On the NP-hardness of scheduling with time restrictions

TL;DR

This paper proves that the single-processor scheduling problem with a fixed time restriction parameter B=2 is NP-hard, and introduces a Polynomial-Time Approximation Scheme (PTAS) for any fixed B ≥ 2, advancing understanding of its computational complexity.

Contribution

It establishes NP-hardness for the problem when B=2 and provides a PTAS for all fixed B ≥ 2, resolving an open question in the field.

Findings

NP-hardness proven for B=2

A PTAS exists for any fixed B ≥ 2

Advances understanding of scheduling with time restrictions

Abstract

In a recent paper, Braun, Chung and Graham [1] have addressed a single-processor scheduling problem with time restrictions. Given a fixed integer $B \geq 2$, there is a set of jobs to be processed by a single processor subject to the following B-constraint. For any real $x$, no unit time interval $[x, x+1)$ is allowed to intersect more than $B$ jobs. The problem has been shown to be NP-hard when $B$ is part of the input and left as an open question whether it remains NP-hard or not if $B$ is fixed [1, 5, 7]. This paper contributes to answering this question that we prove the problem is NP-hard even when $B=2$. A PTAS is also presented for any constant $B \geq 2$.