417 research outputs found
Parametrized Complexity of Weak Odd Domination Problems
Given a graph , a subset of vertices is a weak odd
dominated (WOD) set if there exists such that
every vertex in has an odd number of neighbours in . denotes
the size of the largest WOD set, and the size of the smallest
non-WOD set. The maximum of and , denoted
, plays a crucial role in quantum cryptography. In particular
deciding, given a graph and , whether is of
practical interest in the design of graph-based quantum secret sharing schemes.
The decision problems associated with the quantities , and
are known to be NP-Complete. In this paper, we consider the
approximation of these quantities and the parameterized complexity of the
corresponding problems. We mainly prove the fixed-parameter intractability
(W-hardness) of these problems. Regarding the approximation, we show that
, and admit a constant factor approximation
algorithm, and that and have no polynomial approximation
scheme unless P=NP.Comment: 16 pages, 5 figure
Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms
The local minimum degree of a graph is the minimum degree that can be reached
by means of local complementation. For any n, there exist graphs of order n
which have a local minimum degree at least 0.189n, or at least 0.110n when
restricted to bipartite graphs. Regarding the upper bound, we show that for any
graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n)
for bipartite graphs, improving the known n/2 upper bound. We also prove that
the local minimum degree is smaller than half of the vertex cover number (up to
a logarithmic term). The local minimum degree problem is NP-Complete and hard
to approximate. We show that this problem, even when restricted to bipartite
graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which
W[1]-hardness is a long standing open question. Finally, we show that the local
minimum degree is computed by a O*(1.938^n)-algorithm, and a
O*(1.466^n)-algorithm for the bipartite graphs
The Parameterized Complexity of Domination-type Problems and Application to Linear Codes
We study the parameterized complexity of domination-type problems.
(sigma,rho)-domination is a general and unifying framework introduced by Telle:
a set D of vertices of a graph G is (sigma,rho)-dominating if for any v in D,
|N(v)\cap D| in sigma and for any $v\notin D, |N(v)\cap D| in rho. We mainly
show that for any sigma and rho the problem of (sigma,rho)-domination is W[2]
when parameterized by the size of the dominating set. This general statement is
optimal in the sense that several particular instances of
(sigma,rho)-domination are W[2]-complete (e.g. Dominating Set). We also prove
that (sigma,rho)-domination is W[2] for the dual parameterization, i.e. when
parameterized by the size of the dominated set. We extend this result to a
class of domination-type problems which do not fall into the
(sigma,rho)-domination framework, including Connected Dominating Set. We also
consider problems of coding theory which are related to domination-type
problems with parity constraints. In particular, we prove that the problem of
the minimal distance of a linear code over Fq is W[2] for both standard and
dual parameterizations, and W[1]-hard for the dual parameterization.
To prove W[2]-membership of the domination-type problems we extend the
Turing-way to parameterized complexity by introducing a new kind of non
deterministic Turing machine with the ability to perform `blind' transitions,
i.e. transitions which do not depend on the content of the tapes. We prove that
the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing
Machine is W[2]-complete. We believe that this new machine can be used to prove
W[2]-membership of other problems, not necessarily related to dominationComment: 19 pages, 2 figure
Quantum Secret Sharing with Graph States
Revised Selected Papers - http://www.memics.cz/2012/International audienceWe study the graph-state-based quantum secret sharing protocols [24,17] which are not only very promising in terms of physical implementation, but also resource efficient since every player's share is composed of a single qubit. The threshold of a graph-state-based protocol admits a lower bound: for any graph of order n, the threshold of the corresponding n-player protocol is at least 0.506n. Regarding the upper bound, lexicographic product of the C 5 graph (cycle of size 5) are known to provide n-player protocols which threshold is n − n 0.68. Using Paley graphs we improve this bound to n − n 0.71. Moreover, using probabilistic methods, we prove the existence of graphs which associated threshold is at most 0.811n. Albeit non-constructive, probabilistic methods permit to prove that a random graph G of order n has a threshold at most 0.811n with high probability. However, verifying that the threshold of a given graph is acually smaller than 0.811n is hard since we prove that the corresponding decision problem is NP-Complete. These results are mainly based on the graphical characterization of the graph-state-based secret sharing properties, in particular we point out strong connections with domination with parity constraints
Robustness: a New Form of Heredity Motivated by Dynamic Networks
We investigate a special case of hereditary property in graphs, referred to
as {\em robustness}. A property (or structure) is called robust in a graph
if it is inherited by all the connected spanning subgraphs of . We motivate
this definition using two different settings of dynamic networks. The first
corresponds to networks of low dynamicity, where some links may be permanently
removed so long as the network remains connected. The second corresponds to
highly-dynamic networks, where communication links appear and disappear
arbitrarily often, subject only to the requirement that the entities are
temporally connected in a recurrent fashion ({\it i.e.} they can always reach
each other through temporal paths). Each context induces a different
interpretation of the notion of robustness.
We start by motivating the definition and discussing the two interpretations,
after what we consider the notion independently from its interpretation, taking
as our focus the robustness of {\em maximal independent sets} (MIS). A graph
may or may not admit a robust MIS. We characterize the set of graphs \forallMIS
in which {\em all} MISs are robust. Then, we turn our attention to the graphs
that {\em admit} a robust MIS (\existsMIS). This class has a more complex
structure; we give a partial characterization in terms of elementary graph
properties, then a complete characterization by means of a (polynomial time)
decision algorithm that accepts if and only if a robust MIS exists. This
algorithm can be adapted to construct such a solution if one exists
K-trivial, K-low and MLR-low sequences: a tutorial
A remarkable achievement in algorithmic randomness and algorithmic
information theory was the discovery of the notions of K-trivial, K-low and
Martin-Lof-random-low sets: three different definitions turns out to be
equivalent for very non-trivial reasons. This paper, based on the course taught
by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014,
provides an exposition of the proof of this equivalence and some related
results. We assume that the reader is familiar with basic notions of
algorithmic information theory.Comment: 25 page
CMB power spectrum contribution from cosmic strings using field-evolution simulations of the Abelian Higgs model
We present the first field-theoretic calculations of the contribution made by
cosmic strings to the temperature power spectrum of the cosmic microwave
background (CMB). Unlike previous work, in which strings were modeled as
idealized one-dimensional objects, we evolve the simplest example of an
underlying field theory containing local U(1) strings, the Abelian Higgs model.
Limitations imposed by finite computational volumes are overcome using the
scaling property of string networks and a further extrapolation related to the
lessening of the string width in comoving coordinates. The strings and their
decay products, which are automatically included in the field theory approach,
source metric perturbations via their energy-momentum tensor, the unequal-time
correlation functions of which are used as input into the CMB calculation
phase. These calculations involve the use of a modified version of CMBEASY,
with results provided over the full range of relevant scales. We find that the
string tension required to normalize to the WMAP 3-year data at multipole
is , where we have quoted statistical and systematic errors
separately, and is Newton's constant. This is a factor 2-3 higher than
values in current circulation.Comment: 23 pages, 14 figures; further optimized figures for 1Mb size limit,
appendix added before submission to journal, matches accepted versio
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