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

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 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. Index TermsEvolutionary algorithm; time complexity; approximation ratio; minimum label spanning tree; multi-objective I. INTRODUCTION The minimum label spanning tree (MLST) problem is an issue arising from practice, which seeks a spanning tree with the minimum number of labels in a connected undirected graph with labeled edges. For example, we want to find a spanning tree that uses the minimum number of types of communication channels in a communication networks connected with different types of channels. The MLST problem, proposed by Chang and Leu, is proved to be NP-hard [1].For this problem, Chang and Leu have proposed two heuristic algorithms. One is the edge replacement algorithm, ERA for short, the other is the maximum vertex covering algorithm, MVCA for short. Their experimental results showed that ERA is not stable, and MVCA is more efficient.The genetic algorithm, belonging to the larger class of EAs, is a general purpose optimization algorithm [2]-[4] with a strong globally searching capacity [5]. So, Xiong, Golden, and Wasil proposed a one-parameter genetic algorithm for the MLST problem. The experimental results on extensive instances generated randomly showed that the genetic algorithm outperforms MVCA [6]. Nummela and Julstrom also proposed an efficient genetic algorithm for solving the MLST problem [7].Besides, many methods recently have been proposed for solving this NP-hard problem. Consoli et al. proposed a hybrid local search combining variable neighborhood search and simulated annealing [8]. Chwatal and Raidl presented exact methods including branch-and-cut and branch-and-cut-and-price [9]. Cerulli et al. utilized several metaheuristic methods for this problem, such as simulated annealing, reactive tabu search, the pilot method, and variable neighborhood search [10]. Consoli et al. still proposed a greedy randomized adaptive search procedure and a variable neighborhood search for solving the MLST problem [11].Since both ERA and MVCA are two original heuristic algorithms for the MLST problem, the worst performance analysis of these two algorithms, especially MVCA, is a hot research topic in recent years. Krumke ...

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