In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constant-factor approximation algorithm for TSPN on an arbitrary set of disjoint, connected neighborhoods in the plane. Prior approximation bounds were O(log n), except in special cases. Our approximation algorithm applies to arbitrary connected neighborhoods of any size or shape. 1

Network coordinates, which embed network distance measurements in a coordinate system, were introduced as a method for determining the proximity of nodes for routing table updates in overlay networks. Their power has far broader reach: due to their low overhead and automatic adaptation to changes in the network, network coordinates provide a new paradigm for managing dynamic overlay networks. We compare network coordinates to other proposals for network-aware overlays and show how they permit the lucid expression of a range of distributed systems problems in well-understood geometric terms. 1.

In this paper we discuss a new type of query in Spatial Databases, called the Trip Planning Query (TPQ). Given a set of points of interest P in space, where each point belongs to a specific category, a starting point S and a destination E, TPQ retrieves the best trip that starts at S, passes through at least one point from each category, and ends at E. For example, a driver traveling from Boston to Providence might want to stop to a gas station, a bank and a post office on his way, and the goal is to provide him with the best possible route (in terms of distance, traffic, road conditions, etc.). The difficulty of this query lies in the existence of multiple choices per category. In this paper, we study fast approximation algorithms for TPQ in a metric space. We provide a number of approximation algorithms with approximation ratios that depend on either the number of categories, the maximum number of points

Let K be the worst-case (supremum) ratio of the weight of the minimum degree-K spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that

Abstract We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-restricted MST problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the lowest cost spanning tree with the property that all nodes have degree at most d. We also develop a polynomial time approximation scheme for the Euclidean version of the Red-Blue Separation Problem, again extending Arora's techniques. Given ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the cost of the optimum separating polygon of the input nodes, in nearly linear time.

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.

The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1979. Very recently the problem was shown to be NP-hard by Mulzer and Rote. In this paper, we present a quasi-polynomial time approximation scheme for Minimum Weight Triangulation.

This paper is concerned with polynomial time approximations schemes for the generalized geometric problems with geographic clustering. We illustrate the approach on the generalized traveling salesman problem which is also known as Group-TSP or TSP with neighborhoods. We prove that under the condition that all regions are non-intersecting and have comparable sizes and shapes, the problem admits PTAS. To derive a PTAS we extend the algorithm by Arora [2]. This extension involves the dissection mechanism and solution of the selection problem. We observe that the results are applicable to many generalized geometric problems, to other Minkowski norms, and to other fixed dimensional spaces.

We consider the rooted orienteering problem: Given a set P of n points in the plane, a starting point r ∈ P, and a length constraint B, one needs to find a path starting from r that visits as many points of P as possible and of length not exceeding B. We present a (1 − ε)-approximation algorithm for this problem that runs in n O(1/ε) time; the computed path visits at least (1 − ε)kopt points of P, where kopt is the number of points visited by an optimal solution. This is the first polynomial time approximation scheme (PTAS) for this problem. The algorithm also works in higher dimensions.

We give a randomized O(n polylog n)-time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed ǫ> 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1 + ǫ)-approximation to the minimum-length forest that connects every requested pair of terminals. 1