# Scheduling Maintenance Jobs in Networks

## TL;DR

This paper studies the scheduling of maintenance jobs in networks to minimize outages, revealing polynomial solutions with preemption, NP-hardness without preemption, and approximation strategies for complex cases.

## Contribution

It provides a comprehensive complexity analysis of maintenance scheduling, including polynomial algorithms, NP-hardness proofs, and approximation bounds for different preemption scenarios.

## Key findings

- Polynomial-time solution with preemption in arbitrary networks.
- NP-hardness of non-preemptive scheduling.
- A simple 2-approximation algorithm for mixed preemption cases.

## Abstract

We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and in the non-preemptive case we give strong non-approximability results. Furthermore, we give tight bounds on the power of preemption, that is, the maximum ratio of the values of non-preemptive and preemptive optimal solutions. Interestingly, the preemptive and the non-preemptive problem can be solved efficiently on paths, whereas we show that mixing both leads to a weakly NP-hard problem that allows for a simple 2-approximation.