50,838 research outputs found
A transform of complementary aspects with applications to entropic uncertainty relations
Even though mutually unbiased bases and entropic uncertainty relations play
an important role in quantum cryptographic protocols they remain ill
understood. Here, we construct special sets of up to 2n+1 mutually unbiased
bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry
properties derived from the Clifford algebra. More precisely, we show that
there exists a unitary transformation that cyclically permutes such bases. This
unitary can be understood as a generalization of the Fourier transform, which
exchanges two MUBs, to multiple complementary aspects. We proceed to prove a
lower bound for min-entropic entropic uncertainty relations for any set of
MUBs, and show that symmetry plays a central role in obtaining tight bounds.
For example, we obtain for the first time a tight bound for four MUBs in
dimension d=4, which is attained by an eigenstate of our complementarity
transform. Finally, we discuss the relation to other symmetries obtained by
transformations in discrete phase space, and note that the extrema of discrete
Wigner functions are directly related to min-entropic uncertainty relations for
MUBs.Comment: 16 pages, 2 figures, v2: published version, clarified ref [30
Graver degrees are not polynomially bounded by true circuit degrees
Let be a toric ideal. We prove that the degrees of the elements of the
Graver basis of are not polynomially bounded by the true degrees of the
circuits of .Comment: 8 pages, 1 figur
A note on the shortest common superstring of NGS reads
The Shortest Superstring Problem (SSP) consists, for a set of strings S =
{s_1,...,s_n}, to find a minimum length string that contains all s_i, 1 <= i <=
k, as substrings. This problem is proved to be NP-Complete and APX-hard.
Guaranteed approximation algorithms have been proposed, the current best ratio
being 2+11/23, which has been achieved following a long and difficult quest.
However, SSP is highly used in practice on next generation sequencing (NGS)
data, which plays an increasingly important role in sequencing. In this note,
we show that the SSP approximation ratio can be improved on NGS reads by
assuming specific characteristics of NGS data that are experimentally verified
on a very large sampling set
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Minimum Cycle Basis and All-Pairs Min Cut of a Planar Graph in Subquadratic Time
A minimum cycle basis of a weighted undirected graph is a basis of the
cycle space of such that the total weight of the cycles in this basis is
minimized. If is a planar graph with non-negative edge weights, such a
basis can be found in time and space, where is the size of . We
show that this is optimal if an explicit representation of the basis is
required. We then present an time and space
algorithm that computes a minimum cycle basis \emph{implicitly}. From this
result, we obtain an output-sensitive algorithm that explicitly computes a
minimum cycle basis in time and space,
where is the total size (number of edges and vertices) of the cycles in the
basis. These bounds reduce to and ,
respectively, when is unweighted. We get similar results for the all-pairs
min cut problem since it is dual equivalent to the minimum cycle basis problem
for planar graphs. We also obtain time and
space algorithms for finding, respectively, the weight vector and a Gomory-Hu
tree of . The previous best time and space bound for these two problems was
quadratic. From our Gomory-Hu tree algorithm, we obtain the following result:
with time and space for preprocessing, the
weight of a min cut between any two given vertices of can be reported in
constant time. Previously, such an oracle required quadratic time and space for
preprocessing. The oracle can also be extended to report the actual cut in time
proportional to its size
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