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

Bounding the Price-of-Fair-Sharing using Knapsack-Cover Constraints to guide Near-Optimal Cost-Recovery Algorithms