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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.70

Incremental (k, z)-Clustering on Graphs

Authors: Emilio Cruciani, Sebastian Forster, and Antonis Skarlatos

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

Given a weighted undirected graph, a number of clusters k, and an exponent z, the goal in the (k, z)-clustering problem on graphs is to select k vertices as centers that minimize the sum of the distances raised to the power z of each vertex to its closest center. This problem includes the well-known k-median (z = 1) and k-means (z = 2) clustering problems. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact (k, z)-clustering solution in the induced shortest-path metric. Prior works by Bhattacharya, Costa, Garg, Lattanzi, and Parotsidis [FOCS 2024] and by Bhattacharya, Costa, and Farokhnejad [STOC 2025] consider the dynamