The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1). In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.

The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.

Clustering problems such as $k$-means and $k$-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.

Clustering is a cornerstone of data analysis that is particularly suited to identifying coherent subgroups or substructures in unlabeled data, as are generated continuously in large amounts these days. However, in many cases traditional clustering methods are not applicable, because data are increasingly being produced and stored in a distributed way, e.g. on edge devices, and privacy concerns prevent it from being transferred to a central server. To address this challenge, we present FedDP-KMeans, a new algorithm for $k$-means clustering that is fully-federated as well as differentially private. Our approach leverages (potentially small and out-of-distribution) server-side data to overcome the primary challenge of differentially private clustering methods: the need for a good initialization. Combining our initialization with a simple federated DP-Lloyds algorithm we obtain an algorithm that achieves excellent results on synthetic and real-world benchmark tasks. We also provide a theoretical analysis of our method that provides bounds on the convergence speed and cluster identification success.

We consider coresets for k-median problems, where the goal is to assign points to centers minimizing the sum of distances. Given a point set P, a coreset Ω is a small weighted subset that approximates the cost of P for all candidate solutions up to a (1 ± ε) multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for k-median coreset construction. As a consequence, we obtain improved k-median coreset bounds for the following metrics:

For a set of points in ℝd, the Euclidean k-means problems consists of finding k centers such that the sum of distances squared from each data point to its closest center is minimized. Coresets are one the main tools developed recently to solve this problem in a big data context. They allow to compress the initial dataset while preserving its structure: running any algorithm on the coreset provides a guarantee almost equivalent to running it on the full data. In this work, we study coresets in a fully-dynamic setting: points are added and deleted with the goal to efficiently maintain a coreset with which a k-means solution can be computed. Based on an algorithm from Henzinger and Kale [ESA’20], we present an efficient and practical implementation of a fully dynamic coreset algorithm, that improves the running time by up to a factor of 20 compared to our non-optimized implementation of the algorithm by Henzinger and Kale, without sacrificing more than 7% on the quality of the k-means solution.

We study in this paper the problem of maintaining a solution to k-median and k-means clustering in a fully dynamic setting. To do so, we present an algorithm to efficiently maintain a coreset, a compressed version of the dataset, that allows easy computation of a clustering solution at query time. Our coreset algorithm has near-optimal update time of Õ(k) in general metric spaces, which reduces to Õ(d) in the Euclidean space ℝ^d. The query time is O(k²) in general metrics, and O(kd) in ℝ^d. To maintain a constant-factor approximation for k-median and k-means clustering in Euclidean space, this directly leads to an algorithm with update time Õ(d), and query time Õ(kd + k²). To maintain a O(polylog k)-approximation, the query time is reduced to Õ(kd).

Coresets are arguably the most popular compression paradigm for center-based clustering objectives such as$k$-means. Given a point set$P$, a coreset$\Omega$is a small, weighted summary that preserves the cost of all candidate solutions$S$up to a$(1\pm\varepsilon)$factor. For$k$-means in$d$-dimensional Euclidean space the cost for solution$S$is$\Sigma_{p\in P}{\min}_{s\in S}\Vert p-s\Vert ^2$. A very popular method for coreset construction, both in theory and practice, is Sensitivity Sampling, where points are sampled in proportion to their importance. We show that Sensitivity Sampling yields optimal coresets of size$\widetilde{O}(k/\varepsilon^{2}\min(\sqrt{k},\varepsilon^{-2}))$for worst-case instances. Uniquely among all known coreset algorithms, for well-clusterable data sets with$\Omega(1)$, cost stability, Sensitivity Sampling gives coresets of size$\widetilde{O}(k/\varepsilon^{2})$, improving over the worst-case lower bound. Notably, Sensitivity Sampling does not have to know the cost stability in order to exploit it: it is appropriately sensitive to the clusterability of the data set while being oblivious to it. We also show that any coreset for stable instances consisting of only input points must have size$\Omega(k/\varepsilon^{2})$. Our results for Sensitivity Sampling also extend to the k-median problem, and more general metric spaces.

