Abstract. Given a Euclidean graph G in R d with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm + n 2 log n) time, resulting in a trivial O(n 3 m + n 4 log n)-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running time we consider several approximation algorithms.Key words. computational geometry, approximation algorithms, geometric networks AMS subject classifications. 65D18, 68U05, 68Q25 DOI. 10.1137/050635675 1. Introduction. Consider a set V of n points in R d . A network on V can be modeled as an undirected graph G with vertex set V of size n and an edge set E of size m where every edge (u, v) has a positive weight w (u, v). A Euclidean network is a geometric network where the weight of the edge (u, v) is equal to the Euclidean distance |uv| between its two endpoints u and v.For two vertices u, v in a weighted graph G we use δ G (u, v) to denote a shortest path between u and v in G, and the length of the path is denoted by d G (u, v). Consider a weighted graph G = (V, E) and a graph G = (V, E ) on the same vertex set but with edge set E ⊆ E. We say that G is a t-spanner of G if for each pair of vertices (u, v). The minimum t such that G is a t-spanner for V is called the stretch factor, or dilation, of G.We say that a Euclidean network G = (V, E) is a t-spanner if G = (V, E) is a t-spanner of the complete network on V . In other words, for any two points p, q ∈ V the graph distance in G is at most t times the Euclidean distance between the two points.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas. Recently spanners found interesting practical applications

Let A and B be two sets of m resp. n weighted points in the plane, with m ≤ n. We present (1 + ) and (2+ )-approximation algorithms for the minimum Euclidean Earth Mover's Distance between A and B under translations and rigid motions respectively. In the general case where the sets have unequal total weights the algorithms run in O((n 3 m/ 4 ) log 2 (n/ )) time for translations and O((n 4 m 2 / 4 ) log 2 (n/ )) time for rigid motions. When the sets have equal total weights, the respective running times decrease to O((n 2 / 4 ) log 2 (n/ )) and O((n 3 m/ 4 ) log 2 (n/ )). We also show how to compute a (1 + ) and (2 + )-approximation of the minimum cost Euclidean bipartite matching under translations and rigid motions in O((n 3/2 / 7/2 ) log 5 n) and O((n/ ) 7/2 log 5 n) time respectively.

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

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