Proportional Representation under Single-Crossing Preferences Revisited

We study the complexity of determining a winning committee under Chamberlin-Courant voting rule when voters' preferences are single-crossing on line, or, more generally, tree. For Skowron et al. (2015) describe an O(n^2mk) algorithm (where n, m, k number voters, candidates and size, respectively); we show that simple tweak improves time to O(nmk). then improve this bound for k=Ω(log n) by reducing our problem k-link path DAGs with concave Monge weights, obtaining nm2^O(√(log log n)) general case nearly linear Borda misrepresentation function. trees, point out issue proposed Clearwater, Puppe Slinko (2015), develop O(nmk) as well.