In the k-Kemeny problem, we are given an ordinal election, i.e., a collection of votes ranking the candidates from best to worst, and we seek the smallest number of swaps of adjacent candidates that ensure that the election has at most k different rankings. We study this problem for a number of structured domains, including the single-peaked, single-crossing, group-separable, and Euclidean ones. We obtain two kinds of results: (1) We show that k-Kemeny remains intractable under most of these domains, even for k=2, and (2) we use k-Kemeny to rank these domains in terms of their diversity.

We study the computational complexity of winner determination problems in approval-based committee elections under Thiele voting rules. These form a class of rules parameterized by a fixed weight vector that specifies how a voter's satisfaction depends on the number of approved candidates elected. We first analyze the structure of optimal solutions based on the sets of voters who approve each candidate---that is, how voters' approval ballots induce dependencies between candidates---revealing constraints on a winning committee under any fixed Thiele voting rule. Using this, we design FPT algorithms for Proportional Approval Voting (PAV) and other Thiele rules on a natural restricted domain known as the Voter Interval domain---that is, after a suitable ordering of voters, each candidate is approved by a consecutive interval of voters. In particular, we show that every Thiele rule on Voter Interval is FPT with respect to a parameter for which the problem is NP-hard on general instances, even when the parameter takes constant values. Our results advance the understanding of the computational complexity of PAV on Voter Interval instances, which remains one of the central open questions in this area. We further resolve two open questions from the literature on PAV (and other Thiele voting rules) by providing a polynomial-time algorithm for instances where each candidate is approved by at most two voters, and an FPT algorithm parameterized by the total score of a winning committee.

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and—in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

We study the complexity of candidate control in participatory budgeting elections. The goal of constructive candidate control is to ensure that a given candidate wins by either adding or deleting candidates from the election (in the destructive setting, the goal is to prevent a given candidate from winning). We show that such control problems are NP-hard to solve for many participatory budgeting voting rules, including Phragmén and Equal-Shares, but there are natural cases with polynomial-time algorithms. We also argue that control by deleting candidates is a useful tool for assessing the performance (or, strength) of initially losing projects, and we support this view with experiments on real-life PB instances.

We study approval-based committee voting from a novel perspective. While extant work largely centers around proportional representation of the voters, we shift our focus to the candidates while preserving proportionality. Intuitively, candidates supported by similar voter groups should receive comparable representation. Since deterministic voting rules cannot achieve this ideal, we develop randomized voting rules that satisfy ex-ante neutrality, monotonicity, and continuity, while maintaining strong ex-post proportionality guarantees.

We introduce and study isomorphic distances between ordinal elections (with the same numbers of candidates and voters). The main feature of these distances is that they are invariant to renaming the candidates and voters, and two elections are at distance zero if and only if they are isomorphic. Specifically, we consider isomorphic extensions of distances between preference orders: Given such a distance d , we extend it to distance d - ID between elections by unifying candidate names and finding a matching between the votes, so that the sum of the d -distances between the matched votes is as small as possible. We show that testing isomorphism of two elections can be done in polynomial time so, in principle, such distances can be tractable. Yet, we show that two very natural isomorphic distances are NP-complete and hard to approximate. We attempt to rectify the situation by showing FPT algorithms for several natural parameterizations.

In the Submodular Facility Location problem (SFL) we are given a collection of $n$ clients and $m$ facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility $f$ to which we assign the subset of clients $S^f$, one has to pay the opening cost $g(S^f)$, where $g(\cdot)$ is a monotone submodular function with $g(\emptyset)=0$. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [SODA'06] gave the current-best $O(\log n)$ approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an $O(\log\log n)$ approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the practically relevant generalizations of SFL where the opening cost of each facility $f$ is of the form $p_f+g(S^f)$ or $w_f\cdot g(S^f)$, where $p_f,w_f \geq 0$ are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility $f$ can be modelled with a submodular function of the set of clients assigned to $f$.

We study extensions of the Election Isomorphism problem, focused on the existence of isomorphic subelections. Specifically, we propose the Subelection Isomorphism and the Maximum Common Subelection problems and study their computational complexity and approximability. Using our problems in experiments, we provide some insights into the nature of several statistical models of elections

We consider the notions of agreement, diversity, and polarization in ordinal elections (that is, in elections where voters rank the candidates). While (computational) social choice offers good measures of agreement between the voters, such measures for the other two notions are lacking. We attempt to rectify this issue by designing appropriate measures, providing means of their (approximate) computation, and arguing that they, indeed, capture diversity and polarization well. In particular, we present "maps of preference orders" that highlight relations between the votes in a given election and which help in making arguments about their nature.

In this paper, we experimentally compare major approval based multiwinner voting rules. To this end, we define a measure of similarity between two equal sized committees subject to a given election. Using synthetic elections coming from several distributions, we analyze how similar are the committees provided by prominent voting rules. Our results can be visualized as maps of voting rules, which provide a counterpoint to a purely axiomatic classification of voting rules. The strength of our proposed method is its independence from preimposed classifications (such as the satisfaction of concrete axioms), and that it indeed offers a much finer distinction than the current state of axiomatic analysis.

We study extensions of the Election Isomorphism problem, focused on the existence of isomorphic subelections. Specifically, we propose the Subelection Isomorphism and the Maximum Common Subelection problems and study their computational complexity and approximability. Using our problems in experiments, we provide some insights into the nature of several statistical models of elections.