We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on $k$-means clustering and sensitivity sampling. Assuming access to an embedding representation of the data with respect to which the model loss is Holder continuous, our approach provably allows selecting a set of “typical” $k + 1/\varepsilon^2$ elements whose average loss corresponds to the average loss of the whole dataset, up to a multiplicative $(1\pm\varepsilon)$ factor and an additive $\varepsilon \lambda \Phi_k$, where $\Phi_k$ represents the $k$-means cost for the input embeddings and $\lambda$ is the Holder constant. We furthermore demonstrate the performance and scalability of our approach on fine-tuning foundation models and show that it outperforms state-of-the-art methods. We also show how it can be applied on linear regression, leading to a new sampling strategy that surprisingly matches the performance of leverage score sampling, while being conceptually simpler and more scalable.

As a staple of data analysis and unsupervised learning, the problem of private clustering has been widely studied, under various privacy models. Centralized differential privacy is the first of them, and the problem has also been studied for the local and the shuffle variation. In each case, the goal is to design an algorithm that computes privately a clustering, with the smallest possible error. The study of each variation gave rise to new algorithm: the landscape of private clustering algorithm is therefore quite intricate. In this paper, we show that a 20 year-old algorithm can be slightly modified to work for any of those models. This provides a unified picture: while matching almost all previously known results, it allows us to improve some of them, and extend to a new privacy model, the continual observation setting, where the input is changing over time and the algorithm must output a new solution at each time step.

We study the theoretical and practical runtime limits of k-means and k-median clustering on large datasets. Since effectively all clustering methods are slower than the time it takes to read the dataset, the fastest approach is to quickly compress the data and perform the clustering on the compressed representation. Unfortunately, there is no universal best choice for compressing the number of points -- while random sampling runs in sublinear time and coresets provide theoretical guarantees, the former does not enforce accuracy while the latter is too slow as the numbers of points and clusters grow. Indeed, it has been conjectured that any sensitivity-based coreset construction requires super-linear time in the datase size. We examine this relationship by first showing that there does exist an algorithm that obtains coresets via sensitivity sampling in effectively linear time -- within log-factors of the time it takes to read the data. Any approach that significantly improves on this must then resort to practical heuristics, leading us to consider the spectrum of sampling strategies across both real and artificial datasets in the static and streaming settings. Through this, we show the conditions in which coresets are necessary for preserving cluster validity as well as the settings in which faster, cruder sampling strategies are sufficient. As a result, we provide a comprehensive theoretical and practical blueprint for effective clustering regardless of data size. Our code is publicly available at https://github.com/Andrew-Draganov/Fast-Coreset-Generation and has scripts to recreate the experiments.

In all state-of-the-art sketching and coreset techniques for clustering, as well as in the best known fixed-parameter tractable approximation algorithms, randomness plays a key role. For the classic k-median and k-means problems, there are no known deterministic dimensionality reduction procedure or coreset construction that avoid an exponential dependency on the input dimension d, the precision parameter $\varepsilon^{-1}$ or k. Furthermore, there is no coreset construction that succeeds with probability $1-1/n$ and whose size does not depend on the number of input points, n. This has led researchers in the area to ask what is the power of randomness for clustering sketches [Feldman WIREs Data Mining Knowl. Discov’20].Similarly, the best approximation ratio achievable deterministically without a complexity exponential in the dimension are $1+\sqrt{2}$ for k-median [Cohen-Addad, Esfandiari, Mirrokni, Narayanan, STOC’22] and 6.12903 for k-means [Grandoni, Ostrovsky, Rabani, Schulman, Venkat, Inf. Process. Lett.’22]. Those are the best results, even when allowing a complexity FPT in the number of clusters k: this stands in sharp contrast with the $(1+\varepsilon)$-approximation achievable in that case, when allowing randomization.In this paper, we provide deterministic sketches constructions for clustering, whose size bounds are close to the best-known randomized ones. We show how to compute a dimension reduction onto $\varepsilon^{-O(1)} \log k$ dimensions in time $k^{O\left(\varepsilon^{-O(1)}+\log \log k\right)}$ poly $(n d)$, and how to build a coreset of size $O\left(k^{2} \log ^{3} k \varepsilon^{-O(1)}\right)$ in time $2^{\varepsilon^{O(1)} k \log ^{3} k}+k^{O\left(\varepsilon^{-O(1)}+\log \log k\right)}$ poly $(n d)$. In the case where k is small, this answers an open question of [Feldman WIDM’20] and [Munteanu and Schwiegelshohn, Künstliche Intell. ’18] on whether it is possible to efficiently compute coresets deterministically.We also construct a deterministic algorithm for computing $(1+$ $\varepsilon)$-approximation to k-median and k-means in high dimensional Euclidean spaces in time $2^{k^{2} \log ^{3} k / \varepsilon^{O(1)}}$ poly $(n d)$, close to the best randomized complexity of $2^{(k / \varepsilon)^{O(1)}}$ nd (see [Kumar, Sabharwal, Sen, JACM 10] and [Bhattacharya, Jaiswal, Kumar, TCS’18]).Furthermore, our new insights on sketches also yield a randomized coreset construction that uses uniform sampling, that immediately improves over the recent results of [Braverman et al. FOCS ’22] by a factor k.

