Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest

Integration of Horizontally Transferred Genes into Regulatory Interaction Networks Takes Many Million Years

Adaptation of bacteria to new or changing environments is often associated with the uptake of foreign genes through horizontal gene transfer. However, it has remained unclear how (and how fast) new genes are integrated into their host's cellular networks. Combining the regulatory and protein interaction networks of Escherichia coli with comparative genomics tools, we provide the first systematic analysis of this issue. Genes transferred recently have fewer interaction partners compared to nontransferred genes in both regulatory and protein interaction networks. Thus, horizontally transferred genes involved in complex regulatory and protein-protein interactions are rarely favored by selection. Only few protein-protein interactions are gained after the initial integration of genes following the transfer event. In contrast, transferred genes are gradually integrated into the regulatory network of their host over evolutionary time. During adaptation to the host cellular environment, horizontally transferred genes recruit existing transcription factors of the host, reflected in the fast evolutionary rates of the cis-regulatory regions of transferred genes. Further, genes resulting from increasingly ancient transfer events show increasing numbers of transcriptional regulators as well as improved coregulation with interacting proteins. Fine-tuned integration of horizontally transferred genes into the regulatory network spans more than 8-22 million years and encompasses accelerated evolution of regulatory regions, stabilization of protein-protein interactions, and changes in codon usage

Multisetting Bell-type inequalities for detecting genuine tripartite entanglement

In a recent paper, Bancal et al. put forward the concept of device-independent witnesses of genuine multipartite entanglement. These witnesses are capable of verifying genuine multipartite entanglement produced in a lab without resorting to any knowledge of the dimension of the state space or of the specific form of the measurement operators. As a by-product they found a three-party three-setting Bell inequality which enables to detect genuine tripartite entanglement in a noisy 3-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as 2/3 in a device-independent way. In this paper, we generalize this inequality to an arbitrary number of settings, demonstrating a threshold visibility of 2/pi~0.6366 for number of settings going to infinity. We also present a pseudo-telepathy Bell inequality achieving the same threshold value. We argue that our device-independent witnesses are optimal in the sense that the above value cannot be beaten with three-party-correlation Bell inequalities.Comment: 7 page

The randomly driven Ising ferromagnet, Part II: One and two dimensions

We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. In Part I, we introduced a general formalism for describing such systems and presented the mean field theory. In this article we derive results for the one dimensional case, which can be only partially solved. Monte Carlo simulations performed on a square lattice indicate that the main features of the mean field theory survive the presence of strong fluctuations.Comment: 10 pages in REVTeX/LaTeX format, 17 eps/ps figures. Submitted to Journal of Physics

Maximal violation of the I3322 inequality using infinite dimensional quantum systems

The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the Clauser-Horne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.Comment: 9 pages, 3 figures, 1 tabl

A TESS Search for Distant Solar System Planets: A Feasibility Study

As the NASA Transiting Exoplanet Survey Satellite (TESS) fulfills its primary mission it is executing an unprecedented all-sky survey with the potential to discover distant planets in our own solar system, as well as hundreds of Transneptunian Objects (TNOs) and Centaurs. We demonstrate that shift-and-stack techniques can be used to efficiently search the Full-Frame Image (FFI) data from the TESS mission and survey the entire sky for outer Solar System objects down to $\sim22^{nd}$ magnitude.Comment: Submitted to RNAA

A TESS Search for Distant Solar System Objects: Yield Estimates

As the NASA Transiting Exoplanet Survey Satellite (TESS) fulfills its primary mission, it is executing an unprecedented survey of almost the entire sky: TESS's approved extended mission will likely extend sky coverage to ~94%, including ~60% of the ecliptic. In an accompanying note we demonstrated that `digital tracking' techniques can be used to efficiently `shift-and-stack' TESS full frame images (FFIs) and showed that combining ~1,300 exposures from a TESS sector gives a 50% detection threshold of $I_C\sim 22.0\pm0.5$, raising the possibility that TESS could discover the hypothesized Planet Nine. In this note, we estimate the yield for such a survey and demonstrate that this technique has the potential to discover hundreds of Kuiper Belt Objects, Scattered Disk Objects and Centaurs in TESS FFI data