Performance analysis of evolutionary algorithm for the maximum internal spanning tree problem

The maximum internal spanning tree (MIST) problem is to find a spanning tree with maximum number of internal node for an undirected graph. It is a variation of the well-known minimum spanning tree problem and is NP-hard. Evolutionary algorithms (EAs) have been successfully applied to solve many NP-hard combinatorial optimization problems in numerical empirical studies. However, researchers know little about their performance guarantees from theory aspect. This paper designs a valid fitness function to guide the well-studied evolutionary algorithm, (1+1) EA , to optimize the MIST problem, and presents theoretical analysis to show that it can obtain a performance guarantee of 2 and 5/3 in expected runtime O(nm^2) and O(nm^4) , respectively. Moreover, this paper proves that the (1+1) EA can achieve a performance guarantee of 3 for a variation of the MIST problem, where each vertex is associated with a weight. In addition, comparison analyses on two family instance graphs are presented to show that (1+1) EA is better than two local search algorithms. This theoretical study provides insight into the process of (1+1) EA constructing a performance guarantee solution for the MIST problem.