We consider the problem of recovering communities in a random directed graph with planted communities. To model real-world directed graphs such as the Twitter or Instagram graphs that exhibit very heterogeneous degree sequences, we introduce the Degree-Heterogeneous Stochastic Block Model (DHSBM), a generalization of the classic Stochastic Block Model (SBM), where the vertex set is partitioned into communities and each vertex $u$ has two (unknown) associated probabilities, $p_u$ and $q_u$, $p_u > q_u$. An arc from $u$ to $v$ is generated with probability $p_u$ if $u$ and $v$ are in the same community and with probability $q_u$ otherwise. Given a graph generated from this model, the goal is to retrieve the communities. The DHSBM allows to generate graphs with planted communities while allowing heterogeneous degree distributions, a quite important feature of real-world networks. In the case where there are two communities, we present an iterative greedy linear-time algorithm that recovers them whenever $\min_u \frac{p_u - q_u}{\sqrt{p_u}} = \Omega(\sqrt{\log (n)/n})$. We also show that, up to a constant, this condition is necessary. Our results also extend to the standard (undirected) SBM, where $p_u = p$ and $q_u= q$ for all nodes $u$. Our algorithm presents the first linear-time algorithm that recovers exactly the communities at the asymptotic information-theoretic threshold, improving over previous near-linear time spectral approaches.

We study the private k-median and k-means clustering problem in d dimensional Euclidean space. By leveraging tree embeddings, we give an efficient and easy to implement algorithm, that is empirically competitive with state of the art non private methods. We prove that our method computes a solution with cost at most O(d3/2 log n)⁆ OPT + O(kd2 log2 n/ε2), where ε is the privacy guarantee. (The dimension term, d, can be replaced with O(log k) using standard dimension reduction techniques.) Although the worst-case guarantee is worse than that of state of the art private clustering methods, the algorithm we propose is practical, runs in near-linear, Õ (nkd), time and scales to tens of millions of points. We also show that our method is amenable to parallelization in large-scale distributed computing environments. In particular we show that our private algorithms can be implemented in logarithmic number of MPC rounds in the sublinear memory regime. Finally, we complement our theoretical analysis with an empirical evaluation demonstrating the algorithm's efficiency and accuracy in comparison to other privacy clustering baselines.

Given a set of $n$ points in $d$ dimensions, the Euclidean $k$-means problem (resp. Euclidean $k$-median) consists of finding $k$ centers such that the sum of squared distances (resp. sum of distances) from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a $(1\pm \varepsilon)$ factor. The current state of the art coreset size is $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for Euclidean $k$-means and $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for Euclidean $k$-median. The best known lower bound for both problems is $\Omega(k\varepsilon^{-2})$. In this paper, we improve these bounds to $\tilde O(\min(k^{3/2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for Euclidean $k$-means and $\tilde O(\min(k^{4/3} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for Euclidean $k$-median. In particular, ours is the first provable bound that breaks through the $k^2$ barrier while retaining an optimal dependency on $\varepsilon$.

Graph clustering is one of the most basic and popular unsupervised learning problems. Among the different formulations of the problem, the modularity objective has been particularly successful in helping design impactful algorithms; Most notably, the Louvain algorithm has become one of the most used algorithm for clustering graphs. However, one major limitation of the Louvain algorithm is its sequential nature which makes it impractical in distributed environments and on massive datasets. In this paper, we provide a parallel version of Louvain which works in the massively parallel computation model (MPC). We show that it recovers the ground-truth clusters in the classic stochastic block model in only a constant number of parallel rounds, and so for a wider regime of parameters than the standard Louvain algorithm as shown recently in [Cohen-Addad et al. 2020].

