2,127 research outputs found
Robust Bounds on Choosing from Large Tournaments
Tournament solutions provide methods for selecting the "best" alternatives
from a tournament and have found applications in a wide range of areas.
Previous work has shown that several well-known tournament solutions almost
never rule out any alternative in large random tournaments. Nevertheless, all
analytical results thus far have assumed a rigid probabilistic model, in which
either a tournament is chosen uniformly at random, or there is a linear order
of alternatives and the orientation of all edges in the tournament is chosen
with the same probabilities according to the linear order. In this work, we
consider a significantly more general model where the orientation of different
edges can be chosen with different probabilities. We show that a number of
common tournament solutions, including the top cycle and the uncovered set, are
still unlikely to rule out any alternative under this model. This corresponds
to natural graph-theoretic conditions such as irreducibility of the tournament.
In addition, we provide tight asymptotic bounds on the boundary of the
probability range for which the tournament solutions select all alternatives
with high probability.Comment: Appears in the 14th Conference on Web and Internet Economics (WINE),
201
Hamilton decompositions of regular expanders: applications
In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Robust classification via MOM minimization
We present an extension of Vapnik's classical empirical risk minimizer (ERM)
where the empirical risk is replaced by a median-of-means (MOM) estimator, the
new estimators are called MOM minimizers. While ERM is sensitive to corruption
of the dataset for many classical loss functions used in classification, we
show that MOM minimizers behave well in theory, in the sense that it achieves
Vapnik's (slow) rates of convergence under weak assumptions: data are only
required to have a finite second moment and some outliers may also have
corrupted the dataset.
We propose an algorithm inspired by MOM minimizers. These algorithms can be
analyzed using arguments quite similar to those used for Stochastic Block
Gradient descent. As a proof of concept, we show how to modify a proof of
consistency for a descent algorithm to prove consistency of its MOM version. As
MOM algorithms perform a smart subsampling, our procedure can also help to
reduce substantially time computations and memory ressources when applied to
non linear algorithms.
These empirical performances are illustrated on both simulated and real
datasets
Flashes and Rainbows in Tournaments
Colour the edges of the complete graph with vertex set
with an arbitrary number of colours. What is the smallest integer such
that if then there must exist a monotone monochromatic path of
length or a monotone rainbow path of length ? Lefmann, R\"{o}dl, and
Thomas conjectured in 1992 that and proved this for . We prove the conjecture for
and establish the general upper bound . This reduces the gap between the best lower and upper bounds from
exponential to polynomial in . We also generalise some of these results to
the tournament setting.Comment: 14 page
Error-Correcting Tournaments
We present a family of pairwise tournaments reducing -class classification
to binary classification. These reductions are provably robust against a
constant fraction of binary errors. The results improve on the PECOC
construction \cite{SECOC} with an exponential improvement in computation, from
to , and the removal of a square root in the regret
dependence, matching the best possible computation and regret up to a constant.Comment: Minor wording improvement
Approximate Hamilton decompositions of robustly expanding regular digraphs
We show that every sufficiently large r-regular digraph G which has linear
degree and is a robust outexpander has an approximate decomposition into
edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint
Hamilton cycles. Here G is a robust outexpander if for every set S which is not
too small and not too large, the `robust' outneighbourhood of S is a little
larger than S. This generalises a result of K\"uhn, Osthus and Treglown on
approximate Hamilton decompositions of dense regular oriented graphs. It also
generalises a result of Frieze and Krivelevich on approximate Hamilton
decompositions of quasirandom (di)graphs. In turn, our result is used as a tool
by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G
which has linear degree and is a robust outexpander even has a Hamilton
decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44
pages, 2 figure
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