Approximation guarantees of evolutionary optimization on the minimum crossing spanning tree problem

The minimum crossing spanning tree (MCST) problem is to find a spanning tree in a given graph G and a family of subsets of vertices S such that the maximum number of edges crossing any set in S reaches minimum value. Evolutionary algorithms (EAs) have been widely used in variety of fields due to its simple and powerful search ability. However, the performance analysis of EAs on the NP-hard MCST problem is still rare. In this paper, we investigate the approximation ability of EAs on the MCST problem from a theoretical point of view. For a special case of the MCST problem where the sets in S are pairwise disjoint, we reveal that a simple EA called (1 + 1) EA can efficiently obtain a spanning tree with crossing at most \(2OPT+2\) in expected runtime \(O(n^7)\), where OPT is the maximum crossing for a minimum crossing spanning tree. Moreover, we also find that for the MCST problem a simple multi-objective evolutionary algorithm called GSEMO can achieve an approximation ratio of \(4r\log n\) in expected polynomial runtime \(O(4rm^3\log n)\). The analysis and results further illustrate that EAs are efficient approximation algorithms for some NP-hard problems.