We present a new local-search algorithm for the k-median clustering problem. We show that local optima for this algorithm give a (2.836 + ∊)-approximation; our result improves upon the (3 + ∊)-approximate local-search algorithm of Arya et al. [AGK+01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution.

The (k,z)-clustering problem consists of finding a set of k points called centers, such that the sum of distances raised to the power of z of every data point to its closest center is minimized. Among the most commonly encountered special cases are k-median problem (z=1) and k-means problem (z=2). The k-median and k-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multiplicative (1± ε) factor, hence reducing from large to small scale the input to the problem.

In this paper, we consider the problem of finding high dimensional power means: given a set $A$ of $n$ points in $\R^d$, find the point $m$ that minimizes the sum of Euclidean distance, raised to the power $z$, over all input points. Special cases of problem include the well-known Fermat-Weber problem -- or geometric median problem -- where $z = 1$, the mean or centroid where $z=2$, and the Minimum Enclosing Ball problem, where $z = \infty$.We consider these problem in the big data regime.Here, we are interested in sampling as few points as possible such that we can accurately estimate $m$.More specifically, we consider sublinear algorithms as well as coresets for these problems.Sublinear algorithms have a random query access to the $A$ and the goal is to minimize the number of queries.Here, we show that $\tilde{O}(\varepsilon^{-z-3})$ samples are sufficient to achieve a $(1+\varepsilon)$ approximation, generalizing the results from Cohen, Lee, Miller, Pachocki, and Sidford [STOC '16] and Inaba, Katoh, and Imai [SoCG '94] to arbitrary $z$. Moreover, we show that this bound is nearly optimal, as any algorithm requires at least $\Omega(\varepsilon^{-z+1})$ queries to achieve said approximation.The second contribution are coresets for these problems, where we aim to find find a small, weighted subset of the points which approximate cost of every candidate point $c\in \mathbb{R}^d$ up to a $(1\pm\varepsilon)$ factor. Here, we show that $\tilde{O}(\varepsilon^{-2})$ points are sufficient, improving on the $\tilde{O}(d\varepsilon^{-2})$ bound by Feldman and Langberg [STOC '11] and the $\tilde{O}(\varepsilon^{-4})$ bound by Braverman, Jiang, Krauthgamer, and Wu [SODA 21].

Given a metric space, the (k,z)-clustering problem consists of finding k centers such that the sum of the of distances raised to the power z of every point to its closest center is minimized. This encapsulates the famous k-median (z=1) and k-means (z=2) clustering problems. Designing small-space sketches of the data that approximately preserves the cost of the solutions, also known as coresets, has been an important research direction over the last 15 years. In this paper, we present a new, simple coreset framework that simultaneously improves upon the best known bounds for a large variety of settings, ranging from Euclidean space, doubling metric, minor-free metric, and the general metric cases.

We consider the classic Facility Location, k -Median, and k -Means problems in metric spaces of doubling dimension d . We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is Õ(2 (1/ε) O(d2) n) , making a significant improvement over the state-of-the-art algorithms that run in time n (d/ε) O(d) . Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k -Median and k -Means and efficient bicriteria approximation schemes for k -Median with outliers, k -Means with outliers and k -Center.

We study clustering problems such as k-Median, k-Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) [Feldmann et al. SICOMP 2018] or run in FPT time (parameterized by the number of clusters $k$, the highway dimension, and the approximation factor) [Becker et al. ESA~2018, Braverman et al. 2020]. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1.

A classic problem in machine learning and data analysis is to partition the vertices of a network in such a way that vertices in the same set are densely connected and vertices in different sets are loosely connected. In practice, the most popular approaches rely on local search algorithms; not only for the ease of implementation and the efficiency, but also because of the accuracy of these methods on many real world graphs. For example, the Louvain algorithm -- a local search based algorithm -- has quickly become the method of choice for clustering in social networks. However, explaining the success of these methods remains an open problem: in the worst-case, the runtime can be up to \Omega(n^2), much worse than what is typically observed in practice, and no guarantee on the quality of its output can be established. The goal of this paper is to shed light on the inner-workings of Louvain; only if we understand Louvain, can we rely on it and further improve it. To achieve this goal, we study the behavior of Louvain in the famous two-bloc Stochastic Block Model, which has a clear ground-truth and serves as the standard testbed for graph clustering algorithms. We provide valuable tools for the analysis of Louvain, but also for many other combinatorial algorithms. For example, we show that the probability for a node to have more edges towards its own community is 1/2 + \Omega( \min( \Delta(p-q)/\sqrt{np},1 )) in the SBM(n,p,q), where \Delta is the imbalance. Note that this bound is asymptotically tight and useful for the analysis of a wide range of algorithms (Louvain, Kernighan-Lin, Simulated Annealing etc).

