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.

Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and q. We consider the problem of designing algorithms that, for an arbitrary constant > 0, compute an -approximation to this stretch factor, i.e., a value t such that t t (1 + )t. We give eÆcient solutions for the cases when G is a path, cycle, or tree. The main idea used in all the algorithms is to use well-separated pair decompositions to speed up the computations. 1 Introduction Let S be a set of n points in R d , where d 1 is a small constant, and let G be an undirected connected graph having the points of S as its vertices. The length of any edge (p; q) of G is dened as the Euclidean distance jpqj between the two vertices p and q. The length of a path in G is dened a...

We give an O(n log n) approximation scheme for Steiner tree in planar graphs. 1

Given a set S of n points in the plane, we dene a Manhattan Network on S as a rectilinear network G with the property that for every pair of points in S, the network G contains the shortest rectilinear path between them. A Minimum Manhattan Network on S is a Manhattan network of minimum possible length. A Manhattan network can be thought of as a graph G = (V; E), where the vertex set V corresponds to points from S and a set of Steiner points S 0 , and the edges in E correspond to horizontal or vertical line segments connecting points in S [S 0 .

Abstract. We give an algorithm that, for any ɛ>0, any undirected planar graph G,andanysetS of nodes of G, computes a (1+ɛ)-optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2 poly(1/ɛ) n log n). 1

We present new polynomial-time approximation schemes (PTAS) for several basic minimum-cost multi-connectivity problems in geometrical graphs. We focus on low connectivity requirements. Each of our schemes either significantly improves the previously known upper time-bound or is the first PTAS for the considered problem. We provide a randomized approximation scheme for finding a biconnected graph spanning a set of points in a multi-dimensional Euclidean space and having the expected total cost within (1+ε) of the optimum. For any constant dimension and ε, our scheme runs in time O(n log n). It can be turned into Las Vegas one without affecting its asymptotic time complexity, and also efficiently derandomized. The only previously known truly polynomial-time approximation (randomized) scheme for this problem runs in expected time n · (log n) O((log log n)9) in the simplest planar case. The efficiency of our scheme relies on transformations of nearly optimal low cost special spanners into sub-multigraphs having good decomposition

We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces. We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t> 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal. Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.

We propose two algorithms for the planar Euclidean traveling salesman problem. The first runs in O(k!kn) time and O(k) space, and the second runs in O(2 k k 2 n) time and O(2 k kn) space, where n denotes the number of input points and k denotes the number of points interior to the convex hull.

A geometric spanner with vertex set P ae IR D is a sparse approximation of the complete Euclidean graph determined by P . We introduce the notion of partitioned neighborhood graphs (PNGs), unifying and generalizing most constructions of spanners treated in literature. Two important parameters characterizing their properties are the outdegree k 2 IN and the stretch factor f ? 1 describing the `quality' of the approximation. PNGs have been throughly investigated with respect to small values of f . We present in this work results about small values of k. The aim of minimizing k rather than f arises from two observations: a) k determines the amount of space required for storing PNGs. b) Many algorithms employing a (previously constructed) spanner have running times depending on its outdegree. Our results include, for fixed dimensions D as well as asymptotically, upper and lower bounds on this optimal value of k. The upper bounds are shown constructively and yield efficient algorithms f...

Abstract. We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1 + ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have non-uniform (1 or 2) connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2. 1