Two-stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP-hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max-flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max-flow structure of the subproblems

Combinatorial content of CCL3L and CCL4L gene copy numbers influence HIV-AIDS susceptibility in Ukrainian children

Transmission risk is greatest when mother and offspring both have low CCL3L or CCL4L gene doses. The impact on HIV-AIDS susceptibility of the chemokine gene-rich locus on 17q12 is dependent on the balance between the doses of genes conferring protective (CCL3La and CCL4La) versus detrimental (CCL4Lb) effects. Hence, the combinatorial genomic content of distinct genes within a copy number variable region may determine disease susceptibility