This paper is devoted to a study of single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of a set of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the sense that all the preferences are traversals of G. Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation (ILP) of the problem of minimizing the number of edges in G or the maximum degree of a vertex in G. We prove that both problems are NP-hard in the general case. However, we show that if the optimal number of edges is m-1 (where m is the number of candidates) then any optimal solution of the ILP is integer and thus the integrality constraints can be relaxed. This provides an alternative proof of the polynomial-time complexity of recognizing single-peaked preferences on a tree. We prove the same result for the case of a path (an axis), providing here also an alternative proof of polynomiality of the recognition problem. Furthermore, we provide a polynomial-time procedure to recognize single-peaked preferences on a pseudotree (a connected graph that contains at most one cycle). We also give some experimental results, both on real and synthetic datasets.
Single-peakedness is one of the most important and well-known domain restrictions on preferences. The computational study of single-peaked electorates has largely been restricted to elections with tie-free votes, and recent work that studies the computational complexity of manipulative attacks for single-peaked elections for votes with ties has been restricted to nonstandard models of single-peaked preferences for top orders. We study the computational complexity of manipulation for votes with ties for the standard model of single-peaked preferences and for single-plateaued preferences. We sho…
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 single-peaked 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 alg…
A preference system $\mathcal{I}$ is an undirected graph where vertices have preferences over their neighbors, and $\mathcal{I}$ admits a master list if all preferences can be derived from a single ordering over all vertices. We study the problem of deciding whether a given preference system $\mathcal{I}$ is close to admitting a master list based on three different distance measures. We determine the computational complexity of the following questions: can $\mathcal{I}$ be modified by (i) $k$ swaps in the preferences, (ii) $k$ edge deletions, or (iii) $k$ vertex deletions so that the resulting…
Single-peaked preferences and domains are extensively researched in social science and economics. In this study, we examine the interval property as well as combinatorial structure of single-peaked preferences on a fixed Left-Right social axis. We introduce a sign representation of single-peaked preferences; consequently, some cardinalities of single-peaked domains are easily obtained. Basic operations on the sign representation, which completely define the Bruhat poset, are also provided. The applications to known results and an isomorphic relation with associated rhombus tiling are given. Fi…
In economics and social choice single-peakedness is one of the most important and commonly studied models for preferences. It is well known that single-peaked consistency for total orders is in P. However in practice a preference profile is not always comprised of total orders. Often voters have indifference between some of the candidates. In a weak preference order indifference must be transitive. We show that single-peaked consistency for weak orders is in P for three different variants of single-peakedness for weak orders. Specifically, we consider Black's original definition of single-peak…
Indian Institute of Science（印度科学学院）；Indian Institute of Technology, Gandhinagar（印度理工学院甘地纳加尔分校）
In multiagent systems, we often have a set of agents each of which have a preference ordering over a set of items and one would like to know these preference orderings for various tasks, for example, data analysis, preference aggregation, voting etc. However, we often have a large number of items which makes it impractical to ask the agents for their complete preference ordering. In such scenarios, we usually elicit these agents' preferences by asking (a hopefully small number of) comparison queries --- asking an agent to compare two items. Prior works on preference elicitation focus on unrest…
We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations. We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. We mainly focus on the case where peaks are equidistant from the agents' locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this se…
Much work has been devoted, during the past twenty years, to using complexity to protect elections from manipulation and control. Many results have been obtained showing NP-hardness shields, and recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societal preferences the complexity shield never existed in the first place. In particular, we show that for electorates having …
The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates tha…
We study the cake-cutting problem when agents have single-peaked preferences over the cake. We show that a recently proposed mechanism by Wang-Wu (2019) to obtain envy-free allocations can yield large welfare losses. Using a simplifying assumption, we characterize all Pareto optimal allocations, which have a simple structure: are peak-preserving and non-wasteful. Finally, we provide simple alternative mechanisms that Pareto dominate that of Wang-Wu, and which achieve envy-freeness or Pareto optimality.
Voting is a general method for aggregating the preferences of multiple agents. Each agent ranks all the possible alternatives, and based on this, an aggregate ranking of the alternatives (or at least a winning alternative) is produced. However, when there are many alternatives, it is impractical to simply ask agents to report their complete preferences. Rather, the agents preferences, or at least the relevant parts thereof, need to be elicited. This is done by asking the agents a (hopefully small) number of simple queries about their preferences, such as comparison queries, which ask an agent …
We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. In particular, we show the following. 1. Every preference profile with up to two voters is one-dimensional Euclidean if and only if it is single-peaked. 2. Every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is single-peaked and single-crossing. By [12], we thus obtain that the smallest single-peaked and single-crossing preference profiles that are not one-dimensional Euclidean consist of three voters and six alternatives.
