arXiv subjects

Martín Costa

Publications and source records attributed to Martín Costa.

14 recordsLinked to original sources

arXiv2026

Incremental Submodular Maximization: Better Than Greedy

We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

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arXiv2026

Fully Dynamic Euclidean k-Means

We consider the Euclidean $k$-means clustering problem in a dynamic setting, where we have to explicitly maintain a solution (a set of $k$ centers) $S \subseteq \mathbb{R}^d$ subject to point insertions/deletions in $\mathbb{R}^d$. We present a dynamic algorithm for Euclidean $k$-means with $\mathrm{poly}(1/ε)$-approximation ratio, $\tilde{O}(k^ε)$ update time, and $\tilde{O}(1)$ recourse, for any $ε\in (0,1)$, even when $d$ and $k$ are both part of the input. This is the first algorithm to achieve a constant ratio with $o(k)$ update time for this problem, whereas the previous $O(1)$-approximation runs in $\tilde O(k)$ update time [Bhattacharya, Costa, Farokhnejad; STOC'25]. In fact, previous algorithms cannot go beyond $O(k)$ update time precisely because they are designed for general metrics where an $Ω(k)$ lower bound is known. We break this $O(k)$ barrier by devising new fundamental data structures to utilize Euclidean properties: a structure that (implicitly) maintains a clustering subject to both center and data point updates, and a range query structure that can evaluate a mergeable function over any metric ball range given as a query. To obtain these structures, we devise the first consistent hashing scheme [Czumaj, Jiang, Krauthgamer, Vesel{ý}, Yang; FOCS'22] that achieves $\tilde O(n^ε)$ running time per point evaluation with competitive parameters. Our final algorithm exploits the framework of [Bhattacharya, Costa, Farokhnejad; STOC'25] for general metrics. The key change is to redesign several critical subroutines so that they reduce to our new Euclidean data structures, replacing the general-metric implementations that are unlikely to run efficiently even when Euclidean properties are provided.

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arXiv2026

Deterministic $k$-Median Clustering in Near-Optimal Time

The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total distance from each point in $V$ to its nearest center in $S$. Mettu and Plaxton [UAI'02] gave a randomized algorithm for $k$-median that computes a $O(1)$-approximation in $\tilde O(nk)$ time. They also showed that any algorithm for this problem with a bounded approximation ratio must have a running time of $Ω(nk)$. Thus, the running time of their algorithm is optimal up to polylogarithmic factors. For deterministic $k$-median, Guha et al.~[FOCS'00] gave an algorithm that computes a $\text{poly}(\log (n/k))$-approximation in $\tilde O(nk)$ time, where the degree of the polynomial in the approximation is unspecified. To the best of our knowledge, this remains the state-of-the-art approximation of any deterministic $k$-median algorithm with this running time. This leads us to the following natural question: What is the best approximation of a deterministic $k$-median algorithm with near-optimal running time? We make progress in answering this question by giving a deterministic algorithm that computes a $O(\log(n/k))$-approximation in $\tilde O(nk)$ time. We also provide a lower bound showing that any deterministic algorithm with this running time must have an approximation ratio of $Ω(\log n/(\log k + \log \log n))$, establishing a gap between the randomized and deterministic settings for $k$-median.

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arXiv2025

Vizing's Theorem in Deterministic Almost-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm. This deterministic time bound was subsequently improved to $\tilde O(m \sqrt n)$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. A series of recent papers improved the time bound of $\tilde O(m\sqrt{n})$ using randomization, culminating in the randomized near-linear time $(Δ+1)$-coloring algorithm by [Assadi, Behnezhad, Bhattacharya, Costa, Solomon, and Zhang, 2025]. At the heart of all of these recent improvements, there is some form of a sublinear time algorithm. Unfortunately, sublinear time algorithms as a whole almost always require randomization. This raises a natural question: can the deterministic time complexity of the problem be reduced below the $\tilde O(m\sqrt{n})$ barrier? In this paper, we answer this question in the affirmative. We present a deterministic almost-linear time $(Δ+1)$-coloring algorithm, namely, an algorithm running in $m \cdot 2^{O(\sqrt{\log Δ})} \cdot \log n = m^{1+o(1)}$ time. Our main technical contribution is to entirely forego sublinear time algorithms. We do so by presenting a new deterministic color-type sparsification approach that runs in almost-linear (instead of sublinear) time, but can be used to color a much larger set of edges.

