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For the problem of aggregating several rankings into one ranking, Kemeny (1959) proposed two methods: the median rule which selects the ranking with the smallest total swap distance to the input rankings, and the mean rule which minimizes the squared swap distances to the input rankings. The median rule has been extensively studied since and is now known simply as Kemeny's rule. It exhibits majoritarian properties, so for example if more than half of the input rankings are the same, then the output of the rule is the same ranking. We observe that this behavior is undesirable in many rank aggregation settings. For example, when we rank objects by different criteria (quality, price, etc.) and want to aggregate them with specified weights for the criteria, then a criterion with weight 51% should have 51% influence on the output instead of 100%. We show that the Squared Kemeny rule (i.e., the mean rule) behaves this way, by establishing a bound on the distance of the output ranking to any input rankings, as a function of their weights. Furthermore, we give an axiomatic characterization of the Squared Kemeny rule, which mirrors the existing characterization of the Kemeny rule but replaces the majoritarian Condorcet axiom by a proportionality axiom. Finally, we discuss the computation of the rule and show its behavior in a simulation study., Comment: 45 pages including appendix

We formulate the problem of eliciting agents' preferences with the goal of finding a Kemeny ranking as a Dueling Bandits problem. Here the bandits' arms correspond to alternatives that need to be ranked and the feedback corresponds to a pairwise comparison between alternatives by a randomly sampled agent. We consider both sampling with and without replacement, i.e., the possibility to ask the same agent about some comparison multiple times or not. We find approximation bounds for Kemeny rankings dependant on confidence intervals over estimated winning probabilities of arms. Based on these we state algorithms to find Probably Approximately Correct (PAC) solutions and elaborate on their sample complexity for sampling with or without replacement. Furthermore, if all agents' preferences are strict rankings over the alternatives, we provide means to prune confidence intervals and thereby guide a more efficient elicitation. We formulate several adaptive sampling methods that use look-aheads to estimate how much confidence intervals (and thus approximation guarantees) might be tightened. All described methods are compared on synthetic data., Comment: This is a long version of the AAAI'24 publication under the same title

Traditional clustering algorithms often struggle to capture the complex relationships within graphs and generalise to arbitrary clustering criteria. The emergence of graph neural networks (GNNs) as a powerful framework for learning representations of graph data provides new approaches to solving the problem. Previous work has shown GNNs to be capable of proposing partitionings using a variety of criteria, however, these approaches have not yet been extended to work on Markov chains or kinetic networks. These arise frequently in the study of molecular systems and are of particular interest to the biochemical modelling community. In this work, we propose several GNN-based architectures to tackle the graph partitioning problem for Markov Chains described as kinetic networks. This approach aims to minimize how much a proposed partitioning changes the Kemeny constant. We propose using an encoder-decoder architecture and show how simple GraphSAGE-based GNNs with linear layers can outperform much larger and more expressive attention-based models in this context. As a proof of concept, we first demonstrate the method's ability to cluster randomly connected graphs. We also use a linear chain architecture corresponding to a 1D free energy profile as our kinetic network. Subsequently, we demonstrate the effectiveness of our method through experiments on a data set derived from molecular dynamics. We compare the performance of our method to other partitioning techniques such as PCCA+. We explore the importance of feature and hyperparameter selection and propose a general strategy for large-scale parallel training of GNNs for discovering optimal graph partitionings.

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is NP-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We first present a comprehensive relationship, both theoretical and empirical, among these parameters. Further, we study the problem of computing all distinct Kemeny rankings under the lens of parameterized complexity. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have FPT algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present FPT approximation algorithms for Kemeny rank aggregation with respect to these parameters.

Kemeny's rule is one of the most studied and well-known voting schemes with various important applications in computational social choice and biology. Recently, Kemeny's rule was generalized via a set-wise approach by Gilbert et. al. This paradigm presents interesting advantages in comparison with Kemeny's rule since not only pairwise comparisons but also the discordance between the winners of subsets of three alternatives are also taken into account in the definition of the $3$-wise Kendall-tau distance between two rankings. In spite of the NP-hardness of the 3-wise Kemeny problem which consists of computing the set of $3$-wise consensus rankings, namely rankings whose total $3$-wise Kendall-tau distance to a given voting profile is minimized, we establish in this paper several generalizations of the Major Order Theorems, as obtained by Milosz and Hamel for Kemeny's rule, for the $3$-wise Kemeny voting schemes to achieve a substantial search space reduction by efficiently determining in polynomial time the relative orders of pairs of alternatives. Essentially, our theorems quantify precisely the nontrivial property that if the preference for an alternative over another one in an election is strong enough, not only in the head-to-head competition but even when taking into account one or two more alternatives, then the relative order of these two alternatives in all $3$-wise consensus rankings must be as expected. As an application, we also obtain an improvement of the Major Order Theorems for Kememy's rule. Moreover, we show that the well-known $3/4$-majority rule of Betzler et al. for Kemeny's rule is only valid in general for elections with no more than $5$ alternatives with respect to the $3$-wise Kemeny scheme. Several simulations and tests of our algorithms on real-world and uniform data are provided., Comment: several improvements included

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.

