Parameterized Intractability for Multi-Winner Election under the Chamberlin-Courant Rule and the Monroe Rule
This addresses a theoretical open problem in computational social choice, providing negative results for algorithm designers working on parameterized complexity in voting systems.
The paper resolves the parameterized complexity of multi-winner election problems under the Chamberlin-Courant and Monroe rules, showing they are W[1]-hard with respect to the sum of misrepresentations, which precludes efficient fixed-parameter tractable algorithms.
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 $β$ of misrepresentations, thereby precluding the existence of any $f(β) \cdot |I|^{O(1)}$ -time algorithm, where $|I|$ denotes the size of the input instance.