Let S be a bicolored set of n points on the plane. A subset I ⊆ S is called an island of S, if I is the intersection of S and a convex set C. In this paper we give an O(n 3 )-time algorithm to find a monochromatic island of maximum cardinality. Our approach also optimizes other parameters and gives an approximation to the class cover problem.
Abstract. We study a class of geometric optimization problems closely related to the 2-center problem: Given a set S of n pairs of points, assign to each point a color ("red" or "blue") so that each pair's points are assigned different colors and a function of the radii of the minimum enclosing balls of the red points and the blue points, respectively, is optimized. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii. For each case, minmax and minsum, we consider distances measured in the L2 and in the L∞ metrics. Our problems are motivated by a facility location problem in transportation system design, in which we are given origin/destination pairs of points for desired travel, and our goal is to locate an optimal road/flight segment in order to minimize the travel to/from the endpoints of the segment.
Artículo de publicación ISIGiven a set of n axis-parallel rectangles in the plane, finding a maximum
independent set (MIS), amaximumweighted independent set (WMIS), and aminimum
hitting set (MHS), are basic problems in computational geometry and combinatorics.
They have attracted significant attention since the sixties, when Wegner conjectured
that the duality gap, equal to the ratio between the size of MIS and the size of MHS,
is always bounded by a universal constant. An interesting case is when there exists a
diagonal line that intersects each of the given rectangles. Indeed, Chepoi and Felsner
recently gave a 6-approximation algorithm forMHSin this setting, and showed that the
duality gap is between 3/2 and 6.We consider the same setting and improve upon these results. First, we derive an O(n2)-time algorithm for the WMIS when, in addition,
every pair of intersecting rectangles have a common point below the diagonal. This
improves and extends a classic result of Lubiw, and gives a 2-approximation algorithm
for WMIS. Second, we show that MIS is NP-hard. Finally, we prove that the duality
gap is between 2 and 4. The upper bound, which implies a 4-approximation algorithm
for MHS, follows from simple combinatorial arguments, whereas the lower bound
represents the best known lower bound on the duality gap, even in the general setting
of the rectangles.Núcleo
Milenio Información y Coordinación en Redes ICM/FIC P10-024F; FONDECYT Grant 11110069 and FONDECYT Grant 1113026
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