For continuous-time ergodic Markov processes, the Kemeny time $\tau_*$ is the characteristic time needed to converge towards the steady state $P_*(x)$ : in real-space, the Kemeny time $\tau_*$ corresponds to the average of the Mean-First-Passage-Time $\tau(x,x_0) $ over the final configuration $x$ drawn with the steady state $P_*(x)$, which turns out to be independent of the initial configuration $x_0$; in the spectral domain, the Kemeny time $\tau_*$ corresponds to the sum of the inverses of all the non-vanishing eigenvalues $\lambda_n \ne 0 $ of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition : we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever., Comment: v2= final revised version (73 pages)

The Kemeny Rank Aggregation (KRA) problem is a well-studied problem in the field of Social Choice with a variety of applications in many different areas like databases and search engines. Intuitively, given a set of votes over a set of candidates, the problem asks to find an aggregated ranking of candidates that minimizes the overall dissatisfaction concerning the votes. Recently, a diverse version of KRA was considered which asks for a sufficiently diverse set of sufficiently good solutions. The framework of diversity of solutions is a young and thriving topic in the field of artificial intelligence. The main idea is to provide the user with not just one, but with a set of different solutions, enabling her to pick a sufficiently good solution that satisfies additional subjective criteria that are hard or impossible to model. In this work, we use a quantum annealer to solve the KRA problem and to compute a representative set of solutions. Quantum annealing is a meta search heuristic that does not only show promising runtime behavior on currently existing prototypes but also samples the solutions space in an inherently different way, making use of quantum effects. We describe how KRA instances can be solved by a quantum annealer and provide an implementation as well as experimental evaluations. As existing quantum annealers are still restricted in their number of qubits, we further implement two different data reduction rules that can split an instance into a set of smaller instances. In our evaluation, we compare classical heuristics that allow to sample multiple solutions such as simulated annealing and local search with quantum annealing performed on a physical quantum annealer. We compare runtime, quality of solution, and diversity of solutions, with and without applying preceding data reduction rules., Comment: 5 figures

A new measure $c(e)$ of the centrality of an edge $e$ in an undirected graph $G$ is introduced. It is based on the variation of the Kemeny constant of the graph after removing the edge $e$. The new measure is designed in such a way that the Braess paradox is avoided. A numerical method for computing $c(e)$ is introduced and a regularization technique is designed in order to deal with cut-edges and disconnected graphs. Numerical experiments performed on synthetic tests and on real road networks show that this measure is particularly effective in revealing bottleneck roads whose removal would greatly reduce the connectivity of the network.

The assumption that voters' preferences share some common structure is a standard way to circumvent NP-hardness results in social choice problems. While the Kemeny ranking problem is NP-hard in the general case, it is known to become easy if the preferences are 1-dimensional Euclidean. In this note, we prove that the Kemeny ranking problem remains NP-hard for $k$-dimensional Euclidean preferences with $k\!\ge\!2$ under norms $\ell_1$, $\ell_2$ and $\ell_\infty$, by showing that any weighted tournament (resp. weighted bipartite tournament) with weights of same parity (resp. even weights) is inducible as the weighted majority tournament of a profile of 2-Euclidean preferences under norm $\ell_2$ (resp. $\ell_1,\ell_{\infty}$), computable in polynomial time. More generally, this result regarding weighted tournaments implies, essentially, that hardness results relying on the (weighted) majority tournament that hold in the general case (e.g., NP-hardness of Slater ranking) are still true for 2-dimensional Euclidean preferences., Comment: Accepted at Discrete Applied Mathematics

In its most traditional setting, the main concern of optimization theory is the search for optimal solutions for instances of a given computational problem. A recent trend of research in artificial intelligence, called solution diversity, has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. When combined with techniques from parameterized complexity theory, the paradigm of diversity of solutions offers a powerful algorithmic framework to address problems of practical relevance. In this work, we investigate the impact of this combination in the field of Kemeny Rank Aggregation, a well-studied class of problems lying in the intersection of order theory and social choice theory and also in the field of order theory itself. In particular, we show that the Kemeny Rank Aggregation problem is fixed-parameter tractable with respect to natural parameters providing natural formalizations of the notions of diversity and of the notion of a sufficiently good solution. Our main results work both when considering the traditional setting of aggregation over linearly ordered votes, and in the more general setting where votes are partially ordered., Comment: Accepted to the 30th International Joint Conference on Artificial Intelligence (IJCAI 2021)

The computational study of election problems generally focuses on questions related to the winner or set of winners of an election. But social preference functions such as Kemeny rule output a full ranking of the candidates (a consensus). We study the complexity of consensus-related questions, with a particular focus on Kemeny and its qualitative version Slater. The simplest of these questions is the problem of determining whether a ranking is a consensus, and we show that this problem is coNP-complete. We also study the natural question of the complexity of manipulative actions that have a specific consensus as a goal. Though determining whether a ranking is a Kemeny consensus is hard, the optimal action for manipulators is to simply vote their desired consensus. We provide evidence that this simplicity is caused by the combination of election system (Kemeny), manipulative action (manipulation), and manipulative goal (consensus). In the process we provide the first completeness results at the second level of the polynomial hierarchy for electoral manipulation and for optimal solution recognition.

