arXiv subjects

Vincent Cohen-Addad

Publications and source records attributed to Vincent Cohen-Addad.

At least 73 records · Page 4Linked to original sources

arXiv2023

Handling Correlated Rounding Error via Preclustering: A 1.73-approximation for Correlation Clustering

We consider the classic Correlation Clustering problem: Given a complete graph where edges are labelled either $+$ or $-$, the goal is to find a partition of the vertices that minimizes the number of the \pedges across parts plus the number of the \medges within parts. Recently, Cohen-Addad, Lee and Newman [CLN22] presented a 1.994-approximation algorithm for the problem using the Sherali-Adams hierarchy, hence breaking through the integrality gap of 2 for the classic linear program and improving upon the 2.06-approximation of Chawla, Makarychev, Schramm and Yaroslavtsev [CMSY15]. We significantly improve the state-of-the-art by providing a 1.73-approximation for the problem. Our approach introduces a preclustering of Correlation Clustering instances that allows us to essentially ignore the error arising from the {\em correlated rounding} used by [CLN22]. This additional power simplifies the previous algorithm and analysis. More importantly, it enables a new {\em set-based rounding} that complements the previous roundings. A combination of these two rounding algorithms yields the improved bound.

cs.DS↗

arXiv2023

On Complexity of 1-Center in Various Metrics

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=ω(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=Ω(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+ε)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

cs.CC↗

arXiv2023

Differentially-Private Hierarchical Clustering with Provable Approximation Guarantees

Hierarchical Clustering is a popular unsupervised machine learning method with decades of history and numerous applications. We initiate the study of differentially private approximation algorithms for hierarchical clustering under the rigorous framework introduced by (Dasgupta, 2016). We show strong lower bounds for the problem: that any $ε$-DP algorithm must exhibit $O(|V|^2/ ε)$-additive error for an input dataset $V$. Then, we exhibit a polynomial-time approximation algorithm with $O(|V|^{2.5}/ ε)$-additive error, and an exponential-time algorithm that meets the lower bound. To overcome the lower bound, we focus on the stochastic block model, a popular model of graphs, and, with a separation assumption on the blocks, propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly. Finally, we perform an empirical study of our algorithms and validate their performance.

cs.LG↗

arXiv2023

Correlation Clustering with Sherali-Adams

Given a complete graph $G = (V, E)$ where each edge is labeled $+$ or $-$, the Correlation Clustering problem asks to partition $V$ into clusters to minimize the number of $+$edges between different clusters plus the number of $-$edges within the same cluster. Correlation Clustering has been used to model a large number of clustering problems in practice, making it one of the most widely studied clustering formulations. The approximability of Correlation Clustering has been actively investigated [BBC04, CGW05, ACN08], culminating in a $2.06$-approximation algorithm [CMSY15], based on rounding the standard LP relaxation. Since the integrality gap for this formulation is 2, it has remained a major open question to determine if the approximation factor of 2 can be reached, or even breached. In this paper, we answer this question affirmatively by showing that there exists a $(1.994 + ε)$-approximation algorithm based on $O(1/ε^2$) rounds of the Sherali-Adams hierarchy. In order to round a solution to the Sherali-Adams relaxation, we adapt the {\em correlated rounding} originally developed for CSPs [BRS11, GS11, RT12]. With this tool, we reach an approximation ratio of $2+ε$ for Correlation Clustering. To breach this ratio, we go beyond the traditional triangle-based analysis by employing a global charging scheme that amortizes the total cost of the rounding across different triangles.

cs.DS↗

arXiv2023

Planar and Minor-Free Metrics Embed into Metrics of Polylogarithmic Treewidth with Expected Multiplicative Distortion Arbitrarily Close to 1

