11 research outputs found
Recognising Multidimensional Euclidean Preferences
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 preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges
Computational Social Choice is an interdisciplinary research area involving
Economics, Political Science, and Social Science on the one side, and
Mathematics and Computer Science (including Artificial Intelligence and
Multiagent Systems) on the other side. Typical computational problems studied
in this field include the vulnerability of voting procedures against attacks,
or preference aggregation in multi-agent systems. Parameterized Algorithmics is
a subfield of Theoretical Computer Science seeking to exploit meaningful
problem-specific parameters in order to identify tractable special cases of in
general computationally hard problems. In this paper, we propose nine of our
favorite research challenges concerning the parameterized complexity of
problems appearing in this context
Structure in Dichotomous Preferences
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.Comment: A preliminary version appeared in the proceedings of IJCAI 2015, the
International Joint Conference on Artificial Intelligenc
Are there any nicely structured preference~profiles~nearby?
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
computationally intractable (NP-hard) questions
Collecting, Classifying, Analyzing, and Using Real-World Elections
We present a collection of real-world elections divided into
datasets from various sources ranging from sports competitions over music
charts to survey- and indicator-based rankings. We provide evidence that the
collected elections complement already publicly available data from the PrefLib
database, which is currently the biggest and most prominent source containing
real-world elections from datasets. Using the map of elections
framework, we divide the datasets into three categories and conduct an analysis
of the nature of our elections. To evaluate the practical applicability of
previous theoretical research on (parameterized) algorithms and to gain further
insights into the collected elections, we analyze different structural
properties of our elections including the level of agreement between voters and
election's distances from restricted domains such as single-peakedness. Lastly,
we use our diverse set of collected elections to shed some further light on
several traditional questions from social choice, for instance, on the number
of occurrences of the Condorcet paradox and on the consensus among different
voting rules
A deep exploration of the complexity border of strategic voting problems
Voting has found applications in a variety of areas. Unfortunately, in a voting activity there may exist strategic individuals who have incentives to attack the election by performing some strategic behavior. One possible way to address this issue is to use computational complexity as a barrier against the strategic behavior. The point is that if it is NP-hard to successfully perform a strategic behavior, the strategic individuals may give up their plan of attacking the election.
This thesis is concerned with strategic behavior in restricted elections, in the sense that the given elections are subject to some combinatorial restrictions. The goal is to find out how the complexity of the strategic behavior changes from the very restricted case to the general case.Abstimmungen werden auf verschiedene Gebiete angewendet. Leider kann es bei einer Abstimmung einzelne Teilnehmer geben, die Vorteile daraus ziehen, die Wahl durch strategisches Verhalten zu manipulieren. Eine Möglichkeit diesem Problem zu begegnen ist es, die Berechnungskomplexität als Hindernis gegen strategisches Verhalten zu nutzen. Die Annahme ist, dass falls es NP-schwer ist, um strategisches Verhalten erfolgreich anzuwenden, der strategisch Handelnde vielleicht den Plan aufgibt die Abstimmung zu attackieren.
Diese Arbeit befasst sich mit strategischem Vorgehen in eingeschränkten Abstimmungen in dem Sinne, dass die vorgegebenen Abstimmungen kombinatorischen Einschränkungen unterliegen. Ziel ist es herauszufinden, wie sich die Komplexität des strategischen Handelns von dem sehr eingeschränkten zu dem generellen Fall ändert
A Characterization of the Single−Peaked Single−Crossing Domain
We investigate elections that are simultaneously single-peaked and single-crossing (SPSC). We show that the domain of 1-dimensional Euclidean elections (where voters and candidates are points on the real line, and each voter prefers the candidates that are close to her to the ones that are further away) is a proper subdomain of the SPSC domain, by constructing an election that is single-peaked and single-crossing, but not 1-Euclidean. We then establish a connection between narcissistic elections (where each candidate is ranked first by at least one voter), single-peaked elections and single-crossing elections, by showing that an election is SPSC if and only if it can be obtained from a narcissistic single-crossing election by deleting voters. We show two applications of our characterization