Time Complexity Analysis of Evolutionary Algorithms for 2-Hop (1,2)-Minimum Spanning Tree Problem

TL;DR

This paper analyzes the time complexity of various evolutionary algorithms applied to a constrained, NP-hard version of the Minimum Spanning Tree problem, providing bounds on their efficiency in finding approximate solutions.

Contribution

It offers the first theoretical analysis of evolutionary algorithms' performance on the 2-Hop (1,2)-Minimum Spanning Tree problem, including novel bounds for different representations.

Findings

Vertex-based EA has better expected time bounds than edge-based ones.

Expected time bounds depend on the representation and fitness functions used.

The study advances understanding of evolutionary algorithms for complex combinatorial problems.

Abstract

The Minimum Spanning Tree problem (abbr. MSTP) is a well-known combinatorial optimization problem that has been extensively studied by the researchers in the field of evolutionary computing to theoretically analyze the optimization performance of evolutionary algorithms. Within the paper, we consider a constrained version of the problem named 2-Hop (1,2)-Minimum Spanning Tree problem (abbr. 2H-(1,2)-MSTP) in the context of evolutionary algorithms, which has been shown to be NP-hard. Following how evolutionary algorithms are applied to solve the MSTP, we first consider the evolutionary algorithms with search points in edge-based representation adapted to the 2H-(1,2)-MSTP (including the (1+1) EA, Global Simple Evolutionary Multi-Objective Optimizer and its two variants). More specifically, we separately investigate the upper bounds on their expected time (i.e., the expected number of fitness evaluations) to obtain a $\frac{3}{2}$-approximate solution with respect to different fitness functions. Inspired by the special structure of 2-hop spanning trees, we also consider the (1+1) EA with search points in vertex-based representation that seems not so natural for the problem and give an upper bound on its expected time to obtain a $\frac{3}{2}$-approximate solution, which is better than the above mentioned ones.