On absolutely and simply popular rankings

Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking $\pi$ of the candidates is at least as good as any other ranking $\sigma$ in the following sense: if we compare $\pi$ to $\sigma$, at least half of all voters will always weakly prefer~$\pi$. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity -- as applied to popular matchings, a well-established topic in computational social choice -- is stricter, because it requires at least half of the voters \emph{who are not indifferent between $\pi$ and $\sigma$} to prefer~$\pi$. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zuylen et al. We also point out strong connections to the famous open problem of finding a Kemeny consensus with 3 voters.Comment: full version of the AAMAS 2021 extended abstract 'On weakly and strongly popular rankings

Randomized Search of Graphs in Log Space and Probabilistic Computation

Reingold has shown that L = SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. Several properties that bound the mixing times of random walks on digraphs have been identified, including the digraph conductance and the digraph spectral expansion. However, rapidly mixing digraphs can still have exponential cover time, thus it is important to specifically identify structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric logspace computations), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Next we relate random walks on graphs to the random walks that arise in Monte Carlo methods applied to optimization problems. We introduce the notion of the asymmetric states of Markov chains and use this definition to obtain some results about Markov chains. We also obtain some results on the mixing times for Markov Chain Monte Carlo Methods. Finally, we consider the question of whether a single long random walk or many short walks is a better strategy for exploration. These are walks which reset to the start after a fixed number of steps. We exhibit digraph families for which a few short walks are far superior to a single long walk. We introduce an iterative deepening random search. We use this strategy estimate the cover time for poly-mixing subgraphs. Finally we discuss complexity theoretic implications and future work

Towards completing the puzzle: complexity of control by replacing, adding, and deleting candidates or voters

We investigate the computational complexity of electoral control in elections. Electoral control describes the scenario where the election chair seeks to alter the outcome of the election by structural changes such as adding, deleting, or replacing either candidates or voters. Such control actions have been studied in the literature for a lot of prominent voting rules. We complement those results by solving several open cases for Copelandα, maximin, k-veto, plurality with runoff, veto with runoff, Condorcet, fallback, range voting, and normalized range voting

Rank Aggregation Using Scoring Rules

To aggregate rankings into a social ranking, one can use scoring systems such as Plurality, Veto, and Borda. We distinguish three types of methods: ranking by score, ranking by repeatedly choosing a winner that we delete and rank at the top, and ranking by repeatedly choosing a loser that we delete and rank at the bottom. The latter method captures the frequently studied voting rules Single Transferable Vote (aka Instant Runoff Voting), Coombs, and Baldwin. In an experimental analysis, we show that the three types of methods produce different rankings in practice. We also provide evidence that sequentially selecting winners is most suitable to detect the "true" ranking of candidates. For different rules in our classes, we then study the (parameterized) computational complexity of deciding in which positions a given candidate can appear in the chosen ranking. As part of our analysis, we also consider the Winner Determination problem for STV, Coombs, and Baldwin and determine their complexity when there are few voters or candidates.Comment: 47 pages including appendi

On Weakly and Strongly Popular Rankings

Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking pi of the candidates is at least as good as any other ranking sigma in the following sense: if we compare pi to sigma, at least half of all voters will always weakly prefer pi. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity---as applied to popular matchings, a well-established topic in computational social choice---is stricter, because it requires at least half of the voters who are not indifferent between pi and sigma to prefer pi. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zylen et al. We also point out connections to the famous open problem of finding a Kemeny consensus with 3 voters

Optimal majority rules and quantitative Condorcet properties of setwise Kemeny voting schemes

The important Kemeny problem, which consists of computing median consensus rankings of an election with respect to the Kemeny voting rule, admits important applications in biology and computational social choice and was generalized recently via an interesting setwise approach by Gilbert et. al. Our first results establish optimal quantitative extensions of the Unanimity property and the well-known $3/4$-majority rule of Betzler et al. for the classical Kemeny median problem. Moreover, by elaborating an exhaustive list of quantified axiomatic properties (such as the Condorcet and Smith criteria, the $5/6$-majority rule, etc.) of the $3$-wise Kemeny rule where not only pairwise comparisons but also the discordance between the winners of subsets of three candidates are also taken into account, we come to the conclusion that the $3$-wise Kemeny voting scheme induced by the $3$-wise Kendall-tau distance presents interesting advantages in comparison with the classical Kemeny rule. For example, it satisfies several improved manipulation-proof properties. Since the $3$-wise Kemeny problem is NP-hard, our results also provide some of the first useful space reduction techniques by determining the relative orders of pairs of alternatives. Our works suggest similar interesting properties of higher setwise Kemeny voting schemes which justify and compensate for the more expensive computational cost than the classical Kemeny scheme