3,491 research outputs found
Designing Fair Ranking Schemes
Items from a database are often ranked based on a combination of multiple
criteria. A user may have the flexibility to accept combinations that weigh
these criteria differently, within limits. On the other hand, this choice of
weights can greatly affect the fairness of the produced ranking. In this paper,
we develop a system that helps users choose criterion weights that lead to
greater fairness.
We consider ranking functions that compute the score of each item as a
weighted sum of (numeric) attribute values, and then sort items on their score.
Each ranking function can be expressed as a vector of weights, or as a point in
a multi-dimensional space. For a broad range of fairness criteria, we show how
to efficiently identify regions in this space that satisfy these criteria.
Using this identification method, our system is able to tell users whether
their proposed ranking function satisfies the desired fairness criteria and, if
it does not, to suggest the smallest modification that does. We develop
user-controllable approximation that and indexing techniques that are applied
during preprocessing, and support sub-second response times during the online
phase. Our extensive experiments on real datasets demonstrate that our methods
are able to find solutions that satisfy fairness criteria effectively and
efficiently
A Smooth Transition from Powerlessness to Absolute Power
We study the phase transition of the coalitional manipulation problem for
generalized scoring rules. Previously it has been shown that, under some
conditions on the distribution of votes, if the number of manipulators is
, where is the number of voters, then the probability that a
random profile is manipulable by the coalition goes to zero as the number of
voters goes to infinity, whereas if the number of manipulators is
, then the probability that a random profile is manipulable
goes to one. Here we consider the critical window, where a coalition has size
, and we show that as goes from zero to infinity, the limiting
probability that a random profile is manipulable goes from zero to one in a
smooth fashion, i.e., there is a smooth phase transition between the two
regimes. This result analytically validates recent empirical results, and
suggests that deciding the coalitional manipulation problem may be of limited
computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains
minor changes after comments of reviewer
Ranking patterns of the unfolding model and arrangements
In the unidimensional unfolding model, given m objects in general position
there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking
pattern of the m given objects. By changing these m objects, we can generate
various ranking patterns. It is natural to ask how many ranking patterns can be
generated and what is the probability of each ranking pattern when the objects
are randomly chosen? These problems are studied by introducing a new type of
arrangement called mid-hyperplane arrangement and by counting cells in its
complement.Comment: 29 pages, 2 figure
COMs: Complexes of Oriented Matroids
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured
them as asymmetric counterparts of oriented matroids, both sharing the key
property of strong elimination. Moreover, symmetry of faces holds in both
structures as well as in the so-called affine oriented matroids. These two
fundamental properties (formulated for covectors) together lead to the natural
notion of "conditional oriented matroid" (abbreviated COM). These novel
structures can be characterized in terms of three cocircuits axioms,
generalizing the familiar characterization for oriented matroids. We describe a
binary composition scheme by which every COM can successively be erected as a
certain complex of oriented matroids, in essentially the same way as a lopsided
set can be glued together from its maximal hypercube faces. A realizable COM is
represented by a hyperplane arrangement restricted to an open convex set. Among
these are the examples formed by linear extensions of ordered sets,
generalizing the oriented matroids corresponding to the permutohedra. Relaxing
realizability to local realizability, we capture a wider class of combinatorial
objects: we show that non-positively curved Coxeter zonotopal complexes give
rise to locally realizable COMs.Comment: 40 pages, 6 figures, (improved exposition
RRR: Rank-Regret Representative
Selecting the best items in a dataset is a common task in data exploration.
However, the concept of "best" lies in the eyes of the beholder: different
users may consider different attributes more important, and hence arrive at
different rankings. Nevertheless, one can remove "dominated" items and create a
"representative" subset of the data set, comprising the "best items" in it. A
Pareto-optimal representative is guaranteed to contain the best item of each
possible ranking, but it can be almost as big as the full data. Representative
can be found if we relax the requirement to include the best item for every
possible user, and instead just limit the users' "regret". Existing work
defines regret as the loss in score by limiting consideration to the
representative instead of the full data set, for any chosen ranking function.
However, the score is often not a meaningful number and users may not
understand its absolute value. Sometimes small ranges in score can include
large fractions of the data set. In contrast, users do understand the notion of
rank ordering. Therefore, alternatively, we consider the position of the items
in the ranked list for defining the regret and propose the {\em rank-regret
representative} as the minimal subset of the data containing at least one of
the top- of any possible ranking function. This problem is NP-complete. We
use the geometric interpretation of items to bound their ranks on ranges of
functions and to utilize combinatorial geometry notions for developing
effective and efficient approximation algorithms for the problem. Experiments
on real datasets demonstrate that we can efficiently find small subsets with
small rank-regrets
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