The concept of bounded highway dimension was developed to capture observed properties of the metrics of road networks. We show that a graph with bounded highway dimension, for any vertex, can be embedded into a a graph of bounded treewidth in such a way that the distance between @math and @math is preserved up to an additive error of @math times the distance from @math or @math to the selected vertex. We show that this theorem yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension. In this problem, the input specifies a depot and a set of clients, each with a location and demand; the output is a set of depot-to-depot tours, where each client is visited by some tour and each tour covers at most @math units of client demand. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding result to handle a set @math of distinguished vertices. The treewidth depends on @math , and the distance between @math and @math is preserved up to an additive error of @math times the distance from @math and @math to @math . This embedding result implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing: the tours can go from one depot to another. The embedding result also implies that, for fixed @math , there is a PTAS for @math -Center in graphs of bounded highway dimension. In this problem, the goal is to minimize @math such that there exist @math vertices (the centers) such that every vertex is within distance @math of some center. Similarly, for fixed @math , there is a PTAS for @math -Median in graphs of bounded highway dimension. In this problem, the goal is to minimize the sum of distances to the @math centers.
For @math -Median , constant-factor approximation algorithms have been found for general metric spaces @cite_11 @cite_34 @cite_7 . The best known approximation ratio for @math -Median in general metrics is @math @cite_19 , and it is NP-hard to approximate within a factor of @math @cite_20 . For @math -Median in @math -dimensional Euclidean space, PTAS have been found when @math is fixed (e.g. @cite_24 ) and when @math is fixed (e.g. @cite_23 ) but there exists no PTAS if @math and @math are part of the input @cite_14 . Recently Cohen- @cite_29 gave a local search-based PTAS for @math -Median in edge-weight planar graphs that extends more generally to minor-closed graph families.
@cite_14 @cite_7 @cite_29 @cite_24 @cite_19 @cite_23 @cite_34 @cite_20 @cite_11
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We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs, (2) k-median and k-means in edge-weighted planar graphs, (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the pth power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution S consists of all solutions obtained from S by removing and adding 1 eO(1) centers. In this paper, we show that for several clustering problems one can extract a small set of points, so that using those core-sets enable us to perform approximate clustering efficiently. The surprising property of those core-sets is that their size is independent of the dimension.Using those, we present a (1+ e)-approximation algorithms for the k-center clustering and k-median clustering problems in Euclidean space. The running time of the new algorithms has linear or near linear dependency on the number of points and the dimension, and exponential dependency on 1 e and k. As such, our results are a substantial improvement over what was previously known.We also present some other clustering results including (1+ e)-approximate 1-cylinder clustering, and k-center clustering with outliers. We present a novel approximation algorithm for @math -median that achieves an approximation guarantee of @math , improving upon the decade-old ratio of @math . Our improved approximation ratio is achieved by exploiting the power of pseudo-approximation. More specifically, our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an @math -approximation algorithm for @math -median, it is sufficient to give a pseudo-approximation algorithm that finds an @math -approximate solution by opening @math facilities. This is a rather surprising result as there exist instances for which opening @math facilities may lead to a significantly smaller cost than that of opening only @math facilities. Second, we give such a pseudo-approximation algorithm with @math . Prior to our work, it was not even known whether opening @math facilities would help improve the approximation ratio. In the k-median problem we are given a set S of n points in a metric space and a positive integer k. We desire to locate k medians in space, such that the sum of the distances from each of the points of S to the nearest median is minimized. This paper gives an approximation scheme for the plane that for any c > 0 produces a solution of cost at most 1+ 1 c times the optimum and runs in time O(n). The approximation scheme also generalizes to some problems related to k-median. Our methodology is to extend Arora’s [1, 2] techniques for the TSP, which hitherto seemed inapplicable to problems such as the k-median problem. A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as theuncapacitated facility locationproblem. Application to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser, and Wolsey. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos, and Aardal. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos, and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is max SNP-hard. However, the inapproximability constants derived from the max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio, assumingNP?DTIMEnO(loglogn)]. One of the most flourishing areas of research in the design and analysis of approximation algorithms has been for facility location problems. In particular, for the metric case of two simple models, the uncapacitated facility location and the k-median problems, there are now a variety of techniques that yield constant performance guarantees. These methods include LP rounding, primal-dual algorithms, and local search techniques. Furthermore, the salient ideas in these algorithms and their analyzes are simple-to-explain and reflect a surprising degree of commonality. This note is intended as companion to our lecture at CONF 2000, mainly to give pointers to the appropriate references.