Kemeny's constant of a simple connected graph $G$ is the expected length of a random walk from $i$ to any given vertex $j \neq i$. We provide a simple method for computing Kemeny's constant for 1-separable via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on $n$ vertices maximizes Kemeny's constant for the class of undirected trees on $n$ vertices. Applying this method again, we simplify existing expressions for the Kemeny's constant of barbell graphs and demonstrate which barbell maximizes Kemeny's constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland et.~al.~\cite{kirkland2016kemeny} and Ciardo \cite{ciardo2020braess}.

The computational complexity of winner determination under common voting rules is a classical and fundamental topic in the field of computational social choice. Previous work has established the NP-hardness of winner determination under some commonly-studied voting rules, especially the Kemeny rule and the Slater rule. In a recent blue-sky paper, Baumeister, Hogrebe, and Rothe (2020) questioned the relevance of the worst-case nature of NP-hardness in social choice and proposed to conduct smoothed complexity analysis (Spielman and Teng 2009) under Blaser and Manthey (2015)'s framework. In this paper, we develop the first smoothed complexity results for winner determination in voting. We illustrate the inappropriateness of Blaser and Manthey (2015)'s smoothed complexity framework in social choice contexts by proving a paradoxical result, which states that the exponential-time brute force search algorithm is smoothed poly-time according to their definition. We then prove the smoothed hardness of Kemeny and Slater using the classical smoothed complexity analysis, and prove a parameterized typical-case smoothed easiness result for Kemeny. Overall, our results show that smoothed complexity analysis in computational social choice is a challenging and fruitful topic.

Testing, tracking and tracing abilities have been identified as pivotal in helping countries to safely reopen activities after the first wave of the COVID-19 virus. Contact tracing apps give the unprecedented possibility to reconstruct graphs of daily contacts, so the question is who should be tested? As human contact networks are known to exhibit community structure, in this paper we show that the Kemeny constant of a graph can be used to identify and analyze bridges between communities in a graph. Our "Kemeny indicator" is the change in Kemeny constant when a node or edge is removed from the graph. We show that testing individuals who are associated with large values of the Kemeny indicator can help in efficiently intercepting new virus outbreaks, when they are still in their early stage. Extensive simulations provide promising results in early identification and in blocking possible "super-spreaders" links that transmit disease between different communities.

In this paper, we advocate the use of setwise contests for aggregating a set of input rankings into an output ranking. We propose a generalization of the Kemeny rule where one minimizes the number of k-wise disagreements instead of pairwise disagreements (one counts 1 disagreement each time the top choice in a subset of alternatives of cardinality at most k differs between an input ranking and the output ranking). After an algorithmic study of this k-wise Kemeny aggregation problem, we introduce a k-wise counterpart of the majority graph. This graph reveals useful to divide the aggregation problem into several sub-problems, which enables to speed up the exact computation of a consensus ranking. By introducing a k-wise counterpart of the Spearman distance, we also provide a 2-approximation algorithm for the k-wise Kemeny aggregation problem. We conclude with numerical tests., Comment: 36 pages, extends a work published at AAAI 2020. Compared to the previous version on arXiv, some notations have been changed, and section 5 has been added

Markov processes are widely used models for investigating kinetic networks. Here we collate and present a variety of results pertaining to kinetic network models, in a unified framework. The aim is to lay out explicit links between several important quantities commonly studied in the field, including mean first passage times (MFPTs), correlation functions and the Kemeny constant, and highlight some of the subtleties which are often overlooked in the literature, while providing new insights. Results include (i) a simple physical interpretation of the Kemeny constant, (ii) a recipe to infer equilibrium distributions and rate matrices from measurements of MFPTs, potentially useful in applications, including milestoning in molecular dynamics, and (iii) a protocol to reduce the dimensionality of kinetic networks, based on specific requirements that the MFPTs in the coarse-grained system should satisfy. It is proven that this protocol coincides with the one proposed by Hummer and Szabo in [1] and it leads to a variational principle for the Kemeny constant. We hope that this study will serve as a useful reference for readers interested in theoretical aspects of kinetic networks, some of which underpin useful applications, including milestoning and coarse-graining., Comment: 9 pages