Modern recommendation systems often face the challenge of personalization at scale--learning individual user preferences while simultaneously satisfying global resource allocation constraints. To illustrate, consider a content platform that must decide which content creators to commission daily, where each creator has a different cost and produces ephemeral content on specific topics. Each user has preferences over all creators' content styles and topics. After commissioning a subset of creators that fit the platform's budget, it matches each user to content from one of these creators, where the same creator's content can be recommended to multiple users. The challenge lies in learning individual user preferences for each creator's content while selecting which creators to commission and how to assign their content to maximize user satisfaction. This problem fits the combinatorial multi-armed bandit framework, where the decision-maker must choose structured action sets [8], such as assigning each user to an item. The goal is to maximize cumulative reward, or equivalently, minimize regret by balancing exploration and exploitation. Unfortunately, combinatorial problems like the one in the example above are NP-complete even for offline settings. Therefore, traditional approaches settle for weaker notions of α-regret [8], competing against the best ef-All authors contributed equally to this work.

We take the classic facility location problem and consider a variation, in which each agent's individual cost function is equal to their distance from the facility multiplied by a scaling factor which is determined by the facility placement. In addition to the general class of continuous scaling functions, we also provide results for piecewise linear scaling functions which can effectively approximate or model the scaling of many real world scenarios. We focus on the objectives of total and maximum cost, describing the computation of the optimal solution. We then move to the approximate mechanism design setting, observing that the agents' preferences may no longer be single-peaked. Consequently, we characterize the conditions on scaling functions which ensure that agents have single-peaked preferences. Under these conditions, we find results on the total and maximum cost approximation ratios achievable by strategyproof and anonymous mechanisms.

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.

Recently, the problem of allocating one resource per agent with initial endowments (house markets) has seen a renewed interest: indeed, while in the domain of strict preferences the Top Trading Cycle algorithm is known to be the only procedure guaranteeing Pareto-optimality, individual rationality, and strategy proofness. However, the situation differs in the single-peaked domain. Indeed, Bade presented the Crawler, an alternative procedure enjoying the same properties, with the additional advantage of being implementable in obviously dominant strategies. In this paper we further investigate the Crawler and propose the Diver, a variant which checks optimally whether an allocation is Pareto-optimal for single-peaked preferences, thus improving over known techniques used for checking Pareto-optimality in more general domains. We also prove that the Diver is asymptotically optimal in terms of communication complexity.

Recently, the problem of allocating one resource per agent with initial endowments (\emph{house markets}) has seen a renewed interest: indeed, while in the general domain Top Trading Cycle is known to be the only procedure guaranteeing Pareto-optimality, individual rationality, and strategy proofness, the situation differs in single-peaked domains. Bade (2019) presented the Crawler, an alternative procedure enjoying the same properties (with the additional advantage of being implementable in obviously dominant strategies); while Damamme et al. (2015) showed that allowing mutually beneficial swap-deals among the agents was already enough to guarantee Pareto-optimality. In this paper we significantly deepen our understanding of this decentralized procedures: we show in particular that the single-peaked domains happen to be ``maximal'' if one wishes to guarantee this convergence property. Interestingly, we also observe that the set of allocations reachable by swap-deals always contains the outcome of the Crawler. To further investigate how these different mechanisms compare, we pay special attention to the average and minimum rank of the resource obtained by the agents in the outcome allocation. We provide theoretical bounds on the loss potentially induced by these procedures with respect to these criteria, and complement these results with an extensive experimental study which shows how different variants of swap dynamics behave. In fact, even the simplest dynamics exhibit very good results, and it is possible to further guide the process towards our objectives, if one is ready to sacrifice a bit in terms of decentralization. On our way, we also show that a simple variant of the Crawler allows to check efficiently that an allocation is Pareto-optimal in single-peaked domains.

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.

Manipulation, bribery, and control are well-studied ways of changing the outcome of an election. Many voting rules are, in the general case, computationally resistant to some of these manipulative actions. However when restricted to single-peaked electorates, these rules suddenly become easy to manipulate. Recently, Faliszewski, Hemaspaandra, and Hemaspaandra studied the computational complexity of strategic behavior in nearly single-peaked electorates. These are electorates that are not single-peaked but close to it according to some distance measure. In this paper we introduce several new distance measures regarding single-peakedness. We prove that determining whether a given profile is nearly single-peaked is NP-complete in many cases. For one case we present a polynomial-time algorithm. In case the single-peaked axis is given, we show that determining the distance is always possible in polynomial time. Furthermore, we explore the relations between the new notions introduced in this paper and existing notions from the literature.

This paper considers a mechanism design problem for locating two identical facilities on an interval, in which an agent can pretend to be multiple agents. A mechanism selects a pair of locations on the interval according to the declared single-peaked preferences of agents. An agent's utility is determined by the location of the better one (typically the closer to her ideal point). This model can represent various application domains. For example, assume a company is going to release two models of its product line and performs a questionnaire survey in an online forum to determine their detailed specs. Typically, a customer will buy only one model, but she can answer multiple times by logging onto the forum under several email accounts. We first characterize possible outcomes of mechanisms that satisfy false-name-proofness, as well as some mild conditions. By extending the result, we completely characterize the class of false-name-proof mechanisms when locating two facilities on a circle. We then clarify the approximation ratios of the false-name-proof mechanisms on a line metric for the social and maximum costs.

We must properly model attacks and the preferences of the electorate for the computational study of attacks on elections to give us insight into the hardness of attacks in practice. Theoretical and empirical analysis are equally important methods to understand election attacks. I discuss my recent work on domain restrictions on partial preferences and on new election attacks. I propose further study into modeling realistic election attacks and the advancement of the current state of empirical analysis of their hardness by using more advanced statistical techniques.