Combinations of Some Shop Scheduling Problems and the Shortest Path Problem: Complexity and Approximation Algorithms

TL;DR

This paper investigates complex combinatorial problems combining shop scheduling and shortest path, establishing their NP-hardness and approximation limits, and proposing several algorithms for practical solutions.

Contribution

It introduces the first complexity and approximation results for combined shop scheduling and shortest path problems, along with new algorithms.

Findings

Problems are NP-hard even with two machines.

Cannot be approximated within a factor less than 2 unless P=NP.

Several approximation algorithms are proposed.

Abstract

We consider several combinatorial optimization problems which combine the classic shop scheduling problems, namely open shop scheduling or job shop scheduling, and the shortest path problem. The objective of the obtained problem is to select a subset of jobs that forms a feasible solution of the shortest path problem, and to execute the selected jobs on the open (or job) shop machines to minimize the makespan. We show that these problems are NP-hard even if the number of machines is two, and cannot be approximated within a factor less than 2 if the number of machines is an input unless P=NP. We present several approximation algorithms for these combination problems.