We prove that there is a randomized polynomial-time algorithm that given an edge-weighted graph $G$ excluding a fixed-minor $Q$ on $n$ vertices and an accuracy parameter $\varepsilon>0$, constructs an edge-weighted graph~$H$ and an embedding $η\colon V(G)\to V(H)$ with the following properties: * For any constant size $Q$, the treewidth of $H$ is polynomial in $\varepsilon^{-1}$, $\log n$, and the logarithm of the stretch of the distance metric in $G$. * The expected multiplicative distortion is $(1+\varepsilon)$: for every pair of vertices $u,v$ of $G$, we have $\mathrm{dist}_H(η(u),η(v))\geq \mathrm{dist}_G(u,v)$ always and $\mathrm{Exp}[\mathrm{dist}_H(η(u),η(v))]\leq (1+\varepsilon)\mathrm{dist}_G(u,v)$. Our embedding is the first to achieve polylogarithmic treewidth of the host graph and comes close to the lower bound by Carroll and Goel, who showed that any embedding of a planar graph with $\mathcal{O}(1)$ expected distortion requires the host graph to have treewidth $Ω(\log n)$. It also provides a unified framework for obtaining randomized quasi-polynomial-time approximation schemes for a variety of problems including network design, clustering or routing problems, in minor-free metrics where the optimization goal is the sum of selected distances. Applications include the capacitated vehicle routing problem, and capacitated clustering problems.

cs.DS↗

arXiv2022

Graph Searching with Predictions

Consider an agent exploring an unknown graph in search of some goal state. As it walks around the graph, it learns the nodes and their neighbors. The agent only knows where the goal state is when it reaches it. How do we reach this goal while moving only a small distance? This problem seems hopeless, even on trees of bounded degree, unless we give the agent some help. This setting with ''help'' often arises in exploring large search spaces (e.g., huge game trees) where we assume access to some score/quality function for each node, which we use to guide us towards the goal. In our case, we assume the help comes in the form of distance predictions: each node $v$ provides a prediction $f(v)$ of its distance to the goal vertex. Naturally if these predictions are correct, we can reach the goal along a shortest path. What if the predictions are unreliable and some of them are erroneous? Can we get an algorithm whose performance relates to the error of the predictions? In this work, we consider the problem on trees and give deterministic algorithms whose total movement cost is only $O(OPT + Δ\cdot ERR)$, where $OPT$ is the distance from the start to the goal vertex, $Δ$ the maximum degree, and the $ERR$ is the total number of vertices whose predictions are erroneous. We show this guarantee is optimal. We then consider a ''planning'' version of the problem where the graph and predictions are known at the beginning, so the agent can use this global information to devise a search strategy of low cost. For this planning version, we go beyond trees and give an algorithms which gets good performance on (weighted) graphs with bounded doubling dimension.

cs.DS↗

arXiv2022

Streaming Euclidean MST to a Constant Factor

We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an $n$-point set $X \subset \mathbb{R}^d$. In the streaming model, the points in $X$ can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, $(1+ε)$ approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was $\tilde{O}(\log n)$, due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improving on the prior $O(\log^2 n)$ bound due to [Indyk, STOC '04] and [Andoni, Indyk, Krauthgamer, SODA '08]. In this paper, we break the logarithmic barrier, and give the first constant factor sublinear space approximation to Euclidean MST. For any $ε\geq 1$, our algorithm achieves an $\tilde{O}(ε^{-2})$ approximation in $n^{O(ε)}$ space. We complement this by proving that any single pass algorithm which obtains a better than $1.10$-approximation must use $Ω(\sqrt{n})$ space, demonstrating that $(1+ε)$ approximations are not possible in high-dimensions, and that our algorithm is tight up to a constant. Nevertheless, we demonstrate that $(1+ε)$ approximations are possible in sublinear space with $O(1/ε)$ passes over the stream. More generally, for any $α\geq 2$, we give a $α$-pass streaming algorithm which achieves a $(1+O(\frac{\log α+ 1}{ αε}))$ approximation in $n^{O(ε)} d^{O(1)}$ space. Our streaming algorithms are linear sketches, and therefore extend to the massively-parallel computation model (MPC). Thus, our results imply the first $(1+ε)$-approximation to Euclidean MST in a constant number of rounds in the MPC model.

cs.DS↗

arXiv2022

Improved Coresets for Euclidean $k$-Means

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 $(1\pm \varepsilon)$ 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 $Ω(k \varepsilon^{-2})$. In this paper, we improve the upper bounds $\tilde O(\min(k^{3/2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for $k$-means and $\tilde O(\min(k^{4/3} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for $k$-median. In particular, ours is the first provable bound that breaks through the $k^2$ barrier while retaining an optimal dependency on $\varepsilon$.

