David Saulpic

Publications

• Improved Coresets for Euclidean k-Means
  Joint work with Vincent Cohen-Addad, Kasper Green Larsen, Chris Schwiegelshohn and Omar Ali Sheikh-Omar
  NeurIPS 2022
  [arXiv]
  Click to see the abstract

  Given a set of n points in d dimensions, the Euclidean k-means problem (resp. the Euclidean k-median problem) 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+/- \eps) factor. The current state of the art coreset size is O(min(k^2 \eps^{-2}, k \eps^{-4}) for Euclidean k-means and O(min(k^2\eps^{-2}, k \eps^{-3}))for Euclidean k-median. The best known lower bound for both problems is \Omega(k \eps^{-2}).
  In this paper, we improve the upper bounds to O(min(k^{3/2}\eps^{-2},k\eps^{-4})) for k-means and O(min(k^{4/3}\eps^{-2}, k\eps^{-3})) for k-median. In particular, ours is the first provable bound that breaks through the k^2 barrier while retaining an optimal dependency on \eps.

  
  
• Scalable Differentially Private Clustering via Hierarchically Separated Trees
  Joint work with Vincent Cohen-Addad, Alessandro Epasto, Silvio Lattanzi, Vahab Mirrokni, Andres Munoz Medina, Chris Schwiegelshohn and Sergei Vassilvitskii
  KDD 2022
  [arXiv]
  Click to see the abstract

  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(d^{3/2}\log n)\cdot OPT + O(k d^2 \log^2 n / \epsilon^2), where \epsilon 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, \tilde{O}(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.

  
  
• Community Recovery in the Degree-Heterogeneous Stochastic Block Model
  Joint work with Vincent Cohen-Addad and Frederik Mallmann-Trenn
  COLT 2022
  
  Click to see the abstract

  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.

  
  
• A Massively Parallel Modularity-Maximizing Algorithm With Provable Guarantees
  Joint work with Vincent Cohen-Addad and Frederik Mallmann-Trenn
  PODC 2022
  
  Click to see the abstract

  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. Yet, 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, Kosowski, Mallmann-Trenn and Saulpic [NeurIPS 2020]

  
  
• Towards Optimal Lower Bounds for k-median and k-means Coresets
  Joint work with Vincent Cohen-Addad, Kasper Green Larsen and Chris Schwiegelshohn
  STOC 2022
  [arXiv]
  Click to see the abstract

  Given a set of points in a metric space, 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. Special cases include the famous 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 +/- eps) factor, hence reducing from large to small scale the input to the problem. While there has been an intensive effort to understand what is the best coreset size possible for both problems in various metric spaces, there is still a significant gap between the state- of-the-art upper and lower bounds.
  In this paper, we make progress on both upper and lower bounds, obtaining tight bounds for several cases, namely: in finite n point general metrics, any coreset must consist of Omega(k/eps^2 log n) points. This improves on the Omega(k/eps log n) lower bound of Braverman, Jiang, Krauthgamer, and Wu [ICML'19] and matches the upper bounds proposed for k-median by Feldman and Langberg [STOC'11] and k-means by Cohen-Addad, Saulpic, and Schwiegelshohn [STOC'21] up to polylog(1/eps) factors. The dependency in k, n is therefore optimal. For doubling metrics with doubling constant D, any coreset must consist of Omega(k/eps^2 D) points. This matches the k-median and k-means upper bounds by Cohen-Addad, Saulpic, and Schwiegelshohn [STOC'21] up to polylog(1/eps) factors. The dependency in k, D is therefore optimal. In d-dimensional Euclidean space, any coreset for (k, z) clustering requires Omega(k/eps^2 ) points. This improves on the Omega(k/ eps) lower bound of Baker, Braverman, Huang, Jiang, Krauthgamer, and Wu [ICML'20] for k-median and complements the Omega(k min(d, 2^{z/20} )) lower bound of Huang and Vishnoi [STOC'20].
  We complement our lower bound for d-dimensional Euclidean space with the construction of a coreset of size O(k/eps^2 · min(eps^{-z} , k)). This improves over the O(k^2 eps^{-4}) upper bound for general power of z proposed by Braverman Jiang, Krauthgamer, and Wu [SODA'21] and over the O(k/eps^4 ) upper bound for k-median by Huang and Vishnoi [STOC'20]. In fact, ours is the first construction breaking through the eps^{-2} min(d, eps^-2 ) barrier inherent in all previous coreset constructions. To do this, we employ a novel chaining based analysis that may be of independent interest. Together our upper and lower bounds for k-median in Euclidean spaces are tight up to a factor O(eps^{-1} polylog k/eps).

