65 research outputs found
On Approximability of Bounded Degree Instances of Selected Optimization Problems
In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics
String theory integrands and supergravity divergences
At low energies, interactions of massless particles in type II strings
compactified on a torus are described by an effective Wilsonian action
, consisting of the usual supergravity Lagrangian
supplemented by an infinite series of higher-derivative vertices, including the
much studied gravitational interactions. Using
recent results on the asymptotics of the integrands governing four-graviton
scattering at genus one and two, I determine the -dependence of the
coefficient of the above interaction, and show that the logarithmic terms
appearing in the limit are related to UV divergences in
supergravity amplitudes, augmented by stringy interactions. This provides a
strong consistency check on the expansion of the integrand near the boundaries
of moduli space, in particular it elucidates the appearance of odd zeta values
in these expansions. I briefly discuss how these logarithms are reflected in
non-analytic terms in the low energy expansion of the string scattering
amplitude.Comment: 40 pages; v2: after fixing a factor of 2 mistake in Eq. (2.41), all
divergent terms now agree with SUGRA predictions. Added a note at end of Sec
1 on the definition of the truncated moduli space M_{h,n}(\Lambda
Confinement and Strings in MQCD
We study aspects of confinement in the M theory fivebrane version of QCD
(MQCD). We show heavy quarks are confined in hadrons (which take the form of
membrane-fivebrane bound states) for N=1 and softly broken N=2 SU(Nc) MQCD. We
explore and clarify the transition from the exotic physics of the latter to the
standard physics of the former. In particular, the many strings and
quark-antiquark mesons found in N=2 field theory by Douglas and Shenker are
reproduced. It is seen that in the N=1 limit all but one such meson disappears
while all of the strings survive. The strings of softly broken N=2, N=1, and
even non-supersymmetric SU(Nc) MQCD have a common ratio for their tensions as a
function of the amount of flux they carry. We also comment on the almost BPS
properties of the Douglas-Shenker strings and discuss the brane picture for
monopole confinement on N=2 QCD Higgs branches.Comment: 39 pages, 17 figures, uses harvma
All Loop Topological String Amplitudes From Chern-Simons Theory
We demonstrate the equivalence of all loop closed topological string
amplitudes on toric local Calabi-Yau threefolds with computations of certain
knot invariants for Chern-Simons theory. We use this equivalence to compute the
topological string amplitudes in certain cases to very high degree and to all
genera. In particular we explicitly compute the topological string amplitudes
for P2 up to degree 12 and P1 x P1 up to total degree 10 to all genera. This
also leads to certain novel large N dualities in the context of ordinary
superstrings, involving duals of type II superstrings on local Calabi-Yau
three-folds without any fluxes.Comment: 62 pages, 15 figures, harvma
The chiral ring of AdS3/CFT2 and the attractor mechanism
We study the moduli dependence of the chiral ring in N = (4,4) superconformal
field theories, with special emphasis on those CFTs that are dual to type IIB
string theory on AdS3xS3xX4. The chiral primary operators are sections of
vector bundles, whose connection describes the operator mixing under motion on
the moduli space. This connection can be exactly computed using the constraints
from N = (4,4) supersymmetry. Its curvature can be determined using the tt*
equations, for which we give a derivation in the physical theory which does not
rely on the topological twisting. We show that for N = (4,4) theories the
chiral ring is covariantly constant over the moduli space, a fact which can be
seen as a non-renormalization theorem for the three-point functions of chiral
primaries in AdS3/CFT2. From the spacetime point of view our analysis has the
following applications. First, in the case of a D1/D5 black string, we can see
the matching of the attractor flow in supergravity to RG-flow in the boundary
field theory perturbed by irrelevant operators, to first order away from the
fixed point. Second, under spectral flow the chiral primaries become the Ramond
ground states of the CFT. These ground states represent the microstates of a
small black hole in five dimensions consisting of a D1/D5 bound state. The
connection that we compute can be considered as an example of Berry's phase for
the internal microstates of a supersymmetric black hole.Comment: 72 pages (60 + appendices
Information transfer fidelity in spin networks and ring-based quantum routers
Spin networks are endowed with an Information Transfer Fidelity (ITF), which defines an absolute upper bound on the probability of transmission of an excitation from one spin to another. The ITF is easily computable but the bound can be reached asymptotically in time only under certain conditions. General conditions for attainability of the bound are established and the process of achieving the maximum transfer probability is given a dynamical model, the translation on the torus. The time to reach the maximum probability is estimated using the simultaneous Diophantine approximation, implemented using a variant of the Lenstra-Lenstra-Lov\'asz (LLL) algorithm. For a ring with uniform couplings, the network can be made a metric space by defining a distance (satisfying the triangle inequality) that quantifies the lack of transmission fidelity between two nodes. It is shown that transfer fidelities and transfer times can be improved significantly by means of simple controls taking the form of non-dynamic, spatially localized bias fields, opening up the possibility for intelligent design of spin networks and dynamic routing of information encoded in them, while being more flexible than engineering fixed couplings to favor some transfers, and less demanding than control schemes requiring fast dynamic controls
- …