Two-machine routing open shop on a tree: instance reduction and efficiently solvable subclass

TL;DR

This paper introduces an instance reduction method for the two-machine routing open shop problem on trees, transforming complex instances into simpler, polynomially solvable subclasses, and lays groundwork for efficient approximation algorithms.

Contribution

The paper presents a novel instance reduction procedure that simplifies the problem on trees to a chain with limited jobs, enabling polynomial-time solutions for certain cases.

Findings

Reduction preserves the lower bound on the optimal makespan.

In three of four outcomes, the initial instance can be solved optimally in linear time.

The approach facilitates the development of efficient approximation algorithms.

Abstract

We consider two-machine routing open shop problem on a tree. In this problem a transportation network with a tree-like structure is given, and each node contains some jobs to be processed by two mobile machines. Machines are initially located in the predefined node called the depot, have to traverse the network to perform their operations on each job (and each job has to be processed by both machines in arbitrary order), and machines have to return to the depot after performing all the operations. The goal is to construct a feasible schedule for machines to process all the jobs and to return to the depot in shortest time possible. This problem is known to be NP-hard even in the case when the transportation network consists of just two nodes. We propose an instance reduction procedure which allows to transform any instance of the problem to a simplified instance on a chain with limited number of jobs. The reduction considered preserves the standard lower bound on the optimum. We describe four possible outcomes of this procedure and prove that in three of them the initial instance can be solved to the optimum in linear time, thus introducing a wide polynomially solvable subclass of the problem considered. Our research can be used as a foundation to construct efficient approximation algorithms for the two-machine routing open shop on a tree.