Given a permutation π of {1, . . . , n} and a positive integer k, we give an algorithm with running time 2 2 log n that decides whether π can be partitioned into at most k increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number, pa...

We study a geometric hitting set problem involving unirectional rays and curves in the plane. We show that this problem is hard to approximate within a logarithmic factor even when the curves are convex polygonal x-monotone chains. Additionally, it is hard to approximate within a factor of 76 even when the curves are line segments with bounded slopes. Lastly, we demonstrate that the problem is ...

17 Transvaginal ultrasound-guided pregnancy reduction (TUGR) is a procedure described for the 18 management of twins post-fixation in the horse. Success rates are often disappointing but 19 reported to be more favourable for bilaterally-situated twins, and when intervention takes 20 place before day 35 of gestation. This study aimed to determine whether stabbing the 21 embryo/fetus rather than ...

Computational complexity is studied for the problem of stabbing set of straight line segments with the smallest cardinality set of disks of fixed radii r > 0 where the set of segments forms straight line drawing of planar graph. This problem along with its relatives [3] arise in physical network security analysis for telecommunication, wireless and road networks represented by geometric graphs ...

Let S be a set of n points in Rd in general position. A set H of k-flats is called an mk-stabber of S if the relative interior of any m-simplex with vertices in S is intersected by at least one element of H. In this paper we give lower and upper bounds on the size of minimum mk-stabbers of point sets in Rd and in general position. We study mainly mk-stabbers in the plane and in R3.

Analytic visibility algorithms, for example methods which compute a subdivided mesh to represent shadows, are notoriously unrobust and hard to use in practice. We present a new method based on a generalized definition of extremal stabbing lines, which are the extremities of shadow boundaries. We treat scenes containing multiple edges or vertices in degenerate configurations, (e.g., collinear or...