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Preprint 655-1999

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

Cooperative facility location games
Michel X. Goemans and Martin Skutella
Journal of Algorithms (special issue devoted to SODA 2000), to appear. An extended abstract appeared in Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'00), pp. 76-85
primary: 91A12 Cooperative games
secondary: 90B80 Discrete location and assignment
90C35 Programming involving graphs or networks
90C90 Applications of mathematical programming
facility location, cooperative games, LP relaxation, randomized rounding, core
The location of facilities in order to provide service for customers is a well-studied problem in the operations research literature. In the basic model, there is a predefined cost for opening a facility and also for connecting a customer to a facility, the goal being to minimize the total cost. Often, both in the case of public facilities (such as libraries, municipal swimming pools, fire stations, ...) and private facilities (such as distribution centers, switching stations, ...), we may want to find a "fair" allocation of the total cost to the customers -- this is known as the cost allocation problem. A central question in cooperative game theory is whether the total cost can be allocated to the customers such that no coalition of customers has any incentive to build their own facility or to ask a competitor to service them.
We establish strong connections between fair cost allocations and linear programming relaxations for several variants of the facility location problem. In particular, we show that a fair cost allocation exists if and only if there is no integrality gap for a corresponding linear programming relaxation. We use this insight in order to give proofs for the existence of fair cost allocations for several classes of instances based on a subtle variant of randomized rounding. This also leads to polynomial-time algorithms to solve the facility location problem in these cases. We also prove that it is in general NP-complete to decide whether a fair cost allocation exists and whether a given allocation is fair.
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