Parameterized Intractability for Multi-Winner Election under the Chamberlin-Courant Rule and the Monroe Rule

TL;DR

This paper establishes the parameterized computational complexity of multi-winner election problems under the Chamberlin-Courant and Monroe rules, showing they are W[1]-hard with respect to misrepresentation sum, indicating no fixed-parameter tractable algorithms are likely.

Contribution

It resolves an open question by proving W[1]-hardness of multi-winner determination under two key voting rules, advancing understanding of their computational intractability.

Findings

Both rules are W[1]-hard with respect to misrepresentation sum.

No fixed-parameter tractable algorithms exist based on this parameter.

The result answers an open problem in computational social choice.

Abstract

Answering an open question by Betzler et al. [Betzler et al., JAIR'13], we resolve the parameterized complexity of the multi-winner determination problem under two famous representation voting rules: the Chamberlin-Courant (in short CC) rule [Chamberlin and Courant, APSR'83] and the Monroe rule [Monroe, APSR'95]. We show that under both rules, the problem is W[1]-hard with respect to the sum $\beta$ of misrepresentations, thereby precluding the existence of any $f(\beta) \cdot |I|^{O(1)}$ -time algorithm, where $|I|$ denotes the size of the input instance.