterized Analysis of NP-hard Problems

TL;DR

This paper investigates the parameterized complexity of single-machine scheduling problems involving two agents, focusing on how the problem's difficulty changes with the number of jobs of the second agent, and identifies which variants are tractable or intractable.

Contribution

It bridges scheduling theory and parameterized complexity by analyzing the complexity of two-agent scheduling problems with respect to the size of one agent's job set.

Findings

Identifies tractable and intractable variants based on scheduling criteria and parameter k.

Provides a complexity classification for various two-agent scheduling problems.

Highlights the boundary between feasible and infeasible problem variants.

Abstract

Scheduling theory is an old and well-established area in combinatorial optimization, whereas the much younger area of parameterized complexity has only recently gained the attention of the community. Our aim is to bring these two areas closer together by studying the parameterized complexity of a class of single-machine two-agent scheduling problems. Our analysis focuses on the case where the number of jobs belonging to the second agent is considerably smaller than the number of jobs belonging to the first agent, and thus can be considered as a fixed parameter k. We study a variety of combinations of scheduling criteria for the two agents, and for each such combination we pinpoint its parameterized complexity with respect to the parameter k. The scheduling criteria that we analyze include the total weighted completion time, the total weighted number of tardy jobs, and the total weighted number of just-in-time jobs. Our analysis draws a borderline between tractable and intractable variants of these problems.