363 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
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
We study the problem of allocating indivisible items to agents with additive
valuations, under the additional constraint that bundles must be connected in
an underlying item graph. Previous work has considered the existence and
complexity of fair allocations. We study the problem of finding an allocation
that is Pareto-optimal. While it is easy to find an efficient allocation when
the underlying graph is a path or a star, the problem is NP-hard for many other
graph topologies, even for trees of bounded pathwidth or of maximum degree 3.
We show that on a path, there are instances where no Pareto-optimal allocation
satisfies envy-freeness up to one good, and that it is NP-hard to decide
whether such an allocation exists, even for binary valuations. We also show
that, for a path, it is NP-hard to find a Pareto-optimal allocation that
satisfies maximin share, but show that a moving-knife algorithm can find such
an allocation when agents have binary valuations that have a non-nested
interval structure.Comment: 21 pages, full version of paper at AAAI-201
Simple Causes of Complexity in Hedonic Games
Hedonic games provide a natural model of coalition formation among
self-interested agents. The associated problem of finding stable outcomes in
such games has been extensively studied. In this paper, we identify simple
conditions on expressivity of hedonic games that are sufficient for the problem
of checking whether a given game admits a stable outcome to be computationally
hard. Somewhat surprisingly, these conditions are very mild and intuitive. Our
results apply to a wide range of stability concepts (core stability, individual
stability, Nash stability, etc.) and to many known formalisms for hedonic games
(additively separable games, games with W-preferences, fractional hedonic
games, etc.), and unify and extend known results for these formalisms. They
also have broader applicability: for several classes of hedonic games whose
computational complexity has not been explored in prior work, we show that our
framework immediately implies a number of hardness results for them.Comment: 7+9 pages, long version of a paper in IJCAI 201
Causal Inference on Discrete Data using Additive Noise Models
Inferring the causal structure of a set of random variables from a finite
sample of the joint distribution is an important problem in science. Recently,
methods using additive noise models have been suggested to approach the case of
continuous variables. In many situations, however, the variables of interest
are discrete or even have only finitely many states. In this work we extend the
notion of additive noise models to these cases. We prove that whenever the
joint distribution \prob^{(X,Y)} admits such a model in one direction, e.g.
Y=f(X)+N, N \independent X, it does not admit the reversed model
X=g(Y)+\tilde N, \tilde N \independent Y as long as the model is chosen in a
generic way. Based on these deliberations we propose an efficient new algorithm
that is able to distinguish between cause and effect for a finite sample of
discrete variables. In an extensive experimental study we show that this
algorithm works both on synthetic and real data sets
backShift: Learning causal cyclic graphs from unknown shift interventions
We propose a simple method to learn linear causal cyclic models in the
presence of latent variables. The method relies on equilibrium data of the
model recorded under a specific kind of interventions ("shift interventions").
The location and strength of these interventions do not have to be known and
can be estimated from the data. Our method, called backShift, only uses second
moments of the data and performs simple joint matrix diagonalization, applied
to differences between covariance matrices. We give a sufficient and necessary
condition for identifiability of the system, which is fulfilled almost surely
under some quite general assumptions if and only if there are at least three
distinct experimental settings, one of which can be pure observational data. We
demonstrate the performance on some simulated data and applications in flow
cytometry and financial time series. The code is made available as R-package
backShift
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