Minimum Spanning Tree on Uncertain Graphs

Abstract

In recent years, lots of data in various domain can be represented and described by uncertain graph model, such as protein interaction networks, social networks, wireless sensor networks, etc. This paper investigates the most reliable minimum spanning tree problem, which aims to find the minimum spanning tree (MST) with largest probability among all possible MSTs on uncertain graphs. In fact, the most reliable MST is an optimal choice between stability and cost. Therefore it has wide applications in practice, for example, it can serve as the basic constructs in a telecommunication network, the link of which can be unreliable and may fail with certain probability. A brute-force method needs to enumerate all possible MSTs and the time consumption grows exponentially with edge size. Hence we put forward an approximate algorithm in $O(d^{2}|V|^{2})$, where d is the largest vertex degree and |V| is vertex size. We point out that the algorithm can achieve exact solution with expected probability at least $(1-(\frac{1}{2})^{(d+1)/2})^{|V|-1}$ and the expected approximation ratio is at least $(\frac{1}{2})^{d|V|}$ when edge probability is uniformly distributed. Our extensive experimental results show that our proposed algorithm is both efficient and effective.