36 research outputs found
Geometric Multicut
We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" F, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, where k is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an O(n^4 log^3 n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colors and n corners in total. We then show that the problem is NP-hard for the case of three colors. Finally, we give a (2-4/3k)-approximation algorithm
Geometric Multicut
We study the following separation problem: Given a collection of colored
objects in the plane, compute a shortest "fence" , i.e., a union of curves
of minimum total length, that separates every two objects of different colors.
Two objects are separated if contains a simple closed curve that has one
object in the interior and the other in the exterior. We refer to the problem
as GEOMETRIC -CUT, where is the number of different colors, as it can be
seen as a geometric analogue to the well-studied multicut problem on graphs. We
first give an -time algorithm that computes an optimal fence
for the case where the input consists of polygons of two colors and corners
in total. We then show that the problem is NP-hard for the case of three
colors. Finally, we give a -approximation algorithm.Comment: 24 pages, 15 figure
The Complexity of Separating Points in the Plane
We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles
Fixed-parameter tractability and lower bounds for stabbing problems
We study the following general stabbing problem from a parameterized
complexity point of view: Given a set of translates of an
object in \Rd, find a set of lines with the property that every object in
is ''stabbed'' (intersected) by at least one line.
We show that when consists of axis-parallel unit squares in \Rtwo the
(decision) problem of stabbing with axis-parallel lines is W[1]-hard with
respect to (and thus, not fixed-parameter tractable unless FPT=W[1]) while
it becomes fixed-parameter tractable when the squares are disjoint. We also
show that the problem of stabbing a set of disjoint unit squares in \Rtwo
with lines of arbitrary directions is W[1]--hard with respect to . Several
generalizations to other types of objects and lines with arbitrary directions
are also presented. Finally, we show that deciding whether a set of unit balls
in \Rd can be stabbed by one line is W[1]--hard with respect to the dimension
.Comment: Based on the MSc. Thesis of Daniel Werner, Free University Berlin,
Berlin, German
Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane
We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest âfenceâ F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an O(n4log3n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised 4/3â
1.2965-approximation algorithm for polygons and any number of colours
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