A Constant-Factor Approximation for Multi-Covering with Disks

We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $\kappa :X \rightarrow \mathcal{N}$, and a constant $\alpha \geq 1$. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client $x \in X$ be covered by at least $\kappa(x)$ of the server disks. The objective function we wish to minimize is the sum of the $\alpha$-th powers of the disk radii. We present a polynomial time algorithm for this problem achieving an $O(1)$ approximation

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

Tidal Barrier and the Asymptotic Mass of Proto Gas-Giant Planets

Extrasolar planets found with radial velocity surveys have masses ranging from several Earth to several Jupiter masses. While mass accretion onto protoplanetary cores in weak-line T-Tauri disks may eventually be quenched by a global depletion of gas, such a mechanism is unlikely to have stalled the growth of some known planetary systems which contain relatively low-mass and close-in planets along with more massive and longer period companions. Here, we suggest a potential solution for this conundrum. In general, supersonic infall of surrounding gas onto a protoplanet is only possible interior to both of its Bondi and Roche radii. At a critical mass, a protoplanet's Bondi and Roche radii are equal to the disk thickness. Above this mass, the protoplanets' tidal perturbation induces the formation of a gap. Although the disk gas may continue to diffuse into the gap, the azimuthal flux across the protoplanets' Roche lobe is quenched. Using two different schemes, we present the results of numerical simulations and analysis to show that the accretion rate increases rapidly with the ratio of the protoplanet's Roche to Bondi radii or equivalently to the disk thickness. In regions with low geometric aspect ratios, gas accretion is quenched with relatively low protoplanetary masses. This effect is important for determining the gas-giant planets' mass function, the distribution of their masses within multiple planet systems around solar type stars, and for suppressing the emergence of gas-giants around low mass stars

Absorption line systems in simulated galaxies fed by cold streams

Hydro cosmological simulations reveal that massive galaxies at high redshift are fed by long narrow streams of merging galaxies and a smoother component of cold gas. We post-process seven high-resolution simulated galaxies with radiative transfer to study the absorption characteristics of the gas in galaxies and streams, in comparison with the statistics of observed absorption-line systems. We find that much of the stream gas is ionized by UV radiation from background and local stellar sources, but still optically thick (N_HI > 10^17 cm^-2) so that the streams appear as Lyman-limit systems (LLSs). At z>3, the fraction of neutral gas in streams becomes non-negligible, giving rise to damped Lyman-alpha (DLA) absorbers as well. The gas in the central and incoming galaxies remains mostly neutral, responsible for DLAs. Within one (two) virial radii, the covering factor of optically thick gas is <25% (10%) for LLSs and <5% (1%) for DLAs, slowly declining with time following the universal expansion. Nevertheless, galaxies and their cold streams in the studied mass range, M_vir = 10^10 - 10^12 Msun, account for >30% of the observed absorbers in the foreground of quasars, the rest possibly arising from smaller galaxies or the intergalactic medium. The mean metallicity in the streams is ~1% solar, much lower than in the galaxies. The simulated galaxies reproduce the Lyalpha-absorption equivalent widths observed around Lyman-break galaxies, but they severely underpredict the equivalent widths in metal lines, suggesting that the latter may arise from outflows. We conclude that the observed metal-poor LLSs are likely detections of the predicted cold streams. Revised analysis of the observed LLSs kinematics and simulations with more massive outflows in conjunction with the inflows may enable a clearer distinction between the signatures of the various gas modes.Comment: 27 pages, 20 figures, 7 tables. MNRAS accepte