Abstract. In the Euclidean group Traveling Salesman Problem (TSP), we are given a set of points P in the plane and a set of m connected regions, each containing at least one point of P. We want to find a tour of minimum length that visits at least one point in each region. This unifies the TSP with Neighborhoods and the Group Steiner Tree problem. We give a (9.1α + 1)-approximation algorithm for the case when the regions are disjoint α-fat objects with possibly varying size. This considerably improves the best results known, in this case, for both the group Steiner tree problem and the TSP with Neighborhoods problem. We also give the first O(1)-approximation algorithm for the problem with intersecting regions. 1

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NP-hardness for some other variants.

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.

Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log 3 n), or O(log 2 n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group. 1

Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NP-complete to decide whether it is possible to place the points inside their regions in such a way that the resulting polygon is simple. Furthermore, it is NP-hard to minimize the length of a simple tour visiting the regions in order, when the connections between consecutive regions do not need to be straight line segments. 1

In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We give (i) an O(α)-approximation algorithm for the case when the regions are disjoint and α-fat, with possibly varying size; (ii) an O(α³)-approximation algorithm for intersecting α-fat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(log n)-approximation algorithm for continuous non-fat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low running-time complexities.

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of

Abstract. In this paper we introduce and study cooperative variants of the Traveling Salesperson Problem. In these problems a salesperson has to make deliveries to customers who are willing to help in the process. Customer cooperativeness may be manifested in several modes: they may assist by approaching the salesperson, by reselling the goods they purchased to other customers, or by doing both. Several objectives are of interest: minimizing the total distance traveled by all the participants, minimizing the maximal distance traveled by a participant and minimizing the total time until all the deliveries are made. All the combinations of cooperation-modes and objective functions are considered, both in weighted undirected graphs and in Euclidean space. We show that most of the problems have a constant approximation algorithm, many of the others admit a PTAS, and a few are solvable in polynomial time. On the intractability side we provide NP-hardness proofs and inapproximability factors, some of which are tight. 1

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Given a rectangular boundary partitioned into rectangles, the Minimum-Length Corridor (MLC-R) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the boundary of the rectangles, and must include at least one point from the boundary of every rectangle and from the rectangular boundary. The MLC-R problem has been shown to be NP-hard. In this paper we present the first polynomial time constant ratio approximation algorithm for the MLC-R and MLCn problems. The MLCn problem is a generalization of the the MLC-R problem where the rectangles are rectilinear k-gons, for k ≤ n. We also present a polynomial time constant ratio approximation algorithm for the Group Traveling Salesperson Problem (GTSP) for a rectangle partitioned into rectilinear k-gons as in the MLCn problem.