187,159 research outputs found
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
The number of generalized balanced lines
Let be a set of red points and blue points in general
position in the plane, with . A line determined by them is said
to be balanced if in each open half-plane bounded by the difference
between the number of red points and blue points is . We show that every set
as above has at least balanced lines. The main techniques in the proof
are rotations and a generalization, sliding rotations, introduced here.Comment: 6 pages, 3 figures, several typos fixed, reference adde
The mixing time of the switch Markov chains: a unified approach
Since 1997 a considerable effort has been spent to study the mixing time of
switch Markov chains on the realizations of graphic degree sequences of simple
graphs. Several results were proved on rapidly mixing Markov chains on
unconstrained, bipartite, and directed sequences, using different mechanisms.
The aim of this paper is to unify these approaches. We will illustrate the
strength of the unified method by showing that on any -stable family of
unconstrained/bipartite/directed degree sequences the switch Markov chain is
rapidly mixing. This is a common generalization of every known result that
shows the rapid mixing nature of the switch Markov chain on a region of degree
sequences. Two applications of this general result will be presented. One is an
almost uniform sampler for power-law degree sequences with exponent
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.Comment: Clarification
Generalization Bounds via Information Density and Conditional Information Density
We present a general approach, based on an exponential inequality, to derive
bounds on the generalization error of randomized learning algorithms. Using
this approach, we provide bounds on the average generalization error as well as
bounds on its tail probability, for both the PAC-Bayesian and single-draw
scenarios. Specifically, for the case of subgaussian loss functions, we obtain
novel bounds that depend on the information density between the training data
and the output hypothesis. When suitably weakened, these bounds recover many of
the information-theoretic available bounds in the literature. We also extend
the proposed exponential-inequality approach to the setting recently introduced
by Steinke and Zakynthinou (2020), where the learning algorithm depends on a
randomly selected subset of the available training data. For this setup, we
present bounds for bounded loss functions in terms of the conditional
information density between the output hypothesis and the random variable
determining the subset choice, given all training data. Through our approach,
we recover the average generalization bound presented by Steinke and
Zakynthinou (2020) and extend it to the PAC-Bayesian and single-draw scenarios.
For the single-draw scenario, we also obtain novel bounds in terms of the
conditional -mutual information and the conditional maximal leakage.Comment: Published in Journal on Selected Areas in Information Theory (JSAIT).
Important note: the proof of the data-dependent bounds provided in the paper
contains an error, which is rectified in the following document:
https://gdurisi.github.io/files/2021/jsait-correction.pd
The Limits of Post-Selection Generalization
While statistics and machine learning offers numerous methods for ensuring
generalization, these methods often fail in the presence of adaptivity---the
common practice in which the choice of analysis depends on previous
interactions with the same dataset. A recent line of work has introduced
powerful, general purpose algorithms that ensure post hoc generalization (also
called robust or post-selection generalization), which says that, given the
output of the algorithm, it is hard to find any statistic for which the data
differs significantly from the population it came from.
In this work we show several limitations on the power of algorithms
satisfying post hoc generalization. First, we show a tight lower bound on the
error of any algorithm that satisfies post hoc generalization and answers
adaptively chosen statistical queries, showing a strong barrier to progress in
post selection data analysis. Second, we show that post hoc generalization is
not closed under composition, despite many examples of such algorithms
exhibiting strong composition properties
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