Network Expansion Problem on the Spanning Tree in Graphs

Motivated by various network improvement models, we study the problem to add some new edges to satisfy the increasing information demand and keep the underlying structure of the networks unchanged. In this paper we propose the general network expansion problem on the spanning tree in graphs (GNEST), then we present the polynomial equivalence between the GNEST problem and the constrained minimum spanning tree problem (CST), which indicates the GNEST problem is NP-hard. By utilizing an algorithm to solve the CST problem, we can design a PTAS to solve the GNEST problem, and the computational complexity is the same as that of the algorithm given in. Finally we study two special versions of the GNEST problem:the minimum network expansion on spanning tree problem (MNEST) and the minimum-cost network expansion on spanning tree (MCNEST). We design two polynomial-time algorithms to solve these two new problems. To solve the MNEST problem we use T-exchange method on spanning trees. To find the optimal solution of the MCNEST problem, we utilize lexicographical order and modify Sollinpsilas algorithm to find the minimum spanning tree as required.