High-Multiplicity Election Problems

TL;DR

This paper investigates how high-multiplicity representations of election preferences impact the computational complexity of election problems, revealing that many problems remain polynomial-time solvable, unlike in weighted voting scenarios.

Contribution

It demonstrates that polynomial-time election problems often stay in P under high-multiplicity representation, contrasting with weighted voting complexities, and explores related scheduling problems.

Findings

Most polynomial-time election problems remain in P with high-multiplicity representation.

High-multiplicity scheduling on uniform parallel machines is in P for any fixed set of job lengths.

Winner determination for Kemeny elections becomes NP-hard in high-multiplicity form.

Abstract

The computational study of elections generally assumes that the preferences of the electorate come in as a list of votes. Depending on the context, it may be much more natural to represent the list succinctly, as the distinct votes of the electorate and their counts, i.e., high-multiplicity representation. We consider how this representation affects the complexity of election problems. High-multiplicity representation may be exponentially smaller than standard representation, and so many polynomial-time algorithms for election problems in standard representation become exponential-time. Surprisingly, for polynomial-time election problems, we are often able to either adapt the same approach or provide new algorithms to show that these problems remain polynomial-time in the high-multiplicity case; this is in sharp contrast to the case where each voter has a weight, where the complexity usually increases. In the process we explore the relationship between high-multiplicity scheduling and manipulation of high-multiplicity elections. And we show that for any fixed set of job lengths, high-multiplicity scheduling on uniform parallel machines is in P, which was previously known for only two job lengths. We did not find any natural case where a polynomial-time election problem does not remain in P when moving to high-multiplicity representation. However, we found one natural NP-hard election problem where the complexity does increase, namely winner determination for Kemeny elections.