Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...

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NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NP-hard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), k-TSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), k-MST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum

We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph , with a non-negative integral function that specifies an upper bound on the degree of each vertex in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of nodes called terminals. The goal is to construct a Steiner containing all the terminals such that the degree of any node is at most the specified upper bound and the total cost of the nodes in is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria -- the degree of any node in the output Steiner tree is and the cost of the tree is times that of a minimum-cost Steiner tree that obeys the degree bound for each node . Our result extends to the more general problem of constructing one-connected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees.

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel network-flow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previously obtained. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for our algorithm. The performance guarantee is the following. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 \Gamma min n d(v)\Gamma2 deg T (v)\Gamma2 : deg T (v) ? 2 o ; where deg T (v) is the initial degree of v. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.

In a census, individual respondents give private information to a trusted party (the census bureau), who publishes a sanitized version of the data. There are two fundamentally conflicting requirements: privacy for the respondents and utility of the sanitized data. Note that this framework is inherently noninteractive. Recently, Chawla et al. (TCC’2005) initiated a theoretical study of the census problem and presented an intuitively appealing definition of privacy breach, called isolation, together with a formal specification of what is required from a data sanitization algorithm: access to the sanitized data should not increase an adversary’s ability to isolate any individual. They also showed that if the data are drawn uniformly from a highdimensional hypercube then recursive histogram sanitization can preserve privacy with a high probability. We extend these results in several ways. First, we develop a method for computing a privacy-preserving histogram sanitization of “round ” distributions, such as the uniform distribution over a high-dimensional ball or sphere. This problem is quite challenging because, unlike for the hypercube, the natural histogram over such a distribution may have long and thin cells that hurt the proof of privacy. We then develop techniques for randomizing the histogram constructions both for the hypercube and the hypersphere. These permit us to apply known results for approximating various quantities of interest (e.g., cost of the minimum spanning tree, or the cost of an optimal solution to the facility location problem over the data points) from histogram counts – in a privacy-preserving fashion. 1

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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.