We consider the problem of fairly allocating the cost of providing a service among a set of users, where the service cost is formulated by an NP-hard {\it covering integer program (CIP)}. The central issue is to determine a cost allocation to each user that, in total, recovers as much as possible of the actual cost while satisfying a stabilizing condition known as the {\it core property}. The ratio between the total service cost and the cost recovered from users has been studied previously, with seminal papers of Deng, Ibaraki, \& Nagomochi and Goemans \& Skutella linking this {\it price-of-fair-sharing} to the integrality gap of an associated LP relaxation. Motivated by an application of cost allocation for network design for LPWANs, an emerging IoT technology, we investigate a general class of CIPs and give the first non-trivial price-of-fair-sharing bounds by using the natural LP relaxation strengthened with knapsack-cover inequalities. Furthermore, we demonstrate that these LP-based methods outperform previously known methods on an LPWAN-derived CIP data set. We also obtain analogous results for a more general setting in which the service provider also gets to select the subset of users, and the mechanism to elicit users' private utilities should be group-strategyproof. The key to obtaining this result is a simplified and improved analysis for a cross-monotone cost-allocation mechanism.

翻译：我们考虑在用户集合中公平分配服务提供成本的问题，其中服务成本由一个NP难问题——覆盖整数规划（CIP）——建模。核心问题在于确定每个用户的成本分摊方案，使得在满足称为核心性质的稳定性条件的同时，从用户处回收的成本总额尽可能接近实际总成本。服务总成本与从用户回收成本之间的比率——即公平共享代价——已有先前研究，Deng、Ibaraki与Nagomochi以及Goemans与Skutella的开创性工作将此比率与相关线性规划松弛的整数间隙联系起来。受低功耗广域网（一种新兴物联网技术）网络设计成本分摊应用的启发，我们研究了一类广义的CIP，并首次通过采用经背包覆盖不等式增强的自然线性规划松弛，给出了非平凡的公平共享代价上界。此外，我们在线性规划松弛方法在低功耗广域网衍生的CIP数据集上优于已知方法。对于更一般的场景——服务提供商可同时选择用户子集，且需设计满足群体策略防护性的机制以获取用户私有效用——我们也获得了类似结论。实现该结果的关键在于对交叉单调成本分摊机制进行了简化且改进的分析。