The tolerant Condorcet points

A Condocet point is a location for a facility such that no other location is closer to a strict majority of users. Since a Condorcet point does not always exist, the solution conditions are relaxed by introducing a tolerance distance, say alpha, for the preference. Then, two locations for the facility are indifferent for a user if the distances from him/her to both locations differ in at most alpha. A location is alpha-Condorcet if no other location is preferred for a strict majority of users using the tolerance distance alpha. The ‘Tolerant Condorcet’ points are the alpha-Condorcet points for the smallest value of alpha for which alpha-Condorcet points exist. We formulate the problem of finding the Tolerant Condorcet points on an undirected network and provide a polynomial algorithm to solve it.