Combinatorial Testing Problems

@inproceedings{AddarioBerry2010CombinatorialTP,
  title={Combinatorial Testing Problems},
  author={Louigi Addario-Berry and Nicolas Broutin and Luc Devroye and G{\'a}bor Lugosi},
  year={2010}
}
Presented on October 15, 2018 from 12:00 p.m.-1:00 p.m. in the Groseclose Building, Room 402, Georgia Institute of Technology (Georgia Tech). 

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References

SHOWING 1-10 OF 47 REFERENCES
A sharp concentration inequality with application
We present a new general concentration-of-measure inequality and illustrate its power by applications in random combinatorics. The results find direct applications in some problems of learning theory.
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
TLDR
A polynomial-time randomized algorithm for estimating the permanent of an arbitrary n × n matrix with nonnegative entries computes an approximation that is within arbitrarily small specified relative error of the true value of the permanent.
Finding and certifying a large hidden clique in a semirandom graph
TLDR
This work shows that a different algorithm, based on the Lovasz theta function, almost surely both finds the hidden clique and certifies its optimality and has an additional advantage of being more robust: it also works in a semirandomhidden clique model, in which an adversary can remove edges from the random portion of the graph.
Exact and Approximate Stepdown Methods for Multiple Hypothesis Testing
Consider the problem of testing k hypotheses simultaneously. In this article we discuss finite- and large-sample theory of stepdown methods that provide control of the familywise error rate (FWE). To
The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees
  • D. Aldous
  • Mathematics
    SIAM J. Discret. Math.
  • 1990
TLDR
It is shown how random walk techniques can be applied to the study of several properties of the uniform random spanning tree: the proportion of leaves, the distribution of degrees, and the diameter.
Finding a large hidden clique in a random graph
TLDR
This paper presents an efficient algorithm for finding a hidden clique of vertices of size k that is based on the spectral properties of the graph and improves the trivial case k ) cn log n .
Searching for a trail of evidence in a maze
Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting,
Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities
This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July
Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimension
Higher criticism for detecting sparse heterogeneous mixtures
Higher criticism, or second-level significance testing, is a multiplecomparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance
...
1
2
3
4
5
...