Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions

AbstractIn this work, we consider general facility location and social choice problems, in which sets of agents A and facilities F are located in a metric space, and our goal is to assign agents to facilities (as well as choose which facilities to open) to optimize the social cost. We form new algorithms to do this in the presence of only ordinal information, i.e., when the true costs or distances of the agents from the facilities are unknown, and only the ordinal preferences of the agents for the facilities are available. The main difference between our work and previous work in this area is that, while we assume that only ordinal information about agent preferences is known, we also know the exact locations of the possible facilities F. Due to this extra information about the facilities, we are able to form powerful algorithms that have small distortion, i.e., perform almost as well as omniscient algorithms (which know the true numerical distances between agents and facilities) but use only ordinal information about agent preferences. For example, we present natural social choice mechanisms for choosing a single facility to open with distortion of at most 3 for minimizing both the total and the median social cost; this factor is provably the best possible. We analyze many general problems including matching, k-center, and k-median, and we present black-box reductions from omniscient approximation algorithms with approximation factor β to ordinal algorithms with approximation factor 1+2β doing this gives new ordinal algorithms for many important problems and establishes a toolkit for analyzing such problems in the future.