2,885 research outputs found
Point Line Cover: The Easy Kernel is Essentially Tight
The input to the NP-hard Point Line Cover problem (PLC) consists of a set
of points on the plane and a positive integer , and the question is
whether there exists a set of at most lines which pass through all points
in . A simple polynomial-time reduction reduces any input to one with at
most points. We show that this is essentially tight under standard
assumptions. More precisely, unless the polynomial hierarchy collapses to its
third level, there is no polynomial-time algorithm that reduces every instance
of PLC to an equivalent instance with points, for
any . This answers, in the negative, an open problem posed by
Lokshtanov (PhD Thesis, 2009).
Our proof uses the machinery for deriving lower bounds on the size of kernels
developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients:
We first show, by reduction from Vertex Cover, that PLC---conditionally---has
no kernel of total size bits. This does not directly imply
the claimed lower bound on the number of points, since the best known
polynomial-time encoding of a PLC instance with points requires
bits. To get around this we build on work of Goodman et al.
(STOC 1989) and devise an oracle communication protocol of cost
for PLC; its main building block is a bound of for the order
types of points that are not necessarily in general position, and an
explicit algorithm that enumerates all possible order types of n points. This
protocol and the lower bound on total size together yield the stated lower
bound on the number of points.
While a number of essentially tight polynomial lower bounds on total sizes of
kernels are known, our result is---to the best of our knowledge---the first to
show a nontrivial lower bound for structural/secondary parameters
Probing a Set of Trajectories to Maximize Captured Information
We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set T of trajectories in the plane, and an integer k? 2, we seek to compute a set of k points ("portals") to maximize the total weight of all subtrajectories of T between pairs of portals. This problem naturally arises in trajectory analysis and summarization.
We show that the TCP is NP-hard (even in very special cases) and give some first approximation results. Our main focus is on attacking the TCP with practical algorithm-engineering approaches, including integer linear programming (to solve instances to provable optimality) and local search methods. We study the integrality gap arising from such approaches. We analyze our methods on different classes of data, including benchmark instances that we generate. Our goal is to understand the best performing heuristics, based on both solution time and solution quality. We demonstrate that we are able to compute provably optimal solutions for real-world instances
Cricket antennae shorten when bending (Acheta domesticus L.).
Insect antennae are important mechanosensory and chemosensory organs. Insect appendages, such as antennae, are encased in a cuticular exoskeleton and are thought to bend only between segments or subsegments where the cuticle is thinner, more flexible, or bent into a fold. There is a growing appreciation of the dominating influence of folds in the mechanical behavior of a structure, and the bending of cricket antennae was considered in this context. Antennae will bend or deflect in response to forces, and the resulting bending behavior will affect the sensory input of the antennae. In some cricket antennae, such as in those of Acheta domesticus, there are a large number (>100) of subsegments (flagellomeres) that vary in their length. We evaluated whether these antennae bend only at the joints between flagellomeres, which has always been assumed but not tested. In addition we questioned whether an antenna undergoes a length change as it bends, which would result from some patterns of joint deformation. Measurements using light microscopy and SEM were conducted on both male and female adult crickets (Acheta domesticus) with bending in four different directions: dorsal, ventral, medial, and lateral. Bending occurred only at the joints between flagellomeres, and antennae shortened a comparable amount during bending, regardless of sex or bending direction. The cuticular folds separating antennal flagellomeres are not very deep, and therefore as an antenna bends, the convex side (in tension) does not have a lot of slack cuticle to "unfold" and does not lengthen during bending. Simultaneously on the other side of the antenna, on the concave side in compression, there is an increasing overlap in the folded cuticle of the joints during bending. Antennal shortening during bending would prevent stretching of antennal nerves and may promote hemolymph exchange between the antenna and head
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