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 optics and light localization in periodic photonic lattices

We review the recent developments in the field of photonic lattices emphasizing their unique properties for controlling linear and nonlinear propagation of light. We draw some important links between optical lattices and photonic crystals pointing towards practical applications in optical communications and computing, beam shaping, and bio-sensing.Comment: to appear in Journal of Nonlinear Optical Physics & Materials (JNOPM

Pro-social behavior in rats is modulated by social experience

In mammals, helping is preferentially provided to members of one's own group. Yet, it remains unclear how social experience shapes pro-social motivation. We found that rats helped trapped strangers by releasing them from a restrainer, just as they did cagemates. However, rats did not help strangers of a different strain, unless previously housed with the trapped rat. Moreover, pair-housing with one rat of a different strain prompted rats to help strangers of that strain, evidence that rats expand pro-social motivation from one individual to phenotypically similar others. To test if genetic relatedness alone can motivate helping, rats were fostered from birth with another strain and were not exposed to their own strain. As adults, fostered rats helped strangers of the fostering strain but not rats of their own strain. Thus, strain familiarity, even to one's own strain, is required for the expression of pro-social behavior

Exploiting disorder for perfect focusing

We demonstrate experimentally that disordered scattering can be used to improve, rather than deteriorate, the focusing resolution of a lens. By using wavefront shaping to compensate for scattering, light was focused to a spot as small as one tenth of the diffraction limit of the lens. We show both experimentally and theoretically that it is the scattering medium, rather than the lens, that determines the width of the focus. Despite the disordered propagation of the light, the profile of the focus was always exactly equal to the theoretical best focus that we derived.Comment: 4 pages, 4 figure

Anomalous spectral scaling of light emission rates in low dimensional metallic nanostructures

The strength of light emission near metallic nanostructures can scale anomalously with frequency and dimensionality. We find that light-matter interactions in plasmonic systems confined in two dimensions (e.g., near metal nanowires) strengthen with decreasing frequency owing to strong mode confinement away from the surface plasmon frequency. The anomalous scaling also applies to the modulation speed of plasmonic light sources, including lasers, with modulation bandwidths growing at lower carrier frequencies. This allows developing optical devices that exhibit simultaneously femto-second response times at the nano-meter scale, even at longer wavelengths into the mid IR, limited only by non-local effects and reversible light-matter coupling

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