This paper contains an extensive combinatorial analysis of the single-peaked domain restriction and investigates the likelihood that an election is single-peaked. We provide a very general upper bound result for domain restrictions that can be defined by certain forbidden configurations. This upper bound implies that many domain restrictions (including the single-peaked restriction) are very unlikely to appear in a random election chosen according to the Impartial Culture assumption. For single-peaked elections, this upper bound can be refined and complemented by a lower bound that is asymptot…
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 committ…
A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin-Courant rule for preferences single-peaked on trees. We show that the egalitarian version of this problem admits a polynomial-t…
LIP6, Sorbonne Université（巴黎索邦大学LIP6实验室）；ILLC, University of Amsterdam（阿姆斯特丹大学ILLC研究所）
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 t…
For object reallocation problems, if preferences are strict but otherwise unrestricted, the Top Trading Cycles rule (TTC) is the leading rule: It is the only rule satisfying efficiency, individual rationality, and strategy-proofness. However, on the subdomain of single-peaked preferences, Bade (2019) defines a new rule, the "crawler", which also satisfies these three properties. (i) The crawler selects an allocation by "visiting" agents in a specific order. A natural "dual" rule can be defined by proceeding in the reverse order. Our first theorem states that the crawler and its dual are actual…
We investigate the complexity of {\sc{Constructive Control by Adding/Deleting Votes}} (CCAV/CCDV) for $r$-approval, Condorcet, Maximin and Copeland$^α$ in $k$-axes and $k$-candidates partition single-peaked elections. In general, we prove that CCAV and CCDV for most of the voting correspondences mentioned above are NP-hard even when~$k$ is a very small constant. Exceptions are CCAV and CCDV for Condorcet and CCAV for $r$-approval in $k$-axes single-peaked elections, which we show to be fixed-parameter tractable with respect to~$k$. In addition, we give a polynomial-time algorithm for recognizi…
We consider the problem of locating a single facility on a vertex in a given graph based on agents' preferences, where the domain of the preferences is either single-peaked or single-dipped. Our main interest is the existence of deterministic social choice functions (SCFs) that are Pareto efficient and false-name-proof, i.e., resistant to fake votes. We show that regardless of whether preferences are single-peaked or single-dipped, such an SCF exists (i) for any tree graph, and (ii) for a cycle graph if and only if its length is less than six. We also show that when the preferences are single-…
Understanding when and how computational complexity can be used to protect elections against different manipulative actions has been a highly active research area over the past two decades. A recent body of work, however, has shown that many of the NP-hardness shields, previously obtained, vanish when the electorate has single-peaked or nearly single-peaked preferences. In light of these results, we investigate whether it is possible to reimpose NP-hardness shields for such electorates by allowing the voters to specify partial preferences instead of insisting they cast complete ballots. In par…
In some preference aggregation scenarios, voters' preferences are highly structured: e.g., the set of candidates may have one-dimensional structure (so that voters' preferences are single-peaked) or be described by a binary decision tree (so that voters' preferences are group-separable). However, sometimes a single axis or a decision tree is insufficient to capture the voters' preferences; rather, there is a small number $k$ of axes or decision trees such that each vote in the profile is consistent with one of these axes (resp., trees). In this work, we study the complexity of deciding whether…
Indian Institute of Science（印度科学学院）；IIT, Kanpur（坎普尔印度理工学院）；ISI, Kolkata（加尔各答印度统计研究所）
We study fairness in social choice settings under single-peaked preferences. Construction and characterization of social choice rules in the single-peaked domain has been extensively studied in prior works. In fact, in the single-peaked domain, it is known that unanimous and strategy-proof deterministic rules have to be min-max rules and those that also satisfy anonymity have to be median rules. Further, random social choice rules satisfying these properties have been shown to be convex combinations of respective deterministic rules. We non-trivially add to this body of results by including fa…
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 noti…
We consider opinion diffusion for undirected networks with sequential updates when the opinions of the agents are single-peaked preference rankings. Our starting point is the study of preserving single-peakedness. We identify voting rules that, when given a single-peaked profile, output at least one ranking that is single peaked w.r.t. a single-peaked axis of the input. For such voting rules we show convergence to a stable state of the diffusion process that uses the voting rule as the agents' update rule. Further, we establish an efficient algorithm that maximises the spread of extreme opinio…
In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is a family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show their hierarchical structure, and analyze clone structures in single-peaked and single-crossing elections. We give a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to bec…
We investigate the problem of deciding whether a given preference profile is close to having a certain nice structure, as for instance single-peaked, single-caved, single-crossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or group-separable profiles. We measure this distance by the number of voters or alternatives that have to be deleted to make the profile a nicely structured one. Our results classify the problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial-time solvable) and computa…
We show that one-dimensional Euclidean preference profiles can not be characterized in terms of finitely many forbidden substructures. This result is in strong contrast to the case of single-peaked and single-crossing preference profiles, for which such finite characterizations have been derived in the literature.
