Carsten Grimm

Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts

We consider a lowerand upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds). We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

A note on the unsolvability of the weighted region shortest path problem

Abstract Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s , t ∈ R 2 , where the distances are measured according to the weighted Euclidean metric—the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers ( ACM Q ). In the ACM Q , one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, −, ×, ÷, ⋅ k , for any integer k ⩾ 2 . Our proof uses Galois theory and is based on Bajajs technique.

Network farthest-point diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.

Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut

We augment a tree

Optimal Data Structures for Farthest-Point Queries in Cactus Networks

T

Efficient Farthest-Point Queries in Two-terminalźSeries-parallel Networks

with a shortcut

NETWORK FARTHEST-POINT DIAGRAMS AND THEIR APPLICATION TO FEED-LINK NETWORK EXTENSION

pq

Optimal Data Structures for Farthest-Point Queries in Cactus Networks.

to minimize the largest distance between any two points along the resulting augmented tree

On farthest-point information in networks

T+pq

Realizing Farthest-Point Voronoi Diagrams.

. We study this problem in a continuous and geometric setting where