# Incomplete Preferences in Single-Peaked Electorates

## TL;DR

This paper investigates the complexity of recognizing single-peaked preferences in incomplete preference profiles, proving NP-completeness in general but providing polynomial algorithms for specific cases like weak orders.

## Contribution

It establishes NP-completeness for recognizing single-peakedness in incomplete profiles and offers polynomial-time algorithms for certain restricted preference structures.

## Key findings

- NP-completeness for partial orders
- Polynomial algorithms for weak orders
- Recognition algorithms for restricted settings

## Abstract

Incomplete preferences are likely to arise in real-world preference aggregation scenarios. This paper deals with determining whether an incomplete preference profile is single-peaked. This is valuable information since many intractable voting problems become tractable given single-peaked preferences. We prove that the problem of recognizing single-peakedness is \NP-complete for incomplete profiles consisting of partial orders. Despite this intractability result, we find several polynomial-time algorithms for reasonably restricted settings. In particular, we give polynomial-time recognition algorithms for weak orders, which can be viewed as preferences with indifference.