In a bottleneck combinatorial problem, the objective is to minimize the highest cost of elements of a subset selected from the combinatorial solution space. 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, similar to the strong duality of linear programming, the alternative forms of the bottleneck combinatorial problems using clutters and blocking systems 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. That is, we study 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 conic program. Finally, we extend the proposed models and solution approaches to the distributionally robust \(\varGamma \)—sum bottleneck combinatorial problem (\(\hbox {DR}\varGamma \hbox {BCP}\)), where decision-makers are interested in minimizing the worst-case sum of \(\varGamma \) highest costs of elements.

在瓶颈组合问题中，目标是使从组合解空间中选择的子集元素的最高成本最小化。本文研究具有随机成本的数据驱动的分布鲁棒瓶颈组合问题（DRBCP），其中成本向量的概率分布包含在以Wasserstein距离指定的经验分布为中心的分布球中。我们研究了来自不同应用程序的两个不同版本的DRBCP：（i）受多跳无线网络应用程序的启发，我们首先研究了DRBCP的不确定性量化（用DRBCP-U表示），决策者希望该不确定性量化具有精确性。 DRBCP的最坏情况值的估计。DRBCP-U的困难在于处理其最大-最小-最大形式。幸运的是，与线性编程的强对偶性相似，使用杂波和阻塞系统的瓶颈组合问题的替代形式使我们能够导出等效的确定性公式，可以通过混合整数程序进行计算。此外，通过在经验分布下绘制DRBCP-U及其采样平均近似值之间的联系，我们表明Wasserstein半径可以按样本大小的负平方根顺序进行选择，从而改善了现有的已知结果；（ii）接下来，在乘车共享应用程序的激励下，决策者选择最佳的服务和乘客匹配，以最大程度地减少不公平。也就是说，我们研究以DRBCP-D表示的决策DRBCP。对于DRBCP-D，我们证明了它的最优解对于其采样平均近似值也是最优的，并且Wasserstein半径的选择顺序与DRBCP-U相似。当样本量较小时，我们建议使用DRBCP-D的最佳值来构造一个无差异的解决方案空间，并提出一个替代的决策稳健模型，该模型找到最佳的无差异的解决方案以最小化经验方差。我们进一步表明，可以将决策鲁棒模型改写为混合整数圆锥程序。最后，我们将提出的模型和解决方案方法扩展到分布稳健性 我们建议使用DRBCP-D的最佳值来构造一个无差异的解决方案空间，并提出一个替代的决策稳健模型，该模型找到最佳的无差异解决方案以最小化经验方差。我们进一步表明，决策鲁棒模型可以重铸为混合整数圆锥程序。最后，我们将提出的模型和解决方案方法扩展到分布稳健性 我们建议使用DRBCP-D的最佳值来构造一个无差异的解决方案空间，并提出一个替代的决策稳健模型，该模型找到最佳的无差异解决方案以最小化经验方差。我们进一步表明，可以将决策鲁棒模型改写为混合整数圆锥程序。最后，我们将提出的模型和解决方案方法扩展到分布稳健性\（\ varGamma \）—总和瓶颈组合问题（\（\ hbox {DR} \ varGamma \ hbox {BCP} \）），决策者希望将\（\ varGamma \）的最坏情况总和最小化元素成本最高。