Genetic algorithms for balanced spanning tree problem

Given an undirected weighted connected graph G = (V,E) with vertex set V and edge set E and a designated vertex r ∈ V , we consider the problem of constructing a spanning tree in G that balances both the minimum spanning tree and the shortest paths tree rooted at r. Formally, for any two constants α, β ≥ 1, we consider the problem of computing an (α, β)-balanced spanning tree T in G, in the sense that, (i) for every vertex v ∈ V , the distance between r and v in T is at most a times the shortest distance between the two vertices in G, and (ii) the total weight of T is at most β times that of the minimum tree weight in G. It is well known that, for any α, β ≥ 1, the problem of deciding whether G contains an (α, β)-balanced spanning tree is NP-complete [15]. Consequently, given any α ≥ 1 (resp., β ≥ 1), the problem of finding an (α, β)-balanced spanning tree that minimizes β (resp., α) is NP-complete. In this paper, we present efficient genetic algorithms for these problems. Our experimental results show that the proposed algorithm returns high quality balanced spanning trees.