456 research outputs found
Simple Wriggling is Hard unless You Are a Fat Hippo
We prove that it is NP-hard to decide whether two points in a polygonal
domain with holes can be connected by a wire. This implies that finding any
approximation to the shortest path for a long snake amidst polygonal obstacles
is NP-hard. On the positive side, we show that snake's problem is
"length-tractable": if the snake is "fat", i.e., its length/width ratio is
small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201
Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams
We study a general family of facility location problems defined on planar
graphs and on the 2-dimensional plane. In these problems, a subset of
objects has to be selected, satisfying certain packing (disjointness) and
covering constraints. Our main result is showing that, for each of these
problems, the time brute force algorithm of selecting objects
can be improved to time. The algorithm is based on an idea
that was introduced recently in the design of geometric QPTASs, but was not yet
used for exact algorithms and for planar graphs. We focus on the Voronoi
diagram of a hypothetical solution of objects, guess a balanced separator
cycle of this Voronoi diagram to obtain a set that separates the solution in a
balanced way, and then recurse on the resulting subproblems. We complement our
study by giving evidence that packing problems have time
algorithms for a much more general class of objects than covering problems
have.Comment: 64 pages, 16 figure
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