Scheduling Bidirectional Traffic on a Path

TL;DR

This paper investigates the complex problem of scheduling bidirectional traffic on a path, revealing its NP-hardness, providing approximation schemes, and developing polynomial algorithms for specific cases.

Contribution

It proves NP-hardness of the problem, offers a PTAS for single segments with non-identical jobs, and develops polynomial algorithms for restricted compatibility scenarios.

Findings

NP-hardness of bidirectional traffic scheduling on a path

Existence of a PTAS for a single segment with non-identical jobs

Polynomial algorithms for restricted compatibility cases

Abstract

We study the fundamental problem of scheduling bidirectional traffic along a path composed of multiple segments. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a segment, while jobs in opposing directions cannot cross a segment at the same time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by proving that it is NP-hard even for identical jobs. We complement this result with a PTAS for a single segment and non-identical jobs. If we allow some pairs of jobs traveling in different directions to cross a segment concurrently, the problem becomes APX-hard even on a single segment and with identical jobs. We give polynomial algorithms for the setting with restricted compatibilities between jobs on a single and any constant number of segments, respectively.