We consider the problem of predicting winners in elections, for the case where we are given complete knowledge about all possible candidates, all possible voters (together with their preferences), but where it is uncertain either which candidates exactly register for the election or which voters cast their votes. Under reasonable assumptions, our problems reduce to counting variants of election control problems. We either give polynomial-time algorithms or prove #P-completeness results for counting variants of control by adding/deleting candidates/voters for Plurality, k-Approval, Approval, Co…
We study efficiency in general collective choice problems where agents have ordinal preferences and randomization is allowed. We explore the structure of preference profiles where ex-ante and ex-post efficiency coincide, offer a unifying perspective on the known results, and give several new characterizations. The results have implications for well-studied mechanisms including random serial dictatorship and a number of specific environments, including the dichotomous, single-peaked, and social choice domains.
We investigate the complexity of $r$-Approval control problems in $k$-peaked elections, where at most $k$ peaks are allowed in each vote with respect to an order of the candidates. We show that most NP-hardness results in general elections also hold in k-peaked elections even for $k=2,3$. On the other hand, we derive polynomial-time algorithms for some problems for $k=2$. All our NP-hardness results apply to Approval and sincere-strategy preference-based Approval as well. Our study leads to many dichotomy results for the problems considered in this paper, with respect to the values of $k$ and …
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 of Betzler et al. (2013). 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 $2^{|D|} \cdot poly(n,m)$ t…
When Graph Traversal Meets Structured Preferences: Unified Framework and Complexity Results
School of Computer and Communication Engineering, Changsha University of Science and Technology（长沙理工大学计算机与通信工程学院）；Department of Economics, Saarland University（萨尔兰大学经济学系）
Indian Institute of Science（印度科学学院）；Indian Institute of Technology, Gandhinagar（印度理工学院甘地纳加尔分校）
Eliciting the preferences of a set of agents over a set of alternatives is a problem of fundamental importance in social choice theory. Prior work on this problem has studied the query complexity of preference elicitation for the unrestricted domain and for the domain of single peaked preferences. In this paper, we consider the domain of single crossing preference profiles and study the query complexity of preference elicitation under various settings. We consider two distinct situations: when an ordering of the voters with respect to which the profile is single crossing is known versus when i…
We study the setting of committee elections, where a group of individuals needs to collectively select a given size subset of available objects. This model is relevant for a number of real-life scenarios including political elections, participatory budgeting, and facility-location. We focus on the core -- the classic notion of proportionality, stability and fairness. We show that for a number of restricted domains including voter-interval, candidate-interval, single-peaked, and single-crossing preferences the core is non-empty and can be found in polynomial time. We show that the core might be…
Indian Institute of Technology Kharagpur（印度理工学院卡哈拉格普尔分校）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；Indian Institute of Technology（印度理工学院）
A preferential domain is a collection of sets of preferences which are linear orders over a set of alternatives. These domains have been studied extensively in social choice theory due to both its practical importance and theoretical elegance. Examples of some extensively studied preferential domains include single peaked, single crossing, Euclidean, etc. In this paper, we study the sample complexity of testing whether a given preference profile is close to some specific domain. We consider two notions of closeness: (a) closeness via preferences, and (b) closeness via alternatives. We further …
We consider the stable matching problem when the preference lists are not given explicitly but are represented in a succinct way and ask whether the problem becomes computationally easier and investigate other implications. We give subquadratic algorithms for finding a stable matching in special cases of natural succinct representations of the problem, the $d$-attribute, $d$-list, geometric, and single-peaked models. We also present algorithms for verifying a stable matching in the same models. We further show that for $d = ω(\log n)$ both finding and verifying a stable matching in the $d$-att…
The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barberà and Pattanaik tell us that some desirable axioms - namely dominance and (strict) independence - are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty se…