We consider the classic Facility Location, k-Median, and k-Means problems in metric spaces of constant doubling dimension. We give the first nearly linear-time approximation schemes for each problem, making a significant improvement over the state-of-the-art algorithms. Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k-Medians and k-Means, and efficient bicriteria approximation schemes for k-Medians with outliers, k-Means with outliers and k-Center.

We consider classic clustering problems in fully dynamic data streams, where data elements can be both inserted and deleted. In this context, several parameters are of importance: (1) the quality of the solution after each insertion or deletion, (2) the time it takes to update the solution, and (3) how different consecutive solutions are. The question of obtaining efficient algorithms in this context for facility location, $k$-median and $k$-means has been raised in a recent paper by Hubert-Chan et al. [WWW'18] and also appears as a natural follow-up on the online model with recourse studied by Lattanzi and Vassilvitskii [ICML'17] (i.e.: in insertion-only streams). In this paper, we focus on general metric spaces and mainly on the facility location problem. We give an arguably simple algorithm that maintains a constant factor approximation, with $O(n\log n)$ update time, and total recourse $O(n)$. This improves over the naive algorithm which consists in recomputing a solution at each time step and that can take up to $O(n^2)$ update time, and $O(n^2)$ total recourse. These bounds are nearly optimal: in general metric space, inserting a point take $O(n)$ times to describe the distances to other points, and we give a simple lower bound of $O(n)$ for the recourse. Moreover, we generalize this result for the $k$-medians and $k$-means problems: our algorithm maintains a constant factor approximation in time $\widetilde{O}(n+k^2)$. We complement our analysis with experiments showing that the cost of the solution maintained by our algorithm at any time $t$ is very close to the cost of a solution obtained by quickly recomputing a solution from scratch at time $t$ while having a much better running time.

In this paper we study the dynamic versions of two basic graph problems: Minimum Dominating Set and its variant Minimum Connected Dominating Set. For those two problems, we present algorithms that maintain a solution under edge insertions and edge deletions in time O(Delta * polylog n) per update, where Delta is the maximum vertex degree in the graph. In both cases, we achieve an approximation ratio of O(log n), which is optimal up to a constant factor (under the assumption that P != NP). Although those two problems have been widely studied in the static and in the distributed settings, to the best of our knowledge we are the first to present efficient algorithms in the dynamic setting. As a further application of our approach, we also present an algorithm that maintains a Minimal Dominating Set in O(min(Delta, sqrt{m})) per update.

The concept of bounded highway dimension was developed to capture observed properties of road networks. We show that a graph of bounded highway dimension with a distinguished root vertex can be embedded into a graph of bounded treewidth in such a way that u-to-v distance is preserved up to an additive error of epsilon times the u-to-root plus v-to-root distances. We show that this embedding 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 Q units of client demand. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding result to handle a set S of root vertices. This 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 k, there is a PTAS for k-Center in graphs of bounded highway dimension. In this problem, the goal is to minimize d so that there exist k vertices (the centers) such that every vertex is within distance d of some center. Similarly, for fixed k, there is a PTAS for k-Median in graphs of bounded highway dimension. In this problem, the goal is to minimize the sum of distances to the k centers.

The Capacitated Vehicle Routing problem is a generalization of the Traveling Salesman problem in which a set of clients must be visited by a collection of capacitated tours. Each tour can visit at most Q clients and must start and end at a specified depot. We present the first approximation scheme for Capacitated Vehicle Routing for non-Euclidean metrics. Specifically we give a quasi-polynomial-time approximation scheme for Capacitated Vehicle Routing with fixed capacities on planar graphs. We also show how this result can be extended to bounded-genus graphs and polylogarithmic capacities, as well as to variations of the problem that include multiple depots and charging penalties for unvisited clients.