Approximating Probabilistic Group Steiner Trees in Graphs

Consider an edge-weighted graph, and a number of properties of interests (PoIs). Each vertex has a probability of exhibiting each PoI. The joint probability that a set of vertices exhibits a PoI is the probability that this set contains at least one vertex that exhibits this PoI. The probabilistic group Steiner tree problem is to find a tree such that (i) for each PoI, the joint probability that the set of vertices in this tree exhibits this PoI is no smaller than a threshold value, e.g., 0.97; and (ii) the total weight of edges in this tree is the minimum. Solving this problem is useful for mining various graphs with uncertain vertex properties, but is NP-hard. The existing work focuses on certain cases, and cannot perform this task. To meet this challenge, we propose 3 approximation algorithms for solving the above problem. Let | Γ| be the number of PoIs, and ξ be an upper bound of the number of vertices for satisfying the threshold value of exhibiting each PoI. Algorithms 1 and 2 have tight approximation guarantees proportional to | Γ| and ξ, and exponential time complexities with respect to ξ and | Γ|, respectively. In comparison, Algorithm 3 has a looser approximation guarantee proportional to, and a polynomial time complexity with respect to, both | Γ| and ξ. Experiments on real and large datasets show that the proposed algorithms considerably outperform the state-of-the-art related work for finding probabilistic group Steiner trees in various cases.