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The Complexity of Separating Points in the Plane
, 2013
"... 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 pro ..."
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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 3pathcondition, and arguing that a shortest cycle in the family gives an optimal solution. The 3pathcondition 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 NPhard for natural families of curves, like segments in two directions or unit circles.
Barrier Resilience of Visibility Polygons
, 2015
"... We consider the problem of computing the Barrier Resilience of a set of Visibility Polygons inside a Polygon. We show that in simple polygons the problem is solvable in time linear in the number of edges. In polygons with holes the problem is APXhard, so only for special cases can we provide polyno ..."
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We consider the problem of computing the Barrier Resilience of a set of Visibility Polygons inside a Polygon. We show that in simple polygons the problem is solvable in time linear in the number of edges. In polygons with holes the problem is APXhard, so only for special cases can we provide polynomial time algorithms.
On the Complexity of Barrier Resilience for Fat Regions
, 2013
"... In the barrier resilience problem (introduced Kumar et al., Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, ..."
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In the barrier resilience problem (introduced Kumar et al., Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, we show that the problem is NPhard when the regions are fat (even when they are axisaligned rectangles of aspect ratio 1: (1+ε)). We also show that the problem is fixedparameter tractable (FPT) for such regions. Using our FPT algorithm, we show that if the regions are βfat and their arrangement has bounded ply ∆, there is a (1+ε)approximation that runs in O(2f(∆,ε,β)n7) time, where f ∈ O(∆2β6ε4 log(β∆/ε)).