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