Distributionally robust bottleneck combinatorial problems: uncertainty quantification and robust decision making

In a bottleneck combinatorial problem, the objective is to minimize highest cost of elements subset selected from solution space. This paper studies data-driven distributionally robust problems (DRBCP) with stochastic costs, where probability distribution vector contained in ball distributions centered at empirical specified by Wasserstein distance. We study two distinct versions DRBCP different applications: (i) Motivated multi-hop wireless network application, we first uncertainty quantification (denoted DRBCP-U), decision-makers would like have an accurate estimation worst-case value DRBCP. The difficulty DRBCP-U handle its max–min–max form. Fortunately, similar strong duality linear programming, alternative forms using clutters and blocking systems allow us derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. addition, drawing connection between sampling average approximation counterpart under distribution, show that radius chosen order negative square root sample size, improving existing known results; (ii) Next, motivated ride-sharing choose best service-and-passenger matching minimizes unfairness. That is, decision-making DRBCP, denoted DRBCP-D. For DRBCP-D, optimal also counterpart, as DRBCP-U. When size small, propose use DRBCP-D construct indifferent space decision-robust model, finds variance. further decision model recast conic program. Finally, extend proposed models approaches $$\varGamma $$ —sum problem ( $$\hbox {DR}\varGamma \hbox {BCP}$$ ), are interested minimizing sum costs elements.