Probabilistic Analysis of Edge Elimination for Euclidean TSP

One way to speed up the calculation of optimal TSP tours in practice is eliminating edges that are certainly not in the optimal tour as a preprocessing step. In order to do so several edge elimination approaches have been proposed in the past. In this work we investigate two of them in the scenario where the input consists of $n$ independently distributed random points in the 2-dimensional unit square with bounded density function from above and below by arbitrary positive constants. We show that after the edge elimination procedure of Hougardy and Schroeder the expected number of remaining edges is $\Theta(n)$, while after that the the non-recursive part of Jonker and Volgenant the expected number of remaining edges is $\Theta(n^2)$

On the Approximation Ratio of the k-Opt and Lin-Kernighan Algorithm for Metric and Graph TSP

The k-Opt and Lin-Kernighan algorithm are two of the most important local search approaches for the Metric TSP. Both start with an arbitrary tour and make local improvements in each step to get a shorter tour. We show that for any fixed k ? 3 the approximation ratio of the k-Opt algorithm for Metric TSP is O(?[k]{n}). Assuming the Erd?s girth conjecture, we prove a matching lower bound of ?(?[k]{n}). Unconditionally, we obtain matching bounds for k = 3,4,6 and a lower bound of ?(n^{2/(3k-3)}). Our most general bounds depend on the values of a function from extremal graph theory and are tight up to a factor logarithmic in the number of vertices unconditionally. Moreover, all the upper bounds also apply to a parameterized version of the Lin-Kernighan algorithm with appropriate parameter. We also show that the approximation ratio of k-Opt for Graph TSP is ?(log(n)/(log log(n))) and O({log(n)/(log log(n))}^{log?(9)+?}) for all ? > 0

An optical phase-locking with large and tunable frequency difference based on vertical-cavity surface-emitting laser

We present a novel technique to phase-lock two lasers with controllable frequency difference. In our setup, one sideband of a current modulated Vertical-Cavity Surface-Emitting Laser (VCSEL) is phase locked to the master laser by injection seeding, while another sideband of the VCSEL is used to phase lock the slave laser. The slave laser is therefore locked in phase with the master laser, with a frequency difference tunable up to about 35 GHz. The sideband suppression rate of the slave laser is more than 30dB at 30 uW seed power. The heterodyne spectrum between master and slave has a linewidth of less than 1 Hz. A coherent population trapping resonance of rubidium is achieved using such beams.Comment: 4 pages, 4 Encapsulated PostScript figure

Dual-comb spectroscopy over 100km open-air path

Satellite-based greenhouse gases (GHG) sensing technologies play a critical role in the study of global carbon emissions and climate change. However, none of the existing satellite-based GHG sensing technologies can achieve the measurement of broad bandwidth, high temporal-spatial resolution, and high sensitivity at the same time. Recently, dual-comb spectroscopy (DCS) has been proposed as a superior candidate technology for GHG sensing because it can measure broadband spectra with high temporal-spatial resolution and high sensitivity. The main barrier to DCS's display on satellites is its short measurement distance in open air achieved thus far. Prior research has not been able to implement DCS over 20 km of open-air path. Here, by developing a bistatic setup using time-frequency dissemination and high-power optical frequency combs, we have implemented DCS over a 113 km turbulent horizontal open-air path. Our experiment successfully measured GHG with 7 nm spectral bandwidth and a 10 kHz frequency and achieved a CO2 sensing precision of <2 ppm in 5 minutes and <0.6 ppm in 36 minutes. Our results represent a significant step towards advancing the implementation of DCS as a satellite-based technology and improving technologies for GHG monitoringComment: 24 pages, 6 figure