On the Approximability of Dodgson and Young Elections

The voting rules proposed by Dodgson and Young are both designed to nd the alternative closest to being a Condorcet winner, according to two di erent notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for ap- proximating the Dodgson score: an LP-based randomized rounding algorithm and a deterministic greedy algorithm, both of which yield an O(logm) approximation ratio, where m is the number of alternatives; we observe that this result is asymptotically optimal, and further prove that our greedy algorithm is optimal up to a factor of 2, unless problems in NP have quasi-polynomial time algorithms. Although the greedy algorithm is computationally superior, we argue that the randomized rounding algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson\u27s rule is NP-hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the Social Choice Theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is NP-hard to approximate by any factor. This leads to an inapproximability result for the Young ranking

Improving Dodgson scoring techniques

The Dodgson score problem is part of the Dodgson election scheme invented by Charles Dodgson and presented in his manuscript. One of the system\u27s strengths (and motivations for its study) is that it satisfies the Condorcet criterion (which states that any candidate who defeats all other candidates in pairwise elections will be declared the winner). It is unfortunate, though, that in a given election no Condorcet winner may exist. Dodgson\u27s election system chooses the winner closest to being the Condorcet winner in the sense that it requires the shortest sequence of edits (swapping of adjacent candidates in the voters\u27 preference rankings) to the votes in order to make it one. The length of this sequence is known as the Dodgson score. The problem of finding the Dodgson score of a candidate is computationally intractable. Thus an approximation is necessary. This paper puts forth MCDodgsonScore, a polynomialtime computable (ln(m) + 1)-approximation of that problem. This approximation is optimal, meaning that an approximation with an asymptotically tighter error bound does not exist. MCDodgsonScore builds on a technique introduced by Homan and Hemaspaandra in 2006. A nice feature of MCDodgsonScore is that, when treated as its own voting rule, it will also satisfy the Condorcet criterion

The Complexity of Manipulating kk-Approval Elections

An important problem in computational social choice theory is the complexity of undesirable behavior among agents, such as control, manipulation, and bribery in election systems. These kinds of voting strategies are often tempting at the individual level but disastrous for the agents as a whole. Creating election systems where the determination of such strategies is difficult is thus an important goal. An interesting set of elections is that of scoring protocols. Previous work in this area has demonstrated the complexity of misuse in cases involving a fixed number of candidates, and of specific election systems on unbounded number of candidates such as Borda. In contrast, we take the first step in generalizing the results of computational complexity of election misuse to cases of infinitely many scoring protocols on an unbounded number of candidates. Interesting families of systems include $k$-approval and $k$-veto elections, in which voters distinguish $k$ candidates from the candidate set. Our main result is to partition the problems of these families based on their complexity. We do so by showing they are polynomial-time computable, NP-hard, or polynomial-time equivalent to another problem of interest. We also demonstrate a surprising connection between manipulation in election systems and some graph theory problems

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context