Much work has been devoted, during the past twenty years, to using complexity to protect elections from manipulation and control. Many results have been obtained showing NP-hardness shields, and recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societ al preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate.
As mentioned in the main text, Bartholdi and Trick @cite_27 , Doignon and Falmagne @cite_37 , and Escoffier, Lang, and "O zt "u rk @cite_1 have provided efficient algorithms for testing single-peakedness and producing a valid candidate linear ordering, for the case when votes are linear orders.
@cite_27 @cite_1 @cite_37
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This paper investigates one of the possible weakening of the (too demanding) assumptions of the Gibbard-Satterthwaite theorem. Namely we deal with a class of voting schemes where at the same time the domain of possible preference preordering of any agent is limited to single-peaked preferences, and the message that this agent sends to the central authority is simply its ‘peak’ — his best preferred alternative. In this context we have shown that strategic considerations justify the central role given to the Condorcet procedure which amounts to elect the ‘median’ peak: namely all strategy-proof anonymous and efficient voting schemes can be derived from the Condorcet procedure by simply adding some fixed ballots to the agent's ballots (with the only restriction that the number of fixed ballots is strictly less than the number of agents). Many NP-hard winner determination problems admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives fast algorithms for finding optimal committees under the PAV and Chamberlin-Courant voting rules under single-peaked preferences, as well as for certain OWA-based rules. Under single-crossing preferences, Young scores can also be calculated. An advantage of this technique is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise. 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. We study the complexity of electing a committee under several variants of the Chamberlin-Courant rule when the voters' preferences are single-peaked on a tree. We first show that this problem is easy for the egalitarian, or "minimax" version of this problem, for arbitrary trees and misrepresentation functions. For the standard (utilitarian) version of this problem we provide an algorithm for an arbitrary misrepresentation function whose running time is polynomial in the input size as long as the number of leaves of the underlying tree is bounded by a constant. On the other hand, we prove that our problem remains computationally hard on trees that have bounded degree, diameter, or pathwidth. Finally, we show how to modify Trick's [1989] algorithm to check whether an election is single-peaked on a tree whose number of leaves does not exceed a given parameter λ.