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Documents authored by Costa, Martín

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.62
Deterministic k-Median Clustering in Near-Optimal Time

Authors: Martín Costa and Ermiya Farokhnejad

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
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 ⊆ 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 Õ(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 poly(log (n/k))-approximation in Õ(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 Õ(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.
Cite as

Martín Costa and Ermiya Farokhnejad. Deterministic k-Median Clustering in Near-Optimal Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 62:1-62:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{costa_et_al:LIPIcs.ICALP.2025.62,
  author =	{Costa, Mart{\'\i}n and Farokhnejad, Ermiya},
  title =	{{Deterministic k-Median Clustering in Near-Optimal Time}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{62:1--62:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.62},
  URN =		{urn:nbn:de:0030-drops-234395},
  doi =		{10.4230/LIPIcs.ICALP.2025.62},
  annote =	{Keywords: k-clustering, k-median, deterministic algorithms, approximation algorithms}
}
Document
DOI: 10.4230/DagRep.14.11.92
Graph Algorithms: Distributed Meets Dynamic (Dagstuhl Seminar 24471)

Authors: Keren Censor-Hillel, Yasamin Nazari, Eva Rotenberg, Thatchaphol Saranurak, and Martín Costa

Published in: Dagstuhl Reports, Volume 14, Issue 11 (2025)

Abstract
This report contains the program and outcomes of the Dagstuhl Seminar "Graph Algorithms: Distributed Meets Dynamic". The field of dynamic graph algorithms address recomputation following edge/vertex insertions/deletions in the input graph. Distributed graph algorithms focus on computing when the input resides across multiple machines, that need to communicate for their joint computation. The seminar brought together researchers from the two communities of dynamic graph algorithms and distributed computing. The aim was to transfer knowledge and techniques between the dynamic and distributed settings, build new collaborations and to explore research directions on computational models of the combined distributed dynamic setting.
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Keren Censor-Hillel, Yasamin Nazari, Eva Rotenberg, Thatchaphol Saranurak, and Martín Costa. Graph Algorithms: Distributed Meets Dynamic (Dagstuhl Seminar 24471). In Dagstuhl Reports, Volume 14, Issue 11, pp. 92-107, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{censorhillel_et_al:DagRep.14.11.92,
  author =	{Censor-Hillel, Keren and Nazari, Yasamin and Rotenberg, Eva and Saranurak, Thatchaphol and Costa, Mart{\'\i}n},
  title =	{{Graph Algorithms: Distributed Meets Dynamic (Dagstuhl Seminar 24471)}},
  pages =	{92--107},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2025},
  volume =	{14},
  number =	{11},
  editor =	{Censor-Hillel, Keren and Nazari, Yasamin and Rotenberg, Eva and Saranurak, Thatchaphol and Costa, Mart{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.11.92},
  URN =		{urn:nbn:de:0030-drops-228186},
  doi =		{10.4230/DagRep.14.11.92},
  annote =	{Keywords: distributed algorithms, dynamic algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.23
Density-Sensitive Algorithms for (Δ + 1)-Edge Coloring

Authors: Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
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 Õ(min{m √n, m Δ}), by Gabow, Nishizeki, Kariv, Leven and Terada from 1985, where n and m denote the number of vertices and edges in the graph, respectively. Recently, Sinnamon shaved off a polylog(n) 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 ≤ 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 Õ(min{m √n, m Δ})⋅ α/Δ, thus improving the longstanding time barrier by a factor of α/Δ. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., α = Õ(1)) as well as when α = Õ(Δ/√n). Our algorithm builds on Gabow et al.’s and Sinnamon’s algorithms, and can be viewed as a density-sensitive refinement of them.
Cite as

Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Density-Sensitive Algorithms for (Δ + 1)-Edge Coloring. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2024.23,
  author =	{Bhattacharya, Sayan and Costa, Mart{\'\i}n and Panski, Nadav and Solomon, Shay},
  title =	{{Density-Sensitive Algorithms for (\Delta + 1)-Edge Coloring}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.23},
  URN =		{urn:nbn:de:0030-drops-210945},
  doi =		{10.4230/LIPIcs.ESA.2024.23},
  annote =	{Keywords: Graph Algorithms, Edge Coloring, Arboricity}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.12
Arboricity-Dependent Algorithms for Edge Coloring

Authors: Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
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 Õ(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 Õ(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.
Cite as

Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Arboricity-Dependent Algorithms for Edge Coloring. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.SWAT.2024.12,
  author =	{Bhattacharya, Sayan and Costa, Mart{\'\i}n and Panski, Nadav and Solomon, Shay},
  title =	{{Arboricity-Dependent Algorithms for Edge Coloring}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.12},
  URN =		{urn:nbn:de:0030-drops-200524},
  doi =		{10.4230/LIPIcs.SWAT.2024.12},
  annote =	{Keywords: Dynamic Algorithms, Graph Algorithms, Edge Coloring, Arboricity}
}
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