Identifying risk-averse low-diameter clusters in graphs with stochastic vertex weights

In this work, we study the problem of detecting risk-averse low-diameter clusters in graphs. It is assumed that the clusters represent k-clubs and that uncertain information manifests itself in the form of stochastic vertex weights whose joint distribution is known. The goal is to find a k-club of minimum risk contained in the graph. A stochastic programming framework that is based on the formalism of coherent risk measures is used to quantify the risk of a cluster. We show that the selected representation of risk guarantees that the optimal subgraphs are maximal clusters. A combinatorial branch-and-bound algorithm is proposed and its computational performance is compared with an equivalent mathematical programming approach for instances with \(k=2,3,\) and 4.