# Testing Preferential Domains Using Sampling

## TL;DR

This paper investigates the sample complexity of testing whether preference profiles are close to specific preferential domains, providing theoretical bounds and simulations to understand the efficiency of such testing methods.

## Contribution

It introduces the first analysis of sample complexity for testing preferential domains, considering different notions of closeness and the impact of random outliers.

## Key findings

- High-probability testing with few samples for common domains
- Impossibility results and lower bounds in certain cases
- Simulations to determine practical constants of sample complexity

## Abstract

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 {\em 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 $\Omega(n)$ 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.