arXiv subjects

Vincent Cohen-Addad

Publications and source records attributed to Vincent Cohen-Addad.

At least 37 records · Page 2Linked to original sources

arXiv2026

Breaching the 2 LMP Approximation Barrier for Facility Location with Applications to k-Median

The Uncapacitated Facility Location (UFL) problem is one of the most fundamental clustering problems: Given a set of clients $C$ and a set of facilities $F$ in a metric space $(C \cup F, dist)$ with facility costs $open : F \to \mathbb{R}^+$, the goal is to find a set of facilities $S \subseteq F$ to minimize the sum of the opening cost $open(S)$ and the connection cost $d(S) := \sum_{p \in C} \min_{c \in S} dist(p, c)$. An algorithm for UFL is called a Lagrangian Multiplier Preserving (LMP) $α$ approximation if it outputs a solution $S\subseteq F$ satisfying $open(S) + d(S) \leq open(S^*) + αd(S^*)$ for any $S^* \subseteq F$. The best-known LMP approximation ratio for UFL is at most $2$ by the JMS algorithm of Jain, Mahdian, and Saberi based on the Dual-Fitting technique. We present a (slightly) improved LMP approximation algorithm for UFL. This is achieved by combining the Dual-Fitting technique with Local Search, another popular technique to address clustering problems. From a conceptual viewpoint, our result gives a theoretical evidence that local search can be enhanced so as to avoid bad local optima by choosing the initial feasible solution with LP-based techniques. Using the framework of bipoint solutions, our result directly implies a (slightly) improved approximation for the $k$-Median problem from 2.6742 to 2.67059.

cs.DS↗

arXiv2025

Algorithmic Thinking Theory

Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan for generating and combining a set of solutions can be thought of as an algorithm for reasoning using a probabilistic oracle. We introduce a theoretical framework for analyzing such reasoning algorithms. This framework formalizes the principles underlying popular techniques for iterative improvement and answer aggregation, providing a foundation for designing a new generation of more powerful reasoning methods. Unlike approaches for understanding models that rely on architectural specifics, our model is grounded in experimental evidence. As a result, it offers a general perspective that may extend to a wide range of current and future reasoning oracles.

cs.AI↗

arXiv2025

Complexity of Local Search for CSPs Parameterized by Constraint Difference

In this paper, we study the parameterized complexity of local search, whose goal is to find a good nearby solution from the given current solution. Formally, given an optimization problem where the goal is to find the largest feasible subset $S$ of a universe $U$, the new input consists of a current solution $P$ (not necessarily feasible) as well as an ordinary input for the problem. Given the existence of a feasible solution $S^*$, the goal is to find a feasible solution as good as $S^*$ in parameterized time $f(k) \cdot n^{O(1)}$, where $k$ denotes the distance $|PΔS^*|$. This model generalizes numerous classical parameterized optimization problems whose parameter $k$ is the minimum number of elements removed from $U$ to make it feasible, which corresponds to the case $P = U$. We apply this model to widely studied Constraint Satisfaction Problems (CSPs), where $U$ is the set of constraints, and a subset $U'$ of constraints is feasible if there is an assignment to the variables satisfying all constraints in $U'$. We give a complete characterization of the parameterized complexity of all boolean-alphabet symmetric CSPs, where the predicate's acceptance depends on the number of true literals.

