The Complexity of Proportionality Degree in Committee Elections

TL;DR

This paper investigates the computational complexity of determining and verifying proportionality degrees in committee elections, expanding understanding beyond justified representation to include group cohesion and counting problems.

Contribution

It analyzes the complexity of computing and testing proportionality degrees, and explores group cohesion and counting problems in committee elections.

Findings

Computing committees with a given proportionality degree is computationally complex.

Testing if a committee meets a specific proportionality degree is also complex.

The study extends the analysis to group cohesion and counting problems in elections.

Abstract

Over the last few years, researchers have put significant effort into understanding of the notion of proportional representation in committee election. In particular, recently they have proposed the notion of proportionality degree. We study the complexity of computing committees with a given proportionality degree and of testing if a given committee provides a particular one. This way, we complement recent studies that mostly focused on the notion of (extended) justified representation. We also study the problems of testing if a cohesive group of a given size exists and of counting such groups.