Testing preferential domains using sampling

A preferential domain is a collection of sets of preferences which are linear orders over a set of alternatives These domains have been studied extensively in social choice theory due to both its practical importance and theoretical elegance Examples of some extensively studied preferential domains include single peaked, single crossing Euclidean, etc In this paper, we study the sample complexity of testing whether a given preference profile is close to some specific domain We consider two notions of closeness: (a) closeness via preferences, and (b) closeness via alternatives We further explore the effect of assuming that the outlier preferences/alternatives to be random (instead of arbitrary) on the sample complexity of the testing problem In most cases, we show that the above testing problem can be solved with high probability for all commonly used domains by observing only a small number of samples (independent of the number of preferences, n, and often the number of alternatives, m) In the remaining few cases, we prove either impossibility results or fl(n) lower bound on the sample complexity We complement our theoretical findings with extensive simulations to figure out the actual constant factors of our asymptotic sample complexity bounds.