633 research outputs found
On the Complexity of Searching in Trees: Average-case Minimization
We focus on the average-case analysis: A function w : V -> Z+ is given which
defines the likelihood for a node to be the one marked, and we want the
strategy that minimizes the expected number of queries. Prior to this paper,
very little was known about this natural question and the complexity of the
problem had remained so far an open question.
We close this question and prove that the above tree search problem is
NP-complete even for the class of trees with diameter at most 4. This results
in a complete characterization of the complexity of the problem with respect to
the diameter size. In fact, for diameter not larger than 3 the problem can be
shown to be polynomially solvable using a dynamic programming approach.
In addition we prove that the problem is NP-complete even for the class of
trees of maximum degree at most 16. To the best of our knowledge, the only
known result in this direction is that the tree search problem is solvable in
O(|V| log|V|) time for trees with degree at most 2 (paths).
We match the above complexity results with a tight algorithmic analysis. We
first show that a natural greedy algorithm attains a 2-approximation.
Furthermore, for the bounded degree instances, we show that any optimal
strategy (i.e., one that minimizes the expected number of queries) performs at
most O(\Delta(T) (log |V| + log w(T))) queries in the worst case, where w(T) is
the sum of the likelihoods of the nodes of T and \Delta(T) is the maximum
degree of T. We combine this result with a non-trivial exponential time
algorithm to provide an FPTAS for trees with bounded degree
Detection of Complex Networks Modularity by Dynamical Clustering
Based on cluster de-synchronization properties of phase oscillators, we
introduce an efficient method for the detection and identification of modules
in complex networks. The performance of the algorithm is tested on computer
generated and real-world networks whose modular structure is already known or
has been studied by means of other methods. The algorithm attains a high level
of precision, especially when the modular units are very mixed and hardly
detectable by the other methods, with a computational effort on
a generic graph with nodes and links.Comment: 5 pages, 2 figures. Version accepted for publication on PRE Rapid
Communications: figures changed and text adde
Direct evaluation of pure graph state entanglement
We address the question of quantifying entanglement in pure graph states.
Evaluation of multipartite entanglement measures is extremely hard for most
pure quantum states. In this paper we demonstrate how solving one problem in
graph theory, namely the identification of maximum independent set, allows us
to evaluate three multipartite entanglement measures for pure graph states. We
construct the minimal linear decomposition into product states for a large
group of pure graph states, allowing us to evaluate the Schmidt measure.
Furthermore we show that computation of distance-like measures such as relative
entropy of entanglement and geometric measure becomes tractable for these
states by explicit construction of closest separable and closest product states
respectively. We show how these separable states can be described using
stabiliser formalism as well as PEPs-like construction. Finally we discuss the
way in which introducing noise to the system can optimally destroy
entanglement.Comment: 23 pages, 9 figure
Secure and linear cryptosystems using error-correcting codes
A public-key cryptosystem, digital signature and authentication procedures
based on a Gallager-type parity-check error-correcting code are presented. The
complexity of the encryption and the decryption processes scale linearly with
the size of the plaintext Alice sends to Bob. The public-key is pre-corrupted
by Bob, whereas a private-noise added by Alice to a given fraction of the
ciphertext of each encrypted plaintext serves to increase the secure channel
and is the cornerstone for digital signatures and authentication. Various
scenarios are discussed including the possible actions of the opponent Oscar as
an eavesdropper or as a disruptor
Enhancing Parallel Cooperative Trajectory Based Metaheuristics with Path Relinking
This paper proposes a novel algorithm combining path relinking with a set of cooperating trajectory based parallel algorithms to yield a new metaheuristic of enhanced search features. Algorithms based on the exploration of the neighborhood of a single solution, like simulated annealing (SA), have offered accurate results for a large number of real-world problems in the past. Because of their trajectory based nature, some advanced models such as the cooperative one are competitive in academic problems, but still show many limitations in addressing large scale instances. In addition, the field of parallel models for trajectory methods has not deeply been studied yet (at least in comparison with parallel population based models). In this work, we propose a new hybrid algorithm which improves cooperative single solution techniques by using path relinking, allowing both to reduce the global execution time and to improve the efficacy of the method. We test here this new model using a large benchmark of instances of two well-known NP-hard problems: MAXSAT and QAP, with competitive results.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density
The classical monomer-dimer model in two-dimensional lattices has been shown
to belong to the \emph{``#P-complete''} class, which indicates the problem is
computationally ``intractable''. We use exact computational method to
investigate the number of ways to arrange dimers on
two-dimensional rectangular lattice strips with fixed dimer density . For
any dimer density , we find a logarithmic correction term in the
finite-size correction of the free energy per lattice site. The coefficient of
the logarithmic correction term is exactly -1/2. This logarithmic correction
term is explained by the newly developed asymptotic theory of Pemantle and
Wilson. The sequence of the free energy of lattice strips with cylinder
boundary condition converges so fast that very accurate free energy
for large lattices can be obtained. For example, for a half-filled lattice,
, while and . For , is accurate at least to 10 decimal
digits. The function reaches the maximum value at , with 11 correct digits. This is also
the \md constant for two-dimensional rectangular lattices. The asymptotic
expressions of free energy near close packing are investigated for finite and
infinite lattice widths. For lattices with finite width, dependence on the
parity of the lattice width is found. For infinite lattices, the data support
the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
Optimal Location of Sources in Transportation Networks
We consider the problem of optimizing the locations of source nodes in
transportation networks. A reduction of the fraction of surplus nodes induces a
glassy transition. In contrast to most constraint satisfaction problems
involving discrete variables, our problem involves continuous variables which
lead to cavity fields in the form of functions. The one-step replica symmetry
breaking (1RSB) solution involves solving a stable distribution of functionals,
which is in general infeasible. In this paper, we obtain small closed sets of
functional cavity fields and demonstrate how functional recursions are
converted to simple recursions of probabilities, which make the 1RSB solution
feasible. The physical results in the replica symmetric (RS) and the 1RSB
frameworks are thus derived and the stability of the RS and 1RSB solutions are
examined.Comment: 38 pages, 18 figure
Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses
A formalism capable of handling the first step of hierarchical replica
symmetry breaking in finite-connectivity models is introduced. The emerging
order parameter is claimed to be a probability distribution over the space of
field distributions (or, equivalently magnetisation distributions) inside the
cluster of states. The approach is shown to coincide with the previous works in
the replica symmetric case and in the two limit cases m=0,1 where m is Parisi's
break-point. As an application to the study of optimization problems, the
ground-state properties of the random 3-Satisfiability problem are investigated
and we present a first RSB solution improving replica symmetric results.Comment: 16 pages Revtex file, 1 figure; amended version with two new
appendices; to be published in J.Phys.
Fast Distributed Approximation for Max-Cut
Finding a maximum cut is a fundamental task in many computational settings.
Surprisingly, it has been insufficiently studied in the classic distributed
settings, where vertices communicate by synchronously sending messages to their
neighbors according to the underlying graph, known as the or
models. We amend this by obtaining almost optimal
algorithms for Max-Cut on a wide class of graphs in these models. In
particular, for any , we develop randomized approximation
algorithms achieving a ratio of to the optimum for Max-Cut on
bipartite graphs in the model, and on general graphs in the
model.
We further present efficient deterministic algorithms, including a
-approximation for Max-Dicut in our models, thus improving the best known
(randomized) ratio of . Our algorithms make non-trivial use of the greedy
approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing
an unconstrained (non-monotone) submodular function, which may be of
independent interest
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