arXiv subjects

David García-Soriano

Publications and source records attributed to David García-Soriano.

7 recordsLinked to original sources

arXiv2026

Hermite Brings a Laptop: Analyzing Frieze-Jerrum Rounding Yields Improved Approximations for Clustering Problems

The Frieze-Jerrum rounding is a standard tool for rounding SDP relaxations of graph partitioning and clustering problems, assigning nodes to at most $k$ clusters using $k$ independent Gaussian vectors. Its analysis hinges on the collision probability $P_k(ρ)$ that two nodes whose SDP vectors have inner product $ρ$ are assigned to the same cluster. No tractable closed form for $P_k$ is known for $k\geq 4$, making it difficult to certify approximation guarantees and hindering the systematic search for better algorithms. We develop a Hermite-coefficient certification framework to derive accurate and tractable bounds on $P_k$. Using the Hermite expansion of Gaussian noise stability, we express $P_k$ as a power series with nonnegative coefficients, reduce these coefficients to one-dimensional Gaussian integrals, and certify finitely many of them, yielding rigorous bounds on $P_k$ over the entire correlation range. Our framework yields strengthened polynomial-time approximations for several clustering problems. For MaxAgree Correlation Clustering, we derive a $0.7818$-approximation, the first improvement in two decades over the $0.7666$ ratio of Swamy (2004). On the hardness side, we show that the integrality ratio of the standard SDP relaxation is at most $0.802$, and that approximation beyond that is Unique Games-hard. We also improve the best known ratios for the variant with at most $K$ clusters, MaxAgree$[K]$ (e.g., from $0.77$ to $0.8151$ for $K=3$). For Max $K$-Cut we resolve, via a structural property of the Hermite expansion, a conjecture of de Klerk et al. (2004) characterizing the Frieze--Jerrum approximation ratio for every $K\ge3$; we show that this ratio is tight, and determine it to within $10^{-6}$ accuracy for $K\le16$. Finally, we reduce the additive approximation error for modularity maximization from $0.42084$ (Kawase et al., 2021) to $0.3790$.

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arXiv2026

Approximate Dual Separation for the Cluster LP: a 1.387 approximation for Correlation Clustering

We give a deterministic $(1.3865+ε)$-approximation for correlation clustering on complete graphs, improving the previous best factor of $1.485+ε$ of Cao et al. (STOC'24). Our first main contribution is an efficient weak separation oracle for the cluster-LP dual. Given signed vertex weights $q$, it either finds a set $S$ with $q(S)>cost(S)$ or certifies that $q/(1+ε)$ is dual feasible, where $cost(S)$ measures the correlation clustering disagreements attributed to $S$ in any clustering in which $S$ is a cluster. Via the ellipsoid method, this yields $(1+ε)$-approximate primal and dual solutions for the fractional cluster LP in deterministic time $2^{poly(1/ε)}n^{O(1)}$ . The separator works directly on the original instance, without global preclustering: a localization argument restricts the search to a small universe, where weak regularity handles the resulting dense quadratic minimization. Complementing this result, we prove that exact dual separation and cluster-LP optimization are NP-hard, even for complete unweighted instances. Our second main contribution is a new rounding of the cluster LP. It combines cluster-based rounding with a continuous conditional pivot rule, and its analysis relies on single variance inequality with explicit weights. This bounds the cluster-LP integrality gap by $1.3865$, near its known lower bound of $4/3$, and gives a per-instance primal--dual certificate of approximation. Finally, we extend our separator to obtain a $(1.92+ε)$-approximation for seeded correlation clustering (where each cluster may contain at most one seed from a prescribed seed set), and to bounded-weight instances, yielding a deterministic polynomial-time approximate implementation of the cluster-insertion primitive used in combinatorial correlation clustering.

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arXiv2020

Finding Densest $k$-Connected Subgraphs

Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$, i.e., half the weighted average degree of the induced subgraph $G[S]$. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may not be robust to vertex/edge failure. Indeed, a densest subgraph may not be well-connected, which implies that the subgraph may be disconnected by removing only a few vertices/edges within it. In this paper, we provide an algorithmic framework to find a dense subgraph that is well-connected in terms of vertex/edge connectivity. Specifically, we introduce the following problems: given a graph $G=(V,E,w)$ and a positive integer/real $k$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$ under the constraint that $G[S]$ is $k$-vertex/edge-connected. For both problems, we propose polynomial-time (bicriteria and ordinary) approximation algorithms, using classic Mader's theorem in graph theory and its extensions.