cs.DS↗

arXiv2025

Solving the Correlation Cluster LP in Sublinear Time

Correlation Clustering is a fundamental and widely-studied problem in unsupervised learning and data mining. The input is a graph and the goal is to construct a clustering minimizing the number of inter-cluster edges plus the number of missing intra-cluster edges. CCL+24 introduced the cluster LP for Correlation Clustering, which they argued captures the problem much more succinctly than previous linear programming formulations. However, the cluster LP has exponential size, with a variable for every possible set of vertices in the input graph. Nevertheless, CCL+24 showed how to find a feasible solution for the cluster LP in time $O(n^{\text{poly}(1/ε)})$ with objective value at most $(1+ε)$ times the value of an optimal solution for the respective Correlation Clustering instance. Furthermore, they showed how to round a solution to the cluster LP, yielding a $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. The main technical result of this paper is a new approach to find a feasible solution for the cluster LP with objective value at most $(1+ε)$ of the optimum in time $\widetilde O(2^{\text{poly}(1/ε)} n)$, where $n$ is the number of vertices in the graph. We also show how to implement the rounding within the same time bounds, thus achieving a fast $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. This bridges the gap between state-of-the-art methods for approximating Correlation Clustering and the recent focus on fast algorithms.

cs.DS↗

arXiv2025

Understanding the Cluster LP for Correlation Clustering

In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla (FOCS 2002), the input is a complete graph where edges are labeled either $+$ or $-$, and the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the -edges within parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev (STOC 2015) gave a 2.06-approximation by providing a near-optimal rounding of the standard LP, and Cohen-Addad, Lee, Li, and Newman (FOCS 2022, 2023) finally bypassed the integrality gap of 2 for this LP giving a $1.73$-approximation for the problem. In order to create a simple and unified framework for Correlation Clustering similar to those for typical approximate optimization tasks, we propose the cluster LP as a strong linear program for Correlation Clustering. We demonstrate the power of the cluster LP by presenting new rounding algorithms, and providing two analyses, one analytically proving a 1.56-approximation and the other solving a factor-revealing SDP to show a 1.485-approximation. Both proofs introduce principled methods by which to analyze the performance of the algorithm, resulting in a significantly improved approximation guarantee. Finally, we prove an integrality gap of $4/3$ for the cluster LP, showing our 1.485-upper bound cannot be drastically improved. Our gap instance directly inspires an improved NP-hardness of approximation with a ratio $24/23 \approx 1.042$; no explicit hardness ratio was known before.

cs.DS↗

arXiv2025

Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that preserves the energy of each vector $x \in \mathbb{R}^n$ in $H$, up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a $(1+\varepsilon)$-approximation, using $\frac{rn}{\varepsilon^2} \log^2 n \log r \cdot\text{poly}(\log \log m)$ bits of space, matching the sample complexity of the best known offline algorithm up to $\text{poly}(\log \log m)$ factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a $(1+\varepsilon)$-approximation, using $\frac{n}{\varepsilon^2} \log n \cdot\text{poly}(\log\log n)$ bits of space, improving the current bound by $\log n$ factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for $(1+\varepsilon)$-hypergraph sparsification, which is optimal up to poly-logarithmic factors. As a result, we achieve $(1+\varepsilon)$-hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using $\frac{n}{\varepsilon^2} \cdot\text{poly}(r, \log n, \log r, \log \log m)$ bits of space.

cs.DS↗

arXiv2025

SYNAPSE-G: Bridging Large Language Models and Graph Learning for Rare Event Classification

Scarcity of labeled data, especially for rare events, hinders training effective machine learning models. This paper proposes SYNAPSE-G (Synthetic Augmentation for Positive Sampling via Expansion on Graphs), a novel pipeline leveraging Large Language Models (LLMs) to generate synthetic training data for rare event classification, addressing the cold-start problem. This synthetic data serve as seeds for semi-supervised label propagation on a similarity graph constructed between the seeds and a large unlabeled dataset. This identifies candidate positive examples, subsequently labeled by an oracle (human or LLM). The expanded dataset then trains/fine-tunes a classifier. We theoretically analyze how the quality (validity and diversity) of the synthetic data impacts the precision and recall of our method. Experiments on the imbalanced SST2 and MHS datasets demonstrate SYNAPSE-G's effectiveness in finding positive labels, outperforming baselines including nearest neighbor search.