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arXiv2025

Vizing's Theorem in Near-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.

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arXiv2025

Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse

In this paper, we consider the \emph{metric $k$-center} problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its \emph{approximation ratio}, \emph{recourse} (the number of changes it makes to the solution $S$), and \emph{update time} (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, \emph{obtaining near-optimal approximation, recourse, and update time simultaneously}. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al.~[SODA'23] with the dynamic sparsifier of Bhattacharya et al.~[NeurIPS'23].

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arXiv2024

Fully Dynamic $k$-Median with Near-Optimal Update Time and Recourse

In metric $k$-clustering, we are given as input a set of $n$ points in a general metric space, and we have to pick $k$ centers and cluster the input points around these chosen centers, so as to minimize an appropriate objective function. In recent years, significant effort has been devoted to the study of metric $k$-clustering problems in a dynamic setting, where the input keeps changing via updates (point insertions/deletions), and we have to maintain a good clustering throughout these updates. The performance of such a dynamic algorithm is measured in terms of three parameters: (i) Approximation ratio, which signifies the quality of the maintained solution, (ii) Recourse, which signifies how stable the maintained solution is, and (iii) Update time, which signifies the efficiency of the algorithm. We consider the metric $k$-median problem, where the objective is the sum of the distances of the points to their nearest centers. We design the first dynamic algorithm for this problem with near-optimal guarantees across all three performance measures (up to a constant factor in approximation ratio, and polylogarithmic factors in recourse and update time). Specifically, we obtain a $O(1)$-approximation algorithm for dynamic metric $k$-median with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Prior to our work, the state-of-the-art here was the recent result of [Bhattacharya et al., FOCS'24], who obtained $O(ε^{-1})$-approximation ratio with $\tilde{O}(k^ε)$ recourse and $\tilde{O}(k^{1+ε})$ update time. We achieve our results by carefully synthesizing the concept of robust centers introduced in [Fichtenberger et al., SODA'21] along with the randomized local search subroutine from [Bhattacharya et al., FOCS'24], in addition to several key technical insights of our own.

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arXiv2024

Even Faster $(Δ+ 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains

Vizing's Theorem from 1964 states that any $n$-vertex $m$-edge graph with maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada~[1985], was $\tilde O(m\sqrt{n})$. Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to $\tilde O(mn^{1/3})$, and by Assadi to $\tilde O(n^2)$. In this paper we present an algorithm that computes such a coloring in $\tilde O(mn^{1/4})$ time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time $\tilde O(Δ^2 + \sqrt{Δn})$. The best previous time bound of any color extension subroutine is either the trivial $O(n)$, dominated by the length of a Vizing chain, or the bound $\tilde{O}(Δ^6)$ by Bernshteyn [2022], dominated by the length of {\em multi-step Vizing chains}, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large $Δ$.

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arXiv2024

Density-Sensitive Algorithms for $(Δ+ 1)$-Edge Coloring

Vizing's theorem asserts the existence of a $(Δ+1)$-edge coloring for any graph $G$, where $Δ= Δ(G)$ denotes the maximum degree of $G$. Several polynomial time $(Δ+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $α= α(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's "uniform density". While $α\le Δ$ in any graph, many natural and real-world graphs exhibit a significant separation between $α$ and $Δ$. In this work we design a $(Δ+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})\cdot \fracαΔ$, thus improving the longstanding time barrier by a factor of $\fracαΔ$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $α= \tilde{O}(1)$) as well as when $α= \tilde{O}(\fracΔ{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.

