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The universality of polynomial time Turing equivalence
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees
Automatic Software Repair: a Bibliography
This article presents a survey on automatic software repair. Automatic
software repair consists of automatically finding a solution to software bugs
without human intervention. This article considers all kinds of repairs. First,
it discusses behavioral repair where test suites, contracts, models, and
crashing inputs are taken as oracle. Second, it discusses state repair, also
known as runtime repair or runtime recovery, with techniques such as checkpoint
and restart, reconfiguration, and invariant restoration. The uniqueness of this
article is that it spans the research communities that contribute to this body
of knowledge: software engineering, dependability, operating systems,
programming languages, and security. It provides a novel and structured
overview of the diversity of bug oracles and repair operators used in the
literature
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
Von Neumann Entropy Penalization and Low Rank Matrix Estimation
A problem of statistical estimation of a Hermitian nonnegatively definite
matrix of unit trace (for instance, a density matrix in quantum state
tomography) is studied. The approach is based on penalized least squares method
with a complexity penalty defined in terms of von Neumann entropy. A number of
oracle inequalities have been proved showing how the error of the estimator
depends on the rank and other characteristics of the oracles. The methods of
proofs are based on empirical processes theory and probabilistic inequalities
for random matrices, in particular, noncommutative versions of Bernstein
inequality
On monotone circuits with local oracles and clique lower bounds
We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , under a certain restrictive
assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles,
extends and provides a matching upper bound for the exponential lower bounds on
the monotone circuit size complexity of -clique obtained by Alon and Boppana
(1987).Comment: Updated acknowledgements and funding informatio
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