1,142 research outputs found
Fairness of Exposure in Rankings
Rankings are ubiquitous in the online world today. As we have transitioned
from finding books in libraries to ranking products, jobs, job applicants,
opinions and potential romantic partners, there is a substantial precedent that
ranking systems have a responsibility not only to their users but also to the
items being ranked. To address these often conflicting responsibilities, we
propose a conceptual and computational framework that allows the formulation of
fairness constraints on rankings in terms of exposure allocation. As part of
this framework, we develop efficient algorithms for finding rankings that
maximize the utility for the user while provably satisfying a specifiable
notion of fairness. Since fairness goals can be application specific, we show
how a broad range of fairness constraints can be implemented using our
framework, including forms of demographic parity, disparate treatment, and
disparate impact constraints. We illustrate the effect of these constraints by
providing empirical results on two ranking problems.Comment: In Proceedings of the 24th ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining, London, UK, 201
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Infinite ergodic theory and Non-extensive entropies
We bring into account a series of result in the infinite ergodic theory that
we believe that they are relevant to the theory of non-extensive entropie
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
Random Constraint Satisfaction Problems
Random instances of constraint satisfaction problems such as k-SAT provide
challenging benchmarks. If there are m constraints over n variables there is
typically a large range of densities r=m/n where solutions are known to exist
with probability close to one due to non-constructive arguments. However, no
algorithms are known to find solutions efficiently with a non-vanishing
probability at even much lower densities. This fact appears to be related to a
phase transition in the set of all solutions. The goal of this extended
abstract is to provide a perspective on this phenomenon, and on the
computational challenge that it poses
Characterization of quantum computable decision problems by state discrimination
One advantage of quantum algorithms over classical computation is the
possibility to spread out, process, analyse and extract information in
multipartite configurations in coherent superpositions of classical states.
This will be discussed in terms of quantum state identification problems based
on a proper partitioning of mutually orthogonal sets of states. The question
arises whether or not it is possible to encode equibalanced decision problems
into quantum systems, so that a single invocation of a filter used for state
discrimination suffices to obtain the result.Comment: 9 page
Stable Flags and the Riemann-Hilbert Problem
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page
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