cs.CG↗

arXiv2022

The Power of Uniform Sampling for Coresets

Motivated by practical generalizations of the classic $k$-median and $k$-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of $n$, the number of input points. Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for $1$-median in low-dimensional Euclidean spaces, specifically of size $\tilde{O}(\varepsilon^{-1.5})$ in $\mathbb{R}^2$ and $\tilde{O}(\varepsilon^{-1.6})$ in $\mathbb{R}^3$.

cs.DS↗

arXiv2022

On the Fixed-Parameter Tractability of Capacitated Clustering

We study the complexity of the classic capacitated k-median and k-means problems parameterized by the number of centers, k. These problems are notoriously difficult since the best known approximation bound for high dimensional Euclidean space and general metric space is $Θ(\log k)$ and it remains a major open problem whether a constant factor exists. We show that there exists a $(3+ε)$-approximation algorithm for the capacitated k-median and a $(9+ε)$-approximation algorithm for the capacitated k-means problem in general metric spaces whose running times are $f(ε,k) n^{O(1)}$. For Euclidean inputs of arbitrary dimension, we give a $(1+ε)$-approximation algorithm for both problems with a similar running time. This is a significant improvement over the $(7+ε)$-approximation of Adamczyk et al. for k-median in general metric spaces and the $(69+ε)$-approximation of Xu et al. for Euclidean k-means.

cs.DS↗

arXiv2022

Fitting Metrics and Ultrametrics with Minimum Disagreements

Given $x \in (\mathbb{R}_{\geq 0})^{\binom{[n]}{2}}$ recording pairwise distances, the METRIC VIOLATION DISTANCE (MVD) problem asks to compute the $\ell_0$ distance between $x$ and the metric cone; i.e., modify the minimum number of entries of $x$ to make it a metric. Due to its large number of applications in various data analysis and optimization tasks, this problem has been actively studied recently. We present an $O(\log n)$-approximation algorithm for MVD, exponentially improving the previous best approximation ratio of $O(OPT^{1/3})$ of Fan et al. [ SODA, 2018]. Furthermore, a major strength of our algorithm is its simplicity and running time. We also study the related problem of ULTRAMETRIC VIOLATION DISTANCE (UMVD), where the goal is to compute the $\ell_0$ distance to the cone of ultrametrics, and achieve a constant factor approximation algorithm. The UMVD can be regarded as an extension of the problem of fitting ultrametrics studied by Ailon and Charikar [SIAM J. Computing, 2011] and by Cohen-Addad et al. [FOCS, 2021] from $\ell_1$ norm to $\ell_0$ norm. We show that this problem can be favorably interpreted as an instance of Correlation Clustering with an additional hierarchical structure, which we solve using a new $O(1)$-approximation algorithm for correlation clustering that has the structural property that it outputs a refinement of the optimum clusters. An algorithm satisfying such a property can be considered of independent interest. We also provide an $O(\log n \log \log n)$ approximation algorithm for weighted instances. Finally, we investigate the complementary version of these problems where one aims at choosing a maximum number of entries of $x$ forming an (ultra-)metric. In stark contrast with the minimization versions, we prove that these maximization versions are hard to approximate within any constant factor assuming the Unique Games Conjecture.

cs.DS↗

arXiv2022

A New Coreset Framework for Clustering

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 \emph{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.

cs.DS↗

arXiv2022

Scalable Differentially Private Clustering via Hierarchically Separated Trees

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

cs.DS↗

arXiv2022

Beyond Impossibility: Balancing Sufficiency, Separation and Accuracy

Among the various aspects of algorithmic fairness studied in recent years, the tension between satisfying both \textit{sufficiency} and \textit{separation} -- e.g. the ratios of positive or negative predictive values, and false positive or false negative rates across groups -- has received much attention. Following a debate sparked by COMPAS, a criminal justice predictive system, the academic community has responded by laying out important theoretical understanding, showing that one cannot achieve both with an imperfect predictor when there is no equal distribution of labels across the groups. In this paper, we shed more light on what might be still possible beyond the impossibility -- the existence of a trade-off means we should aim to find a good balance within it. After refining the existing theoretical result, we propose an objective that aims to balance \textit{sufficiency} and \textit{separation} measures, while maintaining similar accuracy levels. We show the use of such an objective in two empirical case studies, one involving a multi-objective framework, and the other fine-tuning of a model pre-trained for accuracy. We show promising results, where better trade-offs are achieved compared to existing alternatives.

