4,032 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
Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results
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-time algorithm. For
the utilitarian version, we prove that winner determination remains NP-hard,
even for the Borda scoring function; however, a winning committee can be found
in polynomial time if either the number of leaves or the number of internal
vertices of the underlying tree is bounded by a constant. To benefit from these
positive results, we need a procedure that can determine whether a given
profile is single-peaked on a tree that has additional desirable properties
(such as, e.g., a small number of leaves). To address this challenge, we
develop a structural approach that enables us to compactly represent all trees
with respect to which a given profile is single-peaked. We show how to use this
representation to efficiently find the best tree for a given profile for use
with our winner determination algorithms: Given a profile, we can efficiently
find a tree with the minimum number of leaves, or a tree with the minimum
number of internal vertices among trees on which the profile is single-peaked.
We also consider several other optimization criteria for trees: for some we
obtain polynomial-time algorithms, while for others we show NP-hardness
results.Comment: 44 pages, extends works published at AAAI 2016 and IJCAI 201
Computational Aspects of Nearly Single-Peaked Electorates
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.Comment: Published in the Journal of Artificial Intelligence Research (JAIR).
A short version of this paper appeared in the proceedings of the
Twenty-Seventh AAAI Conference on Artificial Intelligence (AAAI 2013). An
even earlier version appeared in the proceedings of the Fourth International
Workshop on Computational Social Choice 2012 (COMSOC 2012
Parameterized Complexity of Multi-winner Determination: More Effort Towards Fixed-Parameter Tractability
We study the parameterized complexity of Winners Determination for three
prevalent -committee selection rules, namely the minimax approval voting
(MAV), the proportional approval voting (PAV), and the Chamberlin-Courant's
approval voting (CCAV). It is known that Winners Determination for these rules
is NP-hard. Moreover, these problems have been studied from the parameterized
complexity point of view with respect to some natural parameters recently.
However, many results turned out to be W[1]-hard or W[2]-hard. Aiming at
deriving more fixed-parameter algorithms, we revisit these problems by
considering more natural and important single parameters, combined parameters,
and structural parameters.Comment: 31 pages, 2 figures, AAMAS 201
Resolving the Complexity of Some Fundamental Problems in Computational Social Choice
This thesis is in the area called computational social choice which is an
intersection area of algorithms and social choice theory.Comment: Ph.D. Thesi
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