The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations.We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. This includes the case of partial representation extension.The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs [Balko et al. ISAAC'13]. So unless P = NP, proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.
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.
We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with m > 1 viewpoints (or guards) located on the terrain surface. A point on the terrain is considered visible if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the colored visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting.
We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon P with n vertices. We give a randomized near-linear-time (1 − ε)-approximation algorithm for this problem: in O(n(log 2 n + (1/ε 3 ) log n + 1/ε 4 )) time we find a convex polygon contained in P that, with probability at least 2/3, has area at least (1 − ε) times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside P with maximum perimeter.To achieve these results we provide new results in geometric probability. The first result is a bound relating the probability that two points chosen uniformly at random inside P are mutually visible and the area of the largest convex body inside P . The second result is a bound on the expected value of the difference between the perimeter of any planar convex body K and the perimeter of the convex hull of a uniform random sample inside K.
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