Performance Analysis on Evolutionary Algorithms for the Minimum Label Spanning Tree Problem

TL;DR

This paper provides a theoretical analysis of evolutionary algorithms' performance on the minimum label spanning tree problem, demonstrating their approximation ratios and superiority over local search methods in certain cases.

Contribution

It offers the first theoretical performance bounds for the (1+1) EA and GSEMO on the MLST problem, including approximation ratios and comparisons with local search algorithms.

Findings

(1+1) EA and GSEMO achieve a (b+1)/2-approximation ratio for MLST_b in polynomial time.

GSEMO achieves a (2ln(n))-approximation ratio for MLST in polynomial time.

GSEMO outperforms (1+1) EA and local search algorithms on specific MLST instances.

Abstract

Some experimental investigations have shown that evolutionary algorithms (EAs) are efficient for the minimum label spanning tree (MLST) problem. However, we know little about that in theory. As one step towards this issue, we theoretically analyze the performances of the (1+1) EA, a simple version of EAs, and a multi-objective evolutionary algorithm called GSEMO on the MLST problem. We reveal that for the MLST$_{b}$ problem the (1+1) EA and GSEMO achieve a $\frac{b+1}{2}$-approximation ratio in expected polynomial times of $n$ the number of nodes and $k$ the number of labels. We also show that GSEMO achieves a $(2ln(n))$-approximation ratio for the MLST problem in expected polynomial time of $n$ and $k$. At the same time, we show that the (1+1) EA and GSEMO outperform local search algorithms on three instances of the MLST problem. We also construct an instance on which GSEMO outperforms the (1+1) EA.