Exact Complexity of the Winner Problem for Young Elections

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 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 Hemaspaandra et al. 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.Comment: 10 page

Dichotomy for Voting Systems

Scoring protocols are a broad class of voting systems. Each is defined by a vector $(\alpha_1,\alpha_2,...,\alpha_m)$, $\alpha_1 \geq \alpha_2 \geq >... \geq \alpha_m$, of integers such that each voter contributes $\alpha_1$ points to his/her first choice, $\alpha_2$ points to his/her second choice, and so on, and any candidate receiving the most points is a winner. What is it about scoring-protocol election systems that makes some have the desirable property of being NP-complete to manipulate, while others can be manipulated in polynomial time? We find the complete, dichotomizing answer: Diversity of dislike. Every scoring-protocol election system having two or more point values assigned to candidates other than the favorite--i.e., having ||\{\alpha_i \condition 2 \leq i \leq m\}||\geq 2--is NP-complete to manipulate. Every other scoring-protocol election system can be manipulated in polynomial time. In effect, we show that--other than trivial systems (where all candidates alway tie), plurality voting, and plurality voting's transparently disguised translations--\emph{every} scoring-protocol election system is NP-complete to manipulate

Computational Social Choice and Computational Complexity: BFFs?

We discuss the connection between computational social choice (comsoc) and computational complexity. We stress the work so far on, and urge continued focus on, two less-recognized aspects of this connection. Firstly, this is very much a two-way street: Everyone knows complexity classification is used in comsoc, but we also highlight benefits to complexity that have arisen from its use in comsoc. Secondly, more subtle, less-known complexity tools often can be very productively used in comsoc.Comment: A version of this paper will appear in AAAI-1

Dodgson's Rule Approximations and Absurdity

With the Dodgson rule, cloning the electorate can change the winner, which Young (1977) considers an "absurdity". Removing this absurdity results in a new rule (Fishburn, 1977) for which we can compute the winner in polynomial time (Rothe et al., 2003), unlike the traditional Dodgson rule. We call this rule DC and introduce two new related rules (DR and D&). Dodgson did not explicitly propose the "Dodgson rule" (Tideman, 1987); we argue that DC and DR are better realizations of the principle behind the Dodgson rule than the traditional Dodgson rule. These rules, especially D&, are also effective approximations to the traditional Dodgson's rule. We show that, unlike the rules we have considered previously, the DC, DR and D& scores differ from the Dodgson score by no more than a fixed amount given a fixed number of alternatives, and thus these new rules converge to Dodgson under any reasonable assumption on voter behaviour, including the Impartial Anonymous Culture assumption.Comment: Expanded draft of paper presented at COMSOC 200

Modeling Single-Peakedness for Votes with Ties

Single-peakedness is one of the most important and well-known domain restrictions on preferences. The computational study of single-peaked electorates has largely been restricted to elections with tie-free votes, and recent work that studies the computational complexity of manipulative attacks for single-peaked elections for votes with ties has been restricted to nonstandard models of single-peaked preferences for top orders. We study the computational complexity of manipulation for votes with ties for the standard model of single-peaked preferences and for single-plateaued preferences. We show that these models avoid the anomalous complexity behavior exhibited by the other models. We also state a surprising result on the relation between the societal axis and the complexity of manipulation for single-peaked preferences.Comment: A shorter version of this paper will appear in STAIRS 201

Rationalizations of Condorcet-Consistent Rules via Distances of Hamming Type

The main idea of the {\em distance rationalizability} approach to view the voters' preferences as an imperfect approximation to some kind of consensus is deeply rooted in social choice literature. It allows one to define ("rationalize") voting rules via a consensus class of elections and a distance: a candidate is said to be an election winner if she is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be distance rationalized. In this paper, we provide new results on distance rationalizability of several Condorcet-consistent voting rules. In particular, we distance rationalize Young's rule and Maximin rule using distances similar to the Hamming distance. We show that the claim that Young's rule can be rationalized by the Condorcet consensus class and the Hamming distance is incorrect; in fact, these consensus class and distance yield a new rule which has not been studied before. We prove that, similarly to Young's rule, this new rule has a computationally hard winner determination problem

Frequency of Correctness versus Average-Case Polynomial Time and Generalized Juntas

We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. We also study some features of probability weight of correctness with respect to generalizations of Procaccia and Rosenschein's junta distributions [PR07b]

On Approximating Optimal Weighted Lobbying, and Frequency of Correctness versus Average-Case Polynomial Time

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant--optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a ``frequently self-knowingly correct algorithm.'' We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein's junta distributions [PR07]

On Choosing Committees Based on Approval Votes in the Presence of Outliers

We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval and minisum approval voting rules, we provide a dichotomous result, resolving the parameterized complexity of this problem for all subsets of five natural parameters considered (by showing either FPT or W[1]-hardness for all subsets of parameters). For the approval voting rule, we resolve the parameterized complexity of this problem for all subsets of parameters except one. We also study approximation algorithms for this problem. We show that there does not exist any alpha(.) factor approximation algorithm for approval and net approval voting rules, for any computable function alpha(.), unless P=NP. For the minisum voting rule, we provide a pseudopolynomial (1+eps) factor approximation algorithm

A O(log⁡m)O(\log m), deterministic, polynomial-time computable approximation of Lewis Carroll's scoring rule

We provide deterministic, polynomial-time computable voting rules that approximate Dodgson's and (the ``minimization version'' of) Young's scoring rules to within a logarithmic factor. Our approximation of Dodgson's rule is tight up to a constant factor, as Dodgson's rule is \NP-hard to approximate to within some logarithmic factor. The ``maximization version'' of Young's rule is known to be \NP-hard to approximate by any constant factor. Both approximations are simple, and natural as rules in their own right: Given a candidate we wish to score, we can regard either its Dodgson or Young score as the edit distance between a given set of voter preferences and one in which the candidate to be scored is the Condorcet winner. (The difference between the two scoring rules is the type of edits allowed.) We regard the marginal cost of a sequence of edits to be the number of edits divided by the number of reductions (in the candidate's deficit against any of its opponents in the pairwise race against that opponent) that the edits yield. Over a series of rounds, our scoring rules greedily choose a sequence of edits that modify exactly one voter's preferences and whose marginal cost is no greater than any other such single-vote-modifying sequence