We present an almost optimal algorithm for the classic Chamberlin-Courant multiwinner voting rule (CC) on single-peaked preference profiles. Given n voters and m candidates, it runs in almost linear time in the input size improving the previous best O(nm^2) time algorithm. We also study multiwinner voting rules on nearly single-peaked preference profiles in terms of the candidate-deletion operation. We show a polynomial-time algorithm for CC where a given candidate-deletion set D has logarithmic size. Actually, our algorithm runs in 2^|D| * poly(n,m) time and the base of the power cannot be improved under the Strong Exponential Time Hypothesis. We also adapt these results to all non-constant Thiele rules which generalize CC with approval ballots.

We extend the map-of-elections framework to the case of approval elections. While doing so, we study a number of statistical cultures, including some new ones, and we analyze their properties. We find that approval elections can be understood in terms of the average number of approvals in the votes, and the extent to which the votes are chaotic.

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and---in addition to a global budget limit---there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to being hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates. When the input is the weighted majority graph, to our knowledge, the fastest algorithm for computing all winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in $Øh(m2.69) time, where m is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all winners. This running time is essentially optimal as it is nearly linear in the size of the weighted majority graph. If the input to the Schulze winners problem is not the weighted majority graph but the preference profile, then constructing the weighted majority graph is a bottleneck that increases the running time significantly; in the special case when there are m candidates and n=Øh(m) voters, the running time is Øh(m2.69), or $Øh(m2.5) if there is a nearly-linear time algorithm for multiplying dense square matrices. To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the weighted majority graph. As the Dominance Product problem is believed to require at least time r2.5-o(1) on r x r matrices, our equivalence implies that constructing the weighted majority graph in $Øh(m2.499) time for m candidates and n = Øh(m) voters would imply a breakthrough in the study of "intermediate" problems [Lincoln et al., 2020] in fine-grained complexity. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a winner. Our paper is the first to bring fine-grained complexity into the field of computational social choice. Previous approaches say nothing about lower bounds for problems that already have polynomial time algorithms. By bringing fine-grained complexity into the picture we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic.

Abstract In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial $\left (\frac {3}{2}+\varepsilon \right )$ 3 2 + ε -approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

We study a revenue maximization problem in the context of social networks. Namely, we generalize a model introduced by Alon, Mansour, and Tennenholtz [2] that captures inequity aversion, i.e., it captures the fact that prices offered to neighboring nodes should not differ significantly. We first provide approximation algorithms for a natural class of instances, where the total revenue is the sum of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for instance, to settings where the seller will only consider a fixed number of discount types or special offers. To complement our positive results, we resolve one of the open questions posed in [2] by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be large, or even when the number of allowed distinct prices is as small as three. Finally, we study extensions of the model regarding the demand type of the clients.

Participatory budgeting systems allow city residents to jointly decide on projects they wish to fund using public money, by letting residents vote on such projects. While participatory budgeting is gaining popularity, existing aggregation methods do not take into account the natural possibility of project interactions, such as substitution and complementarity effects. Here we take a step towards fixing this issue: First, we augment the standard model of participatory budgeting by introducing a partition over the projects and model the type and extent of project interactions within each part using certain functions. We study the computational complexity of finding bundles that maximize voter utility, as defined with respect to such functions. Motivated by the desire to incorporate project interactions in real-world participatory budgeting systems, we identify certain cases that admit efficient aggregation in the presence of such project interactions.

In approval-based multiwinner elections, we are given a set of voters, a set of candidates, and, for each voter, a set of candidates approved by the voter. The goal is to find a committee of size k that maximizes the total utility of the voters. In this paper, we study approximability of Thiele rules, which are known to be NP-hard to solve exactly. We provide a tight polynomial time approximation algorithm for a natural class of geometrically dominant weights that includes such voting rules as Proportional Approval Voting or p-Geometric. The algorithm is relatively simple: first we solve a linear program and then we round a solution by employing a framework called pipage rounding due to Ageev and Sviridenko (2004) and Calinescu et al. (2011). We provide a matching lower bound via a reduction from the Label Cover problem. Moreover, assuming a conjecture called Gap-ETH, we show that better approximation ratio cannot be obtained even in time f(k)*pow(n,o(k)).

In the SHIFT-BRIBERY problem we are given an election, a preferred candidate, and the costs of shifting this preferred candidate up the voters’ preference orders. The goal is to find such a set of shifts that ensures that the preferred candidate wins the election. We give the first polynomial-time approximation scheme for the case of positional scoring rules, and for the Copeland rule we show strong inapproximability results.

We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables.

We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. Obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev (2017) were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular weight vector.

We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance d from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time O*(2o(d log d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O*(d2d) algorithm of Misra, Nabeel and Singh is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O*((3/ε)2d), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time nO(1/ε2⋅log(1/ε))⋅poly(m), where n is a number of voters and m is a number of alternatives. It almost matches the running time of the fastest known PTAS for Closest String due to Ma and Sun.

We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O⋆(2o(d log d), unless the Exponential Time Hypothesis (ETH) fails. This means that the O⋆(d2d) algorithm of Misra et al. (AAMAS 2015) is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O⋆((3/ε)2d), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs in time nO(1/ε2·log(1/ε))· poly(m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun (SIAM J. Comp. 2009).