The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)

Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Local algorithms : Self-stabilization on speed

Non peer reviewe

On the well-posedness of uncalibrated photometric stereo under general lighting

Uncalibrated photometric stereo aims at estimating the 3D-shape of a surface, given a set of images captured from the same viewing angle, but under unknown, varying illumination. While the theoretical foundations of this inverse problem under directional lighting are well-established, there is a lack of mathematical evidence for the uniqueness of a solution under general lighting. On the other hand, stable and accurate heuristical solutions of uncalibrated photometric stereo under such general lighting have recently been proposed. The quality of the results demonstrated therein tends to indicate that the problem may actually be well-posed, but this still has to be established. The present paper addresses this theoretical issue, considering first-order spherical harmonics approximation of general lighting. Two important theoretical results are established. First, the orthographic integrability constraint ensures uniqueness of a solution up to a global concave-convex ambiguity , which had already been conjectured, yet not proven. Second, the perspective integrability constraint makes the problem well-posed, which generalizes a previous result limited to directional lighting. Eventually, a closed-form expression for the unique least-squares solution of the problem under perspective projection is provided , allowing numerical simulations on synthetic data to empirically validate our findings

A Dual Channel X-ray Spectrometer for Fast Ignition Research

A new Dual Channel Highly Ordered Pyrolytic Graphite (DC-HOPG) x-ray spectrometer was developed to study laser-generated electron beam transport. The instrument uses a pair of graphite crystals and has the advantage of simultaneously detecting self emission from low-Z materials in first diffraction order and high-Z materials in second order. The emissions from the target are detected using a pair of parallel imaging plates positioned in a such way that the noise from background is minimized and the mosaic focusing is achieved. Initial tests of the diagnostic on Titan laser (I {approx} 10{sup 20} W/cm{sup 2}, {tau} = 0.7 ps) show excellent signal-to-noise ratio (SNR) > 1000 for the low energy channel and SNR > 400 for the high energy channel

Controlling waves in space and time for imaging and focusing in complex media

In complex media such as white paint and biological tissue, light encounters nanoscale refractive-index inhomogeneities that cause multiple scattering. Such scattering is usually seen as an impediment to focusing and imaging. However, scientists have recently used strongly scattering materials to focus, shape and compress waves by controlling the many degrees of freedom in the incident waves. This was first demonstrated in the acoustic and microwave domains using time reversal, and is now being performed in the optical realm using spatial light modulators to address the many thousands of spatial degrees of freedom of light. This approach is being used to investigate phenomena such as optical super-resolution and the time reversal of light, thus opening many new avenues for imaging and focusing in turbid medi

Altered Arterial Stiffness and Subendocardial Viability Ratio in Young Healthy Light Smokers after Acute Exercise

Studies showed that long-standing smokers have stiffer arteries at rest. However, the effect of smoking on the ability of the vascular system to respond to increased demands (physical stress) has not been studied. The purpose of this study was to estimate the effect of smoking on arterial stiffness and subendocardial viability ratio, at rest and after acute exercise in young healthy individuals.Healthy light smokers (n = 24, pack-years = 2.9) and non-smokers (n = 53) underwent pulse wave analysis and carotid-femoral pulse wave velocity measurements at rest, and 2, 5, 10, and 15 minutes following an exercise test to exhaustion. Smokers were tested, 1) after 12h abstinence from smoking (chronic condition) and 2) immediately after smoking one cigarette (acute condition). At rest, chronic smokers had higher augmentation index and lower aortic pulse pressure than non-smokers, while subendocardial viability ratio was not significantly different. Acute smoking increased resting augmentation index and decreased subendocardial viability ratio compared with non-smokers, and decreased subendocardial viability ratio compared with the chronic condition. After exercise, subendocardial viability ratio was lower, and augmentation index and aortic pulse pressure were higher in non-smokers than smokers in the chronic and acute conditions. cfPWV rate of recovery of was greater in non-smokers than chronic smokers after exercise. Non-smokers were also able to achieve higher workloads than smokers in both conditions.Chronic and acute smoking appears to diminish the vascular response to physical stress. This can be seen as an impaired 'vascular reserve' or a blunted ability of the blood vessels to accommodate the changes required to achieve higher workloads. These changes were noted before changes in arterial stiffness or subendocardial viability ratio occurred at rest. Even light smoking in young healthy individuals appears to have harmful effects on vascular function, affecting the ability of the vascular bed to respond to increased demands