Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

Ruling out Stellar Companions and Resolving the Innermost Regions of Transitional Disks with the Keck Interferometer

With the Keck Interferometer, we have studied at 2 um the innermost regions of several nearby, young, dust depleted "transitional" disks. Our observations target five of the six clearest cases of transitional disks in the Taurus/Auriga star-forming region (DM Tau, GM Aur, LkCa 15, UX Tau A, and RY Tau) to explore the possibility that the depletion of optically thick dust from the inner disks is caused by stellar companions rather than the more typical planet-formation hypothesis. At the 99.7% confidence level, the observed visibilities exclude binaries with flux ratios of at least 0.05 and separations ranging from 2.5 to 30 mas (0.35 - 4 AU) over >= 94% of the area covered by our measurements. All targets but DM Tau show near-infrared excess in their SED higher than our companion flux ratio detection limits. While a companion has previously been detected in the candidate transitional disk system CoKu Tau/4, we can exclude similar mass companions as the typical origin for the clearing of inner dust in transitional disks and of the near-infrared excess emission. Unlike CoKu Tau/4, all our targets show some evidence of accretion. We find that all but one of the targets are clearly spatially resolved, and UX Tau A is marginally resolved. Our data is consistent with hot material on small scales (0.1 AU) inside of and separated from the cooler outer disk, consistent with the recent SED modeling. These observations support the notion that some transitional disks have radial gaps in their optically thick material, which could be an indication for planet formation in the habitable zone (~ a few AU) of a protoplanetary disk.Comment: 36 pages, 7 figures. Accepted for publication in Ap