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arXiv2024

Fully Dynamic $k$-Clustering with Fast Update Time and Small Recourse

In the dynamic metric $k$-median problem, we wish to maintain a set of $k$ centers $S \subseteq V$ in an input metric space $(V, d)$ that gets updated via point insertions/deletions, so as to minimize the objective $\sum_{x \in V} \min_{y \in S} d(x, y)$. The quality of a dynamic algorithm is measured in terms of its approximation ratio, "recourse" (the number of changes in $S$ per update) and "update time" (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic $O(1)$ approximation algorithm with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Dynamic $k$-median is a canonical example of a class of problems known as dynamic $k$-clustering, that has received significant attention in recent years. To the best of our knowledge, however, previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic $k$-median, we come arbitrarily close to resolving the main open question on this topic, with the following results. (I) We develop a new framework of randomized local search that is suitable for adaptation in a dynamic setting. For every $ε> 0$, this gives us a dynamic $k$-median algorithm with $O(1/ε)$ approximation ratio, $\tilde{O}(k^ε)$ recourse and $\tilde{O}(k^{1+ε})$ update time. This framework also generalizes to dynamic $k$-clustering with $\ell^p$-norm objectives, giving similar bounds for the dynamic $k$-means and a new trade-off for dynamic $k$-center. (II) If it suffices to maintain only an estimate of the value of the optimal $k$-median objective, then we obtain a $O(1)$ approximation algorithm with $\tilde{O}(k)$ update time. We achieve this result via adapting the Lagrangian Relaxation framework to the dynamic setting.

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arXiv2024

Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years.

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arXiv2024

Arboricity-Dependent Algorithms for Edge Coloring

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(Δ+ O(Δ^{1 - μ}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $μ> 0$, where $Δ$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(Δ+ O(α))$-edge coloring in $\tilde O(1)$ amortized update time, where $α$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.

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arXiv2023

Nibbling at Long Cycles: Dynamic (and Static) Edge Coloring in Optimal Time

We consider the problem of maintaining a $(1+ε)Δ$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $Δ$. The state-of-the-art update time is $O_ε(\text{polylog}(n))$, by Duan, He and Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7 n/ε^2)$, where $Δ= Ω(\log^2 n / ε^2)$. The following natural question arises: What is the best possible update time of an algorithm for this task? More specifically, \textbf{ can we bring it all the way down to some constant} (for constant $ε$)? This question coincides with the \emph{static} time barrier for the problem: Even for $(2Δ-1)$-coloring, there is only a naive $O(m \log Δ)$-time algorithm. We answer this fundamental question in the affirmative, by presenting a dynamic $(1+ε)Δ$-edge coloring algorithm with $O(\log^4 (1/ε)/ε^9)$ update time, provided $Δ= Ω_ε(\text{polylog}(n))$. As a corollary, we also get the first linear time (for constant $ε$) \emph{static} algorithm for $(1+ε)Δ$-edge coloring; in particular, we achieve a running time of $O(m \log (1/ε)/ε^2)$. We obtain our results by carefully combining a variant of the \textsc{Nibble} algorithm from Bhattacharya, Grandoni and Wajc [SODA'21] with the subsampling technique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

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arXiv2023

Fully Dynamic $k$-Clustering in $\tilde O(k)$ Update Time

We present a $O(1)$-approximate fully dynamic algorithm for the $k$-median and $k$-means problems on metric spaces with amortized update time $\tilde O(k)$ and worst-case query time $\tilde O(k^2)$. We complement our theoretical analysis with the first in-depth experimental study for the dynamic $k$-median problem on general metrics, focusing on comparing our dynamic algorithm to the current state-of-the-art by Henzinger and Kale [ESA'20]. Finally, we also provide a lower bound for dynamic $k$-median which shows that any $O(1)$-approximate algorithm with $\tilde O(\text{poly}(k))$ query time must have $\tilde Ω(k)$ amortized update time, even in the incremental setting.

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