  
  
• An Improved Local Search Algorithm for k-Median
  Joint work with Vincent Cohen-Addad, Anupam Gupta, Lunjia Hu and Hoon Oh
  SODA 2022
  [arXiv]
  Click to see the abstract

  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+\epsilon)-approximation; our result improves upon the (3+\epsilon)-approximate local-search algorithm of Arya et al. [STOC'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.

  
  
• Improved Coresets and Sublinear Algorithms for Power Means in Euclidean Spaces.
  Joint work with Vincent Cohen-Addad and Chris Schwiegelshohn
  Spotlight presentation at NeurIPS 2021
  [Paper], [Online presentation at NeurIPS]
  Click to see the abstract

  We study in this paper the geometric (1, z)-clustering problem: given n points in R^d, find the point x that minimizes the sum of Euclidean distance, raised to the power z, over all input points. This problem interpolates between the well-known Fermat-Weber problem -- or geometric median problem-- where z = 1, and the Minimum Enclosing Ball problem, where z = infinity.
  
  Our contribution is the design of a precise estimator that sample only a constant number of points. Namely, for any \epsilon > 0, we show that sampling uniformly at random O(\epsilon^{-z-3}) input points is enough to find a center such that the sum of distances to the power z to that center is within a (1+\epsilon)-factor of the optimum. We also provide a lower bound, showing that any such algorithm must sample at least \Omega(\epsilon^{-z+1}) points.
  
  This implies an algorithm that computes a (1+\epsilon)-approximation running in time O(d \epsilon^{-z-3}), generalizing the result from Cohen et al [STOC '16] to arbitrary z. This also implies a (1+\epsilon)-approximation in the streaming setting, with memory independent of the number of input points.

  
  
• A New Coreset Framework for Clustering.
  Joint work with Vincent Cohen-Addad and Chris Schwiegelshohn
  STOC 2021
  [arXiv] [Long presentation at IRIF] [Short presentation at STOC]
  Click to see the abstract

  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.

  
  
• On the Power of Louvain for Graph Clustering.
  Joint work with Vincent Cohen-Addad, Adrian Kosowski and Frederik Mallmann-Trenn
  NeurIPS 2020
  [Paper]
  Click to see the abstract

  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).

  
  
• Polynomial Time Approximation Schemes for Clustering in Low Highway Dimension Graphs.
  Joint work with Andreas Emil Feldmann
  ESA 2020, invited to JCSS
  [arXiv], [Online presentation at ESA]
  Click to see the abstract

  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.

  
  
• Fully Dynamic Consistent Facility Location
  Joint work with Vincent Cohen-Addad, Niklas Hjuler, Nikos Parotsidis and Chris Schwiegelshohn
  NeurIPS 2019
  [Paper]
  Click to see the abstract

  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(nlog 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 O((n+k^2) polylog n). 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.

  
  
• Linear-Time Approximation Schemes for Clustering in Doubling Metrics
  Joint work with Vincent Cohen-Addad and Andreas Emil Feldmann
  FOCS 2019, JACM
  [arXiv], [Vincent's talk at ICERM]
  Click to see the abstract

  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 near-linear in n, making a significant improvement over the state-of-the-art algorithms which run in time $n^(d/\eps)^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-Medians and k-Means, and efficient bicriteria approximation schemes for k-Medians with outliers, k-Means with outliers and k-Center

  
  
• Dominating Sets and Connected Dominating Sets in Dynamic Graphs.
  Joint work with Niklas Hjuler, Giuseppe F. Italiano and Nikos Parotsidis
  STACS 2019
  [arXiv]
  Click to see the abstract

  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(Δ polylog n) per update, where Δ 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(Δ,√m))

  
  
• Polynomial-Time Approximation Schemes for k-center, k-median, and Capacitated Vehicle Routing in Bounded Highway Dimension.
  Joint work with Amariah Becker and Phil Klein
  ESA 2018
  [arXiv]
  Click to see the abstract

  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 u and v is preserved up to an additive error of ε times the distance from u or v 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 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 distinguished vertices. The treewidth depends on |S|, and the distance between u and v is preserved up to an additive error of ε times the distance from u and v to S. 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 k, there is a PTAS for k-Center in graphs of bounded highway dimension. In this problem, the goal is to minimize d such 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.

  
  
• A Quasi-Polynomial-Time Approximation Scheme for Vehicle Routing on Planar and Bounded-Genus Graphs
  Joint work with Amariah Becker and Phil Klein
  ESA 2017
  [Paper]
  Click to see the abstract

  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.