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Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 8 (1 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Minimum weight convex Steiner partitions
, 2009
"... New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n / loglog n) times longer than a ..."
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Cited by 1 (1 self)
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New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n / loglog n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudotriangle. We also show that the minimum length convex Steiner partition of n points along a pseudotriangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worstcase bounds in both cases.
On a Linear Program for MinimumWeight Triangulation EXTENDED ABSTRACT ∗
"... Minimumweight triangulation (MWT) is NPhard. It has a polynomialtime constantfactor approximation algorithm, and a variety of effective polynomialtime heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are wellstudied, but previously no connection was kno ..."
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Minimumweight triangulation (MWT) is NPhard. It has a polynomialtime constantfactor approximation algorithm, and a variety of effective polynomialtime heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are wellstudied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. 1