Chance-Constrained Programming Models and Approximation Algorithms for General Stochastic Bottleneck Spanning Tree Problems

This paper considers a balance-constrained stochastic bottleneck spanning tree problem (BCSBSTP) in which edge weights are independently distributed but may follow arbitrary random distributions. The problem minimizes a scalar and seeks a spanning tree, of which the maximum edge weight is bounded by the scalar for a given certain probability, and meanwhile the minimum edge weight is lower bounded in another chance constraint. The paper formulates the BCSBSTP as a mixed-integer nonlinear program, and develops two mixed-integer linear programming approximations by using special ordered set of type one (SOS1) and special ordered set of type two (SOS2) variables. By relaxing the chance constraint on the minimum edge weight, the BCSBSTP is simplified as a stochastic bottleneck spanning tree problem, for which a bisection algorithm is developed to approximate optimal solutions in polynomial time. Based on the properties derived in the development of the bisection algorithm, we show that the BCSBSTP is NP-Complete. We demonstrate the computational results by testing all models and algorithms on a diverse set of graph instances with edge weights that are independently distributed.