Given a weighted graph G = ( V , E ) and a real number t ⩾ 1 , a t-spanner of G is a spanning subgraph G ′ with the property that for every edge xy in G, there exists a path between x and y in G ′ whose weight is no more than t times the weight of the edge xy. We review results and present open problems on different variants of the problem of constructing plane geometric t-spanners.

/pdf/on-plane-geometric-spanners-a-survey-and-open-problems-3bytq11rre.pdf

On plane geometric spanners: A survey and open problems

We study several geometric set cover and set packing problems involving configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an @e-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algorithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems.

Exact algorithms and APX-hardness results for geometric packing and covering problems

Given a set S of n points in the plane, and an integer k such that 0 ≤ k < n, we show that a geometric graph with vertex set S, at most n – 1 + k edges, and dilation O(n / (k + 1)) can be computed in time O(n log n). We also construct n–point sets for which any geometric graph with n – 1 + k edges has dilation Ω(n / (k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position.

/pdf/sparse-geometric-graphs-with-small-dilation-52acpyeny2.pdf

Sparse geometric graphs with small dilation

Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worst-case ratio of these two values, for all pairs of vertices, is called the vertex-to-vertex dilation of G.

We prove that computing a minimum-dilation graph that connects a given n-point set in the plane, using not more than a given number m of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. In addition, we show that the minimum dilation tree over a given point set may in fact contain edge crossings.

/pdf/computing-geometric-minimum-dilation-graphs-is-np-hard-37duyj0lxb.pdf

Computing geometric minimum-dilation graphs is NP-hard

In this paper, we study asymmetric power assignments that induce a low-energy k-strongly connected communication graph with spanner properties. We address two spanner models: energy and distance. The former serves as an indicator for the energy consumed in a message propagation between two nodes, while the latter reflects the geographic properties of routing in the induced communication graph. We consider a random wireless ad hoc network with IVI = n nodes distributed uniformly and independently in a unit square. For k ∈ {1, 2}, we propose several power assignments that obtain a good bicriteria approximation on the total cost and stretch factor under the two models. For k > 2, we analyze a power assignment developed by Carmi et al. and derive some interesting bounds on the stretch factor for both models as well. We also describe how to compute all the power assignments distributively, and we provide simulation results. To the best of our knowledge, these are the first provable theoretical bounds for low-cost spanners in wireless ad hoc networks.

Near-optimal multicriteria spanner constructions in wireless ad hoc networks

We study several set cover problems in low dimensional geometric settings. Specifically, we describe a PTAS for the problem of computing a minimum cover of given points by a set of weighted fat objects. Here, we allow the objects to expand by some prespecified δ-fraction of their diameter. Next, we show that the problem of computing a minimum weight cover of points by weighted halfplanes (without expansion) can be solved exactly in the plane. We also study the problem of covering ℝ d by weighted halfspaces, and provide approximation algorithms and hardness results. We also investigate the dual settings of computing minimum weight simplex that covers a given target point. Finally, we provide a near linear time algorithm for the problem of solving a LP minimizing the total weight of violated constraints needed to be removed to make it feasible.

https://jocg.org/index.php/jocg/article/download/2950/2644

Weighted geometric set cover problems revisited

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(nlog^3n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region.

https://geometrica.saclay.inria.fr/team/Marc.Glisse/publications/voronoi/FPVD.pdf

Farthest-polygon Voronoi diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region.

Farthest-Polygon Voronoi Diagrams

In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists.

Computing a Minimum-Dilation Spanning Tree is NP-hard

In a geometric network G=(S,E), the graph distance between two vertices u,[email protected]?S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation @d>1, it is NP-hard to determine whether a spanning tree of S with dilation at most @d exists.

Mira Lee

Papers

Weighted geometric set cover problems revisited

Farthest-polygon Voronoi diagrams

Farthest-Polygon Voronoi Diagrams

Computing a Minimum-Dilation Spanning Tree is NP-hard

Computing a minimum-dilation spanning tree is NP-hard