cs.LG↗

arXiv2025

Approximating High-Dimensional Earth Mover's Distance as Fast as Closest Pair

We give a reduction from $(1+\varepsilon)$-approximate Earth Mover's Distance (EMD) to $(1+\varepsilon)$-approximate Closest Pair (CP). As a consequence, we improve the fastest known approximation algorithm for high-dimensional EMD. Here, given $p\in [1, 2]$ and two sets of $n$ points $X,Y \subseteq (\mathbb R^d,\ell_p)$, their EMD is the minimum cost of a perfect matching between $X$ and $Y$, where the cost of matching two vectors is their $\ell_p$ distance. Further, CP is the basic problem of finding a pair of points realizing $\min_{x \in X, y\in Y} ||x-y||_p$. Our contribution is twofold: we show that if a $(1+\varepsilon)$-approximate CP can be computed in time $n^{2-ϕ}$, then a $1+O(\varepsilon)$ approximation to EMD can be computed in time $n^{2-Ω(ϕ)}$; plugging in the fastest known algorithm for CP [Alman, Chan, Williams FOCS'16], we obtain a $(1+\varepsilon)$-approximation algorithm for EMD running in time $n^{2-\tildeΩ(\varepsilon^{1/3})}$ for high-dimensional point sets, which improves over the prior fastest running time of $n^{2-Ω(\varepsilon^2)}$ [Andoni, Zhang FOCS'23]. Our main technical contribution is a sublinear implementation of the Multiplicative Weights Update framework for EMD. Specifically, we demonstrate that the updates can be executed without ever explicitly computing or storing the weights; instead, we exploit the underlying geometric structure to perform the updates implicitly.

cs.DS↗

arXiv2025

Scalable Private Partition Selection via Adaptive Weighting

In the differentially private partition selection problem (a.k.a. private set union, private key discovery), users hold subsets of items from an unbounded universe. The goal is to output as many items as possible from the union of the users' sets while maintaining user-level differential privacy. Solutions to this problem are a core building block for many privacy-preserving ML applications including vocabulary extraction in a private corpus, computing statistics over categorical data and learning embeddings over user-provided items. We propose an algorithm for this problem, MaxAdaptiveDegree (MAD), which adaptively reroutes weight from items with weight far above the threshold needed for privacy to items with smaller weight, thereby increasing the probability that less frequent items are output. Our algorithm can be efficiently implemented in massively parallel computation systems allowing scalability to very large datasets. We prove that our algorithm stochastically dominates the standard parallel algorithm for this problem. We also develop a two-round version of our algorithm, MAD2R, where results of the computation in the first round are used to bias the weighting in the second round to maximize the number of items output. In experiments, our algorithms provide the best results among parallel algorithms and scale to datasets with hundreds of billions of items, up to three orders of magnitude larger than those analyzed by prior sequential algorithms.

cs.DS↗

arXiv2025

Scalable contribution bounding to achieve privacy

In modern datasets, where single records can have multiple owners, enforcing user-level differential privacy requires capping each user's total contribution. This "contribution bounding" becomes a significant combinatorial challenge. Existing sequential algorithms for this task are computationally intensive and do not scale to the massive datasets prevalent today. To address this scalability bottleneck, we propose a novel and efficient distributed algorithm. Our approach models the complex ownership structure as a hypergraph, where users are vertices and records are hyperedges. The algorithm proceeds in rounds, allowing users to propose records in parallel. A record is added to the final dataset only if all its owners unanimously agree, thereby ensuring that no user's predefined contribution limit is violated. This method aims to maximize the size of the resulting dataset for high utility while providing a practical, scalable solution for implementing user-level privacy in large, real-world systems.

cs.DS↗

arXiv2025

Less is More: Adaptive Coverage for Synthetic Training Data

Synthetic training data generation with Large Language Models (LLMs) like Google's Gemma and OpenAI's GPT offer a promising solution to the challenge of obtaining large, labeled datasets for training classifiers. When rapid model deployment is critical, such as in classifying emerging social media trends or combating new forms of online abuse tied to current events, the ability to generate training data is invaluable. While prior research has examined the comparability of synthetic data to human-labeled data, this study introduces a novel sampling algorithm, based on the maximum coverage problem, to select a representative subset from a synthetically generated dataset. Our results demonstrate that training a classifier on this contextually sampled subset achieves superior performance compared to training on the entire dataset. This "less is more" approach not only improves model accuracy but also reduces the volume of data required, leading to potentially more efficient model fine-tuning.