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arXiv2020

Query-Efficient Correlation Clustering

Correlation clustering is arguably the most natural formulation of clustering. Given n objects and a pairwise similarity measure, the goal is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. A main drawback of correlation clustering is that it requires as input the $Θ(n^2)$ pairwise similarities. This is often infeasible to compute or even just to store. In this paper we study \emph{query-efficient} algorithms for correlation clustering. Specifically, we devise a correlation clustering algorithm that, given a budget of $Q$ queries, attains a solution whose expected number of disagreements is at most $3\cdot OPT + O(\frac{n^3}{Q})$, where $OPT$ is the optimal cost for the instance. Its running time is $O(Q)$, and can be easily made non-adaptive (meaning it can specify all its queries at the outset and make them in parallel) with the same guarantees. Up to constant factors, our algorithm yields a provably optimal trade-off between the number of queries $Q$ and the worst-case error attained, even for adaptive algorithms. Finally, we perform an experimental study of our proposed method on both synthetic and real data, showing the scalability and the accuracy of our algorithm.

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arXiv2020

Fair-by-design matching

Matching algorithms are used routinely to match donors to recipients for solid organs transplantation, for the assignment of medical residents to hospitals, record linkage in databases, scheduling jobs on machines, network switching, online advertising, and image recognition, among others. Although many optimal solutions may exist to a given matching problem, when the elements that shall or not be included in a solution correspond to individuals, it becomes of paramount importance that the solution be selected fairly. In this paper we study individual fairness in matching problems. Given that many maximum matchings may exist, each one satisfying a different set of individuals, the only way to guarantee fairness is through randomization. Hence we introduce the distributional maxmin fairness framework which provides, for any given input instance, the strongest guarantee possible simultaneously for all individuals in terms of satisfaction probability (the probability of being matched in the solution). Specifically, a probability distribution over feasible solutions is maxmin-fair if it is not possible to improve the satisfaction probability of any individual without decreasing it for some other individual which is no better off. In the special case of matchings in bipartite graphs, our framework is equivalent to the egalitarian mechanism of Bogomolnaia and Mouline. Our main contribution is a polynomial-time algorithm for fair matching building on techniques from minimum cuts, and edge-coloring algorithms for regular bipartite graphs, and transversal theory. For bipartite graphs, our algorithm runs in $O((|V|^2 + |E||V|^{2/3}) \cdot (\log |V|)^2)$ expected time and scales to graphs with tens of millions of vertices and hundreds of millions of edges. To the best of our knowledge, this provides the first large-scale implementation of the egalitarian mechanism.

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arXiv2015

The Power of Both Choices: Practical Load Balancing for Distributed Stream Processing Engines

We study the problem of load balancing in distributed stream processing engines, which is exacerbated in the presence of skew. We introduce Partial Key Grouping (PKG), a new stream partitioning scheme that adapts the classical "power of two choices" to a distributed streaming setting by leveraging two novel techniques: key splitting and local load estimation. In so doing, it achieves better load balancing than key grouping while being more scalable than shuffle grouping. We test PKG on several large datasets, both real-world and synthetic. Compared to standard hashing, PKG reduces the load imbalance by up to several orders of magnitude, and often achieves nearly-perfect load balance. This result translates into an improvement of up to 60% in throughput and up to 45% in latency when deployed on a real Storm cluster.

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arXiv2013

Local correlation clustering

Correlation clustering is perhaps the most natural formulation of clustering. Given $n$ objects and a pairwise similarity measure, the goal is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its theoretical appeal, the practical relevance of correlation clustering still remains largely unexplored, mainly due to the fact that correlation clustering requires the $Θ(n^2)$ pairwise similarities as input. In this paper we initiate the investigation into \emph{local} algorithms for correlation clustering. In \emph{local correlation clustering} we are given the identifier of a single object and we want to return the cluster to which it belongs in some globally consistent near-optimal clustering, using a small number of similarity queries. Local algorithms for correlation clustering open the door to \emph{sublinear-time} algorithms, which are particularly useful when the similarity between items is costly to compute, as it is often the case in many practical application domains. They also imply $(i)$ distributed and streaming clustering algorithms, $(ii)$ constant-time estimators and testers for cluster edit distance, and $(iii)$ property-preserving parallel reconstruction algorithms for clusterability. Specifically, we devise a local clustering algorithm attaining a $(3, \varepsilon)$-approximation in time $O(1/\varepsilon^2)$ independently of the dataset size. An explicit approximate clustering for all objects can be produced in time $O(n/\varepsilon)$ (which is provably optimal). We also provide a fully additive $(1,\varepsilon)$-approximation with local query complexity $poly(1/\varepsilon)$ and time complexity $2^{poly(1/\varepsilon)}$. The latter yields the fastest polynomial-time approximation scheme for correlation clustering known to date.

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