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

Sander Aarts, Jacob Dentes, Manxi Wu, David Shmoys

TL;DR

This work investigates a general class of CIPs and gives the first non-trivial price-of-fair-sharing bounds by using the natural LP relaxation strengthened with knapsack-cover inequalities and demonstrates that these LP-based methods outperform previously known methods on an LPWAN-derived CIP data set.

Abstract

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.

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

TL;DR

This work investigates a general class of CIPs and gives the first non-trivial price-of-fair-sharing bounds by using the natural LP relaxation strengthened with knapsack-cover inequalities and demonstrates that these LP-based methods outperform previously known methods on an LPWAN-derived CIP data set.

Abstract

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.
Paper Structure (12 sections, 4 theorems, 22 equations, 1 figure, 2 algorithms)

This paper contains 12 sections, 4 theorems, 22 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Optimizing coverage for LPWAN networks and IoT
  3. Cost-sharing fundamentals
  4. Knapsack-Cover Constraints to the Rescue
  5. Cost-shares and the strengthened LP
  6. Proof of Theorem 1
  7. A case study on LoRaWAN coverage
  8. Group-strategyproof KC-LP cost-shares
  9. Proof of Theorem 2.
  10. Acknowledgments.
  11. Disclosure of Interests.
  12. LoRaWAN, network planning, and the CIP

Key Result

theorem thmcountertheorem

Let $\mathbf{y} = (y^S_j)_{S \subseteq \mathcal{F}, j \in \mathcal{U}}$ be a KC-LP dual-feasible solution. Then, cost-shares satisfy the core property. We say the cost-shares in (eq:cost-shares) are induced by$\mathbf{y}$.

Figures (1)

  • Figure 1: Costs incurred and recovery for 10 instances. IP-OPT is the cost of the IP optimum, KC-LP the cost of the KC-LP optimum. Prefixes PD, Gr, and GR+, represent the PrimalDual, Greedy, and Greedy+, respectively. Suffixes Obj and Rev represent the integer objective cost, and cost-share revenue, respectively.

Theorems & Definitions (4)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • theorem thmcountertheorem: Carnes and Shmoys carnes2008primal