Multiwinner voting captures a wide variety of settings, from parliamentary elections in democratic systems to product placement in online shopping platforms. There is a large body of work dealing with axiomatic characterizations, computational complexity, and algorithmic analysis of multiwinner voting rules. Although many challenges remain, significant progress has been made in showing existence of fair and representative outcomes as well as efficient algorithmic solutions for many commonly studied settings. However, much of this work focuses on single-shot elections, even though in numerous real-world settings elections are held periodically and repeatedly. Hence, it is imperative to extend the study of multiwinner voting to temporal settings. Recently, there have been several efforts to address this challenge. However, these works are difficult to compare, as they model multi-period voting in very different ways. We propose a unified framework for studying temporal fairness in this domain, drawing connections with various existing bodies of work, and consolidating them within a general framework. We also identify gaps in existing literature, outline multiple opportunities for future work, and put forward a vision for the future of multiwinner voting in temporal settings.

In this survey, we review the literature investigating participatory budgeting as a social choice problem. Participatory Budgeting (PB) is a democratic process in which citizens are asked to vote on how to allocate a given amount of public money to a set of projects. From a social choice perspective, it corresponds then to the problem of aggregating opinions about which projects should be funded, into a budget allocation satisfying a budget constraint. This problem has received substantial attention in recent years and the literature is growing at a fast pace. In this survey, we present the most important research directions from the literature, each time presenting a large set of representative results. We only focus on the indivisible case, that is, PB problems in which projects can either be fully funded or not at all. The aim of the survey is to present a comprehensive overview of the state of the research on PB. We aim at providing both a general overview of the main research questions that are being investigated, and formal and unified definitions of the most important technical concepts from the literature.

Computing minimal models is an important task in Knowledge Representation and Reasoning that appears in formalisms such as circumscription, diagnosis and answer set programming. Even the most basic question of whether there exists a minimal model containing a given variable is known to be $\Sigma_2 P$-complete. In this work we study the problem of computing minimal models from the viewpoint of parameterized complexity theory. We perform an extensive complexity analysis of this problem with respect to eleven parameters. Tractable fragments based on combinations of these parameters are identified by giving several fixed-parameter algorithms. For the remaining combinations we show parameterized hardness results and thus prove that under usual complexity theoretic assumptions no further fixed-parameter algorithms exist for these parameters.

We introduce the domain of preferences that are single-peaked on a circle, which is a generalization of the well-studied single-peaked domain. This preference restriction is useful, e.g., for scheduling decisions, certain facility location problems, and for one-dimensional decisions in the presence of extremist preferences. We give a fast recognition algorithm of this domain, provide a characterisation by finitely many forbidden subprofiles, and show that many popular single- and multi-winner voting rules are polynomial-time computable on this domain. In particular, we prove that Proportional Approval Voting can be computed in polynomial time for profiles that are single-peaked on a circle. In contrast, Kemeny's rule remains hard to evaluate, and several impossibility results from social choice theory can be proved using only profiles in this domain.

Incomplete preferences are likely to arise in real-world preference aggregation scenarios. This paper deals with determining whether an incomplete preference profile is single-peaked. This is valuable information since many intractable voting problems become tractable given singlepeaked preferences. We prove that the problem of recognizing single-peakedness is NP-complete for incomplete profiles consisting of partial orders. Despite this intractability result, we find several polynomial-time algorithms for reasonably restricted settings. In particular, we give polynomial-time recognition algorithms for weak orders, which can be viewed as preferences with indifference.

The winner determination problems of many attractive multi-winner voting rules are NP-complete. However, they often admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives efficient algorithms for finding optimal committees under Proportional Approval Voting (PAV) and the Chamberlin-Courant rule with single-peaked preferences, as well as for certain OWA-based rules. For PAV, this is the first technique able to efficiently find an optimal committee when preferences are single-peaked. An advantage of our approach is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise.

If voters' preferences are one-dimensional, many hard problems in computational social choice become tractable. A preference profile can be classified as one-dimensional if it has the single-crossing property, which requires that the voters can be ordered from left to right so that their preferences are consistent with this order. In practice, preferences may exhibit some one-dimensional structure, despite not being single-crossing in the formal sense. Hence, we ask whether one can identify preference profiles that are close to being single-crossing. We consider three distance measures, which are based on partitioning voters or candidates or performing a small number of swaps in each vote. We prove that it can be efficiently decided if voters can be split into two single-crossing groups. Also, for every fixed k >= 1 we can decide in polynomial time if a profile can be made single-crossing by performing at most k candidate swaps per vote. In contrast, for each k >= 3 it is NP-complete to decide whether candidates can be partitioned into k sets so that the restriction of the input profile to each set is single-crossing.

Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d -Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d > 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d > 1. We also study dichotomous preferences and the behaviour of other metrics, and survey a variety of related work.

Many hard computational social choice problems are known to become tractable when voters' preferences belong to a restricted domain, such as those of single-peaked or single-crossing preferences. However, to date, all algorithmic results of this type have been obtained for the setting where each voter's preference list is a total order of candidates. The goal of this paper is to extend this line of research to the setting where voters' preferences are dichotomous, i.e., each voter approves a subset of candidates and disapproves the remaining candidates. We propose several analogues of the notions of single-peaked and single-crossing preferences for dichotomous profiles and investigate the relationships among them. We then demonstrate that for some of these notions the respective restricted domains admit efficient algorithms for computationally hard approval-based multi-winner rules.