On the Computation of Fully Proportional Representation

We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the “sum of misrepresentations” is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these “minimax” versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

💡 Research Summary

The paper investigates the computational complexity of winner determination for two classic fully proportional representation (FPR) rules—Chamberlin‑Courant (CC) and Monroe (M)—and introduces two new “min‑max” variants that aim to minimize the worst‑case misrepresentation rather than the total sum. All four rules (CC‑sum, M‑sum, CC‑max, M‑max) are known to be NP‑hard in the general case, so the authors explore whether specific parameterizations or restricted preference domains make the problems tractable.

First, the authors conduct a parameterized complexity analysis with respect to two natural parameters: the number of candidates (m) and the number of winners (representatives) (k). They prove that when (m) is treated as a parameter, every rule becomes fixed‑parameter tractable (FPT). The algorithmic approach is essentially a bounded‑search over subsets of candidates combined with dynamic programming to assign voters optimally; the running time is (f(m)\cdot \text{poly}(n)), where (n) is the number of voters. In stark contrast, parameterizing by (k) yields W