cs.LG↗

arXiv2022

Improved Approximations for Euclidean $k$-means and $k$-median, via Nested Quasi-Independent Sets

Motivated by data analysis and machine learning applications, we consider the popular high-dimensional Euclidean $k$-median and $k$-means problems. We propose a new primal-dual algorithm, inspired by the classic algorithm of Jain and Vazirani and the recent algorithm of Ahmadian, Norouzi-Fard, Svensson, and Ward. Our algorithm achieves an approximation ratio of $2.406$ and $5.912$ for Euclidean $k$-median and $k$-means, respectively, improving upon the 2.633 approximation ratio of Ahmadian et al. and the 6.1291 approximation ratio of Grandoni, Ostrovsky, Rabani, Schulman, and Venkat. Our techniques involve a much stronger exploitation of the Euclidean metric than previous work on Euclidean clustering. In addition, we introduce a new method of removing excess centers using a variant of independent sets over graphs that we dub a "nested quasi-independent set". In turn, this technique may be of interest for other optimization problems in Euclidean and $\ell_p$ metric spaces.

cs.DS↗

arXiv2022

Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor

We consider the numerical taxonomy problem of fitting a positive distance function ${D:{S\choose 2}\rightarrow \mathbb R_{>0}}$ by a tree metric. We want a tree $T$ with positive edge weights and including $S$ among the vertices so that their distances in $T$ match those in $D$. A nice application is in evolutionary biology where the tree $T$ aims to approximate the branching process leading to the observed distances in $D$ [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees, and for the special case of ultrametrics with a root having the same distance to all vertices in $S$. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was $O((\log n)(\log \log n))$ by Ailon and Charikar [2005] who wrote "Determining whether an $O(1)$ approximation can be obtained is a fascinating question".

cs.DS↗

arXiv2022

Improved Approximation Algorithms and Lower Bounds for Search-Diversification Problems

We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds. - We give a polynomial-time approximation scheme (PTAS) for a diversified search ranking problem [Bansal et al., ICALP 2010] whose objective is to minimizes the discounted cumulative gain. Our PTAS runs in time $n^{2^{O(\log(1/ε)/ε)}} \cdot m^{O(1)}$ where $n$ denotes the number of elements in the databases. Complementing this, we show that no PTAS can run in time $f(ε) \cdot (nm)^{2^{o(1/ε)}}$ assuming Gap-ETH; therefore our running time is nearly tight. Both of our bounds answer open questions of Bansal et al. - We next consider the Max-Sum Dispersion problem, whose objective is to select $k$ out of $n$ elements that maximizes the dispersion, which is defined as the sum of the pairwise distances under a given metric. We give a quasipolynomial-time approximation scheme for the problem which runs in time $n^{O_ε(\log n)}$. This improves upon previously known polynomial-time algorithms with approximate ratios 0.5 [Hassin et al., Oper. Res. Lett. 1997; Borodin et al., ACM Trans. Algorithms 2017]. Furthermore, we observe that known reductions rule out approximation schemes that run in $n^{\tilde{o}_ε(\log n)}$ time assuming ETH. - We consider a generalization of Max-Sum Dispersion called Max-Sum Diversification. In addition to the sum of pairwise distance, the objective includes another function $f$. For monotone submodular $f$, we give a quasipolynomial-time algorithm with approximation ratio arbitrarily close to $(1 - 1/e)$. This improves upon the best polynomial-time algorithm which has approximation ratio $0.5$ by Borodin et al. Furthermore, the $(1 - 1/e)$ factor is tight as achieving better-than-$(1 - 1/e)$ approximation is NP-hard [Feige, J. ACM 1998].

cs.DS↗

arXiv2022

Near-Optimal Correlation Clustering with Privacy

Correlation clustering is a central problem in unsupervised learning, with applications spanning community detection, duplicate detection, automated labelling and many more. In the correlation clustering problem one receives as input a set of nodes and for each node a list of co-clustering preferences, and the goal is to output a clustering that minimizes the disagreement with the specified nodes' preferences. In this paper, we introduce a simple and computationally efficient algorithm for the correlation clustering problem with provable privacy guarantees. Our approximation guarantees are stronger than those shown in prior work and are optimal up to logarithmic factors.

cs.LG↗