We study the complexity of (approximate) winner determination under the Monroe and Chamberlin--Courant multiwinner voting rules, which determine the set of representatives by optimizing the total (dis)satisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions (the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitarian case). We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin--Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of them and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show that our simple and fast algorithms can in many cases find near-perfect solutions.
There are several single-winner voting rules for which winner determination is known to be @math -hard. These rules include, for example, Dodgson's rule @cite_2 @cite_9 @cite_21 , Young's rule @cite_32 @cite_21 , and Kemeny's rule @cite_2 @cite_22 @cite_14 . For the single-transferable vote rule (STV), the winner determination problem becomes @math -hard if we use the so-called parallel-universes tie-breaking @cite_13 . Many of these hardness results hold even in the sense of parameterized complexity theory (however, there also is a number of fixed-parameter tractability results; see the references above for details).
@cite_14 @cite_22 @cite_9 @cite_21 @cite_32 @cite_2 @cite_13
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The Kemeny Score problem is central to many applications in the context of rank aggregation. Given a set of permutations (votes) over a set of candidates, one searches for a "consensus permutation" that is "closest" to the given set of permutations. Computing an optimal consensus permutation is NP-hard. We provide first, encouraging fixed-parameter tractability results for computing optimal scores (that is, the overall distance of an optimal consensus permutation). Our fixed-parameter algorithms employ the parameters "score of the consensus", "maximum distance between two input permutations", and "number of candidates". We extend our results to votes with ties and incomplete votes, thus, in both cases having no longer permutations as input. Kemeny proposed a voting scheme which is distinguished by the fact that it is the unique voting scheme that is neutral, consistent, and Condorcet. Bartholdi, Tovey, and Trick showed that determining the winner in Kemeny's system is NP-hard. We provide a stronger lower bound and an upper bound matching the lower bound, namely, we show that determining the winner in Kemeny's system is complete for P||NP, the class of sets solvable via parallel access to NP. In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound—NP-hardness—on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, that is, it is complete for Theta_ 2 p for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses. Abstract We show that the two NP-complete problems of Dodgson Score and Young Score have differing computational complexities when the winner is close to being a Condorcet winner. On the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-completeness results depending on the cost model for switching ties. In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for || NP , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll's 1876 voting scheme were recently established by In contrast, we prove that the winner and ranking problems in Fishburn's homogeneous variant of Carroll's voting scheme can be solved efficiently by linear programming. We show that a voting scheme suggested by Lewis Carroll can be impractical in that it can be computationally prohibitive (specifically, NP-hard) to determine whether any particular candidate has won an election. We also suggest a class of “impracticality theorems” which say that any fair voting scheme must, in the worst-case, require excessive computation to determine a winner. In social choice, a preference function (PF) takes a set of votes (linear orders over a set of alternatives) as input, and produces one or more rankings (also linear orders over the alternatives) as output. Such functions have many applications, for example, aggregating the preferences of multiple agents, or merging rankings (of, say, webpages) into a single ranking. The key issue is choosing a PF to use. One natural and previously studied approach is to assume that there is an unobserved "correct" ranking, and the votes are noisy estimates of this. Then, we can use the PF that always chooses the maximum likelihood estimate (MLE) of the correct ranking. In this paper, we define simple ranking scoring functions (SRSFs) and show that the class of neutral SRSFs is exactly the class of neutral PFs that are MLEs for some noise model. We also define composite ranking scoring functions (CRSFs) and show a condition under which these coincide with SRSFs. We study key properties such as consistency and continuity, and consider some example PFs. In particular, we study Single Transferable Vote (STV), a commonly used PF, showing that it is a CRSF but not an SRSF, thereby clarifying the extent to which it is an MLE function. This also gives a new perspective on how ties should be broken under STV. We leave some open questions.