A Polynomial Time Approximation Scheme for a Single Machine Scheduling Problem Using a Hybrid Evolutionary Algorithm

TL;DR

This paper presents a mathematically rigorous polynomial time approximation scheme for the NP-hard single machine scheduling problem using a hybrid evolutionary algorithm, bridging heuristic success with theoretical analysis.

Contribution

It introduces a parameterized family of evolutionary algorithms that provide a polynomial time approximation scheme for a classic NP-hard scheduling problem.

Findings

Provides a rigorous polynomial time approximation scheme.

Bridges heuristic methods with theoretical guarantees.

Potential applicability to other NP-hard problems.

Abstract

Nowadays hybrid evolutionary algorithms, i.e, heuristic search algorithms combining several mutation operators some of which are meant to implement stochastically a well known technique designed for the specific problem in question while some others playing the role of random search, have become rather popular for tackling various NP-hard optimization problems. While empirical studies demonstrate that hybrid evolutionary algorithms are frequently successful at finding solutions having fitness sufficiently close to the optimal, many fewer articles address the computational complexity in a mathematically rigorous fashion. This paper is devoted to a mathematically motivated design and analysis of a parameterized family of evolutionary algorithms which provides a polynomial time approximation scheme for one of the well-known NP-hard combinatorial optimization problems, namely the "single machine scheduling problem without precedence constraints". The authors hope that the techniques and ideas developed in this article may be applied in many other situations.