354 research outputs found
Finding the Minimum-Weight k-Path
Given a weighted -vertex graph with integer edge-weights taken from a
range , we show that the minimum-weight simple path visiting
vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k
M). If the weights are reals in , we provide a
-approximation which has a running time of \tilde{O}(2^k
\poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem
of -tree, in which we wish to find a minimum-weight copy of a -node tree
in a given weighted graph , under the same restrictions on edge weights
respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k)
M n^3) and a -approximate solution of running time
\tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above
algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201
Bounded Counter Languages
We show that deterministic finite automata equipped with two-way heads
are equivalent to deterministic machines with a single two-way input head and
linearly bounded counters if the accepted language is strictly bounded,
i.e., a subset of for a fixed sequence of symbols . Then we investigate linear speed-up for counter machines. Lower
and upper time bounds for concrete recognition problems are shown, implying
that in general linear speed-up does not hold for counter machines. For bounded
languages we develop a technique for speeding up computations by any constant
factor at the expense of adding a fixed number of counters
Spin-charge separation in ultra-cold quantum gases
We investigate the physical properties of quasi-1D quantum gases of fermion
atoms confined in harmonic traps. Using the fact that for a homogeneous gas,
the low energy properties are exactly described by a Luttinger model, we
analyze the nature and manifestations of the spin-charge separation. Finally we
discuss the necessary physical conditions and experimental limitations
confronting possible experimental implementations.Comment: 4 pages, revtex4, 2 eps figure
An Improved Exact Algorithm for the Exact Satisfiability Problem
The Exact Satisfiability problem, XSAT, is defined as the problem of finding
a satisfying assignment to a formula in CNF such that exactly one
literal in each clause is assigned to be "1" and the other literals in the same
clause are set to "0". Since it is an important variant of the satisfiability
problem, XSAT has also been studied heavily and has seen numerous improvements
to the development of its exact algorithms over the years.
The fastest known exact algorithm to solve XSAT runs in time,
where is the number of variables in the formula. In this paper, we propose
a faster exact algorithm that solves the problem in time. Like
many of the authors working on this problem, we give a DPLL algorithm to solve
it. The novelty of this paper lies on the design of the nonstandard measure, to
help us to tighten the analysis of the algorithm further
NMR and Neutron Scattering Experiments on the Cuprate Superconductors: A Critical Re-Examination
We show that it is possible to reconcile NMR and neutron scattering
experiments on both LSCO and YBCO, by making use of the Millis-Monien-Pines
mean field phenomenological expression for the dynamic spin-spin response
function, and reexamining the standard Shastry-Mila-Rice hyperfine Hamiltonian
for NMR experiments. The recent neutron scattering results of Aeppli et al on
LSCO (x=14%) are shown to agree quantitatively with the NMR measurements of
and the magnetic scaling behavior proposed by Barzykin and Pines.
The reconciliation of the relaxation rates with the degree of
incommensuration in the spin fluctuation spectrum seen in neutron experiments
is achieved by introducing a new transferred hyperfine coupling between
oxygen nuclei and their next nearest neighbor spins; this leads to a
near-perfect cancellation of the influence of the incommensurate spin
fluctuation peaks on the oxygen relaxation rates of LSCO. The inclusion of the
new term also leads to a natural explanation, within the one-component
model, the different temperature dependence of the anisotropic oxygen
relaxation rates for different field orientations, recently observed by
Martindale . The measured significant decrease with doping of the
anisotropy ratio, in LSCO system, from
for to for LSCO (x=15%) is made compatible with the
doping dependence of the shift in the incommensurate spin fluctuation peaks
measured in neutron experiments, by suitable choices of the direct and
transferred hyperfine coupling constants and B.Comment: 24 pages in RevTex, 9 figures include
Charge transfer fluctuation, wave superconductivity, and the Raman phonon in the Cuprates: A detailed analysis
The Raman spectrum of the phonon in the superconducting cuprate
materials is investigated theoretically in detail in both the normal and
superconducting phases, and is contrasted with that of the phonon. A
mechanism involving the charge transfer fluctuation between the two oxygen ions
in the CuO plane coupled to the crystal field perpendicular to the plane is
discussed and the resulting electron-phonon coupling is evaluated. Depending on
the symmetry of the phonon the weight of different parts of the Fermi surface
in the coupling is different. This provides the opportunity to obtain
information on the superconducting gap function at certain parts of the Fermi
surface. The lineshape of the phonon is then analyzed in detail both in the
normal and superconducting states. The Fano lineshape is calculated in the
normal state and the change of the linewidth with temperature below T is
investigated for a pairing symmetry. Excellent agreement is
obtained for the phonon lineshape in YBaCuO. These
experiments, however, can not distinguish between and a
highly anisotropic -wave pairing.Comment: Revtex, 21 pages + 4 postscript figures appended, tp
Kernel Bounds for Structural Parameterizations of Pathwidth
Assuming the AND-distillation conjecture, the Pathwidth problem of
determining whether a given graph G has pathwidth at most k admits no
polynomial kernelization with respect to k. The present work studies the
existence of polynomial kernels for Pathwidth with respect to other,
structural, parameters. Our main result is that, unless NP is in coNP/poly,
Pathwidth admits no polynomial kernelization even when parameterized by the
vertex deletion distance to a clique, by giving a cross-composition from
Cutwidth. The cross-composition works also for Treewidth, improving over
previous lower bounds by the present authors. For Pathwidth, our result rules
out polynomial kernels with respect to the distance to various classes of
polynomial-time solvable inputs, like interval or cluster graphs. This leads to
the question whether there are nontrivial structural parameters for which
Pathwidth does admit a polynomial kernelization. To answer this, we give a
collection of graph reduction rules that are safe for Pathwidth. We analyze the
success of these results and obtain polynomial kernelizations with respect to
the following parameters: the size of a vertex cover of the graph, the vertex
deletion distance to a graph where each connected component is a star, and the
vertex deletion distance to a graph where each connected component has at most
c vertices.Comment: This paper contains the proofs omitted from the extended abstract
published in the proceedings of Algorithm Theory - SWAT 2012 - 13th
Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
On the Bilayer Coupling in the Yttrium-Barium Family of High Temperature Superconductors
We present and solve a model for the susceptibility of two CuO2 planes
coupled by an interplane coupling J_perp and use the results to analyze a
recent "cross-relaxation" NMR experiment on Y2Ba4Cu7O15. We deduce that in this
material the product of J_perp and the maximum value of the in-plane
susceptibility chi_max varies from approximately 0.2 at T = 200 K to 0.4 at T =
120 K and that this implies the existence of a temperature dependent in-plane
spin correlation length. Using estimates of chi_max from the literature we find
5 meV < J_perp < 20 meV. We discuss the relation of the NMR results to neutron
scattering results which have been claimed to imply that in YBa2Cu3O_{6+x} the
two planes of a bilayer are perfectly anticorrelated. We also propose that the
recently observed 41 meV excitation in YBa2Cu3O7 is an exciton pulled down
below the superconducting gap by J_perp.Comment: 11 pages, 3 postscript figures (uuencoded and compressed
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