The Parameterized Complexity of Scheduling with Precedence Delays: Shuffle Product and Directed Bandwidth

TL;DR

This paper investigates the parameterized complexity of scheduling problems with precedence constraints, establishing XNLP-completeness results for Shuffle Product and Directed Bandwidth, and exploring their implications.

Contribution

It provides new XNLP-completeness results for classic scheduling problems, including the case of trees, and highlights Shuffle Product as a notable addition to XNLP-complete problems.

Findings

Both Shuffle Product and Directed Bandwidth are XNLP-complete.

Directed Bandwidth remains XNLP-complete on trees.

Shuffle Product is a promising new addition to XNLP-complete problems.

Abstract

In this paper, we study the parameterized complexity of several variants of scheduling with precedence constraints between jobs. Namely, we consider the single machine setting with delay values on top of the precedence constraints. Such scheduling problems are related to several decades-old problems with open parameterized complexity status, notably Shuffle Product and Directed Bandwidth. We obtain XNLP-completeness results for both problems, and derive implications to scheduling with minimum (resp. maximum) delays parameterized by the width of the directed acyclic graph giving the precedence constraints, and/or by the maximum delay value in the input. Regarding Directed Bandwidth, we also settle the case of trees by showing XNLP-completeness parameterized by the target value. Beyond these results, we believe that Shuffle Product is an unusual and promising addition to the list of XNLP-complete problems.