cs.LG↗

arXiv2025

An Efficient Massively Parallel Constant-Factor Approximation Algorithm for the $k$-Means Problem

In this paper, we present an efficient massively parallel approximation algorithm for the $k$-means problem. Specifically, we provide an MPC algorithm that computes a constant-factor approximation to an arbitrary $k$-means instance in $O(\log\log n \cdot \log\log\log n)$ rounds. The algorithm uses $O(n^σ)$ bits of memory per machine, where $σ> 0$ is a constant that can be made arbitrarily small. The global memory usage is $O(n^{1+\varepsilon})$ bits for an arbitrarily small constant $\varepsilon > 0$, and is thus only slightly superlinear. Recently, Czumaj, Gao, Jiang, Krauthgamer, and Veselý showed that a constant-factor bicriteria approximation can be computed in $O(1)$ rounds in the MPC model. However, our algorithm is the first constant-factor approximation for the general $k$-means problem that runs in $o(\log n)$ rounds in the MPC model. Our approach builds upon the foundational framework of Jain and Vazirani. The core component of our algorithm is a constant-factor approximation for the related facility location problem. While such an approximation was already achieved in constant time in the work of Czumaj et al.\ mentioned above, our version additionally satisfies the so-called Lagrangian Multiplier Preserving (LMP) property. This property enables the transformation of a facility location approximation into a comparably good $k$-means approximation.

cs.DS↗

arXiv2025

Correlation Clustering Beyond the Pivot Algorithm

We study the classic correlation clustering in the dynamic setting. Given $n$ objects and a complete labeling of the object-pairs as either similar or dissimilar, the goal is to partition the objects into arbitrarily many clusters while minimizing disagreements with the labels. In the dynamic setting, an update consists of a flip of a label of an edge. In a breakthrough result, [BDHSS, FOCS'19] showed how to maintain a 3-approximation with polylogarithmic update time by providing a dynamic implementation of the Pivot algorithm of [ACN, STOC'05]. Since then, it has been a major open problem to determine whether the 3-approximation barrier can be broken in the fully dynamic setting. In this paper, we resolve this problem. Our algorithm, Modified Pivot, locally improves the output of Pivot by moving some vertices to other existing clusters or new singleton clusters. We present an analysis showing that this modification does indeed improve the approximation to below 3. We also show that its output can be maintained in polylogarithmic time per update.

cs.DS↗

arXiv2025

Private Training & Data Generation by Clustering Embeddings

Deep neural networks often use large, high-quality datasets to achieve high performance on many machine learning tasks. When training involves potentially sensitive data, this process can raise privacy concerns, as large models have been shown to unintentionally memorize and reveal sensitive information, including reconstructing entire training samples. Differential privacy (DP) provides a robust framework for protecting individual data and in particular, a new approach to privately training deep neural networks is to approximate the input dataset with a privately generated synthetic dataset, before any subsequent training algorithm. We introduce a novel principled method for DP synthetic image embedding generation, based on fitting a Gaussian Mixture Model (GMM) in an appropriate embedding space using DP clustering. Our method provably learns a GMM under separation conditions. Empirically, a simple two-layer neural network trained on synthetically generated embeddings achieves state-of-the-art (SOTA) classification accuracy on standard benchmark datasets. Additionally, we demonstrate that our method can generate realistic synthetic images that achieve downstream classification accuracy comparable to SOTA methods. Our method is quite general, as the encoder and decoder modules can be freely substituted to suit different tasks. It is also highly scalable, consisting only of subroutines that scale linearly with the number of samples and/or can be implemented efficiently in distributed systems.

