77,627 research outputs found
A Note on the Unsolvability of the Weighted Region Shortest Path Problem
Let S be a subdivision of the plane into polygonal regions, where each region
has an associated positive weight. The weighted region shortest path problem is
to determine a shortest path in S between two points s, t in R^2, where the
distances are measured according to the weighted Euclidean metric-the length of
a path is defined to be the weighted sum of (Euclidean) lengths of the
sub-paths within each region. We show that this problem cannot be solved in the
Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one
can compute exactly any number that can be obtained from the rationals Q by
applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for
any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's
technique.Comment: 6 pages, 1 figur
Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
We seek to augment a geometric network in the Euclidean plane with shortcuts
to minimize its continuous diameter, i.e., the largest network distance between
any two points on the augmented network. Unlike in the discrete setting where a
shortcut connects two vertices and the diameter is measured between vertices,
we take all points along the edges of the network into account when placing a
shortcut and when measuring distances in the augmented network.
We study this network augmentation problem for paths and cycles. For paths,
we determine an optimal shortcut in linear time. For cycles, we show that a
single shortcut never decreases the continuous diameter and that two shortcuts
always suffice to reduce the continuous diameter. Furthermore, we characterize
optimal pairs of shortcuts for convex and non-convex cycles. Finally, we
develop a linear time algorithm that produces an optimal pair of shortcuts for
convex cycles. Apart from the algorithms, our results extend to rectifiable
curves.
Our work reveals some of the underlying challenges that must be overcome when
addressing the discrete version of this network augmentation problem, where we
minimize the discrete diameter of a network with shortcuts that connect only
vertices
Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours
In this paper, we study problems of connecting classes of points via
noncrossing structures. Given a set of colored terminal points, we want to find
a graph for each color that connects all terminals of its color with the
restriction that no two graphs cross each other. We consider these problems
both on the Euclidean plane and in planar graphs.
On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored
traveling salesman problem as well as for the red-blue-green separation problem
(in which we want to separate terminals of three colors with two noncrossing
polygons of minimum length), both on the Euclidean plane. This improves the
work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler
red-blue separation problem. For the case of unweighted plane graphs, we also
show a PTAS for the two-colored traveling salesman problem. All these results
are based on our new patching procedure that might be of independent interest.
On the negative side, we show that the problem of connecting terminal pairs
with noncrossing paths is NP-hard on the Euclidean plane, and that the problem
of finding two noncrossing spanning trees is NP-hard in plane graphs.Comment: 36 pages, 15 figures (colored
Computing optimal shortcuts for networks
We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts.Peer ReviewedPostprint (published version
Computing Optimal Shortcuts for Networks
We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts
Prizing on Paths: A PTAS for the Highway Problem
In the highway problem, we are given an n-edge line graph (the highway), and
a set of paths (the drivers), each one with its own budget. For a given
assignment of edge weights (the tolls), the highway owner collects from each
driver the weight of the associated path, when it does not exceed the budget of
the driver, and zero otherwise. The goal is choosing weights so as to maximize
the profit.
A lot of research has been devoted to this apparently simple problem. The
highway problem was shown to be strongly NP-hard only recently
[Elbassioni,Raman,Ray-'09]. The best-known approximation is O(\log n/\log\log
n) [Gamzu,Segev-'10], which improves on the previous-best O(\log n)
approximation [Balcan,Blum-'06].
In this paper we present a PTAS for the highway problem, hence closing the
complexity status of the problem. Our result is based on a novel randomized
dissection approach, which has some points in common with Arora's quadtree
dissection for Euclidean network design [Arora-'98]. The basic idea is
enclosing the highway in a bounding path, such that both the size of the
bounding path and the position of the highway in it are random variables. Then
we consider a recursive O(1)-ary dissection of the bounding path, in subpaths
of uniform optimal weight. Since the optimal weights are unknown, we construct
the dissection in a bottom-up fashion via dynamic programming, while computing
the approximate solution at the same time. Our algorithm can be easily
derandomized. We demonstrate the versatility of our technique by presenting
PTASs for two variants of the highway problem: the tollbooth problem with a
constant number of leaves and the maximum-feasibility subsystem problem on
interval matrices. In both cases the previous best approximation factors are
polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]
Computing optimal shortcuts for networks
We augment a plane Euclidean network with a segment or shortcut to minimize the largest distance between any two points along the edges of the resulting network. In this continuous setting, the problem of computing distances and placing a shortcut is much harder as all points on the network, instead of only the vertices, must be taken into account. Our main result for general networks states that it is always possible to determine in polynomial time whether the network has an optimal shortcut and compute one in case of existence. We also improve this general method for networks that are paths, restricted to using two types of shortcuts: those of any fixed direction and shortcuts that intersect the path only on its endpoints.Peer ReviewedPostprint (published version
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