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Answering an open question by Betzler et al. [Betzler et al., JAIR'13], weresolve the parameterized complexity of the multi-winner determination problemunder two famous representation voting rules: the Chamberlin-Courant (in shortCC) 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 respectto the sum $\beta$ of misrepresentations, thereby precluding the existence ofany $f(\beta) \cdot |I|^{O(1)}$ -time algorithm, where $|I|$ denotes the sizeof the input instance.

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