cs.LG↗

arXiv2025

Efficient Data Selection at Scale via Influence Distillation

Effective data selection is critical for efficient training of modern Large Language Models (LLMs). This paper introduces Influence Distillation, a novel, mathematically-justified framework for data selection that employs second-order information to optimally weight training samples. By distilling each sample's influence on a target distribution, our method assigns model-specific weights that are used to select training data for LLM fine-tuning, guiding it toward strong performance on the target domain. We derive these optimal weights for both Gradient Descent and Adam optimizers. To ensure scalability and reduce computational cost, we propose a $\textit{landmark-based approximation}$: influence is precisely computed for a small subset of "landmark" samples and then efficiently propagated to all other samples to determine their weights. We validate Influence Distillation by applying it to instruction tuning on the Tulu V2 dataset, targeting a range of tasks including GSM8k, SQuAD, and MMLU, across several models from the Llama and Qwen families. Experiments show that Influence Distillation matches or outperforms state-of-the-art performance while achieving up to $3.5\times$ faster selection.

cs.CL↗

arXiv2025

Metric Embeddings Beyond Bi-Lipschitz Distortion via Sherali-Adams

Metric embeddings are a widely used method in algorithm design, where generally a ``complex'' metric is embedded into a simpler, lower-dimensional one. Historically, the theoretical computer science community has focused on bi-Lipschitz embeddings, which guarantee that every pairwise distance is approximately preserved. In contrast, alternative embedding objectives that are commonly used in practice avoid bi-Lipschitz distortion; yet these approaches have received comparatively less study in theory. In this paper, we focus on Multi-dimensional Scaling (MDS), where we are given a set of non-negative dissimilarities $\{d_{i,j}\}_{i,j\in [n]}$ over $n$ points, and the goal is to find an embedding $\{x_1,\dots,x_n\} \subset R^k$ that minimizes $$\textrm{OPT}=\min_{x}\mathbb{E}_{i,j\in [n]}\left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2.$$ Despite its popularity, our theoretical understanding of MDS is extremely limited. Recently, Demaine et. al. (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for this objective, which achieves an embedding in constant dimensional Euclidean space with cost $\textrm{OPT} +ε$ in $n^2\cdot 2^{\textrm{poly}(Δ/ε)}$ time, where $Δ$ is the aspect ratio of the input dissimilarities. For metrics that admit low-cost embeddings, $Δ$ scales polynomially in $n$. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on $Δ$: for constant dimensional Euclidean space, we achieve a solution with cost $O(\log Δ)\cdot \textrm{OPT}^{Ω(1)}+ε$ in time $n^{O(1)} \cdot 2^{\text{poly}((\log(Δ)/ε))}$. Our algorithms are based on a novel geometry-aware analysis of a conditional rounding of the Sherali-Adams LP Hierarchy, allowing us to avoid exponential dependency on the aspect ratio, which would typically result from this rounding.

cs.DS↗

arXiv2025

Fast, Space-Optimal Streaming Algorithms for Clustering and Subspace Embeddings

We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $Δ$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(nΔ))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(nΔ))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.

cs.DS↗

arXiv2025

Fair Clustering in the Sliding Window Model

We study streaming algorithms for proportionally fair clustering, a notion originally suggested by Chierichetti et. al. (2017), in the sliding window model. We show that although there exist efficient streaming algorithms in the insertion-only model, surprisingly no algorithm can achieve finite multiplicative ratio without violating the fairness constraint in the sliding window. Hence, the problem of fair clustering is a rare separation between the insertion-only streaming model and the sliding window model. On the other hand, we show that if the fairness constraint is relaxed by a multiplicative $(1+\varepsilon)$ factor, there exists a $(1 + \varepsilon)$-approximate sliding window algorithm that uses $\text{poly}(k\varepsilon^{-1}\log n)$ space. This achieves essentially the best parameters (up to degree in the polynomial) provided the aforementioned lower bound. We also implement a number of empirical evaluations on real datasets to complement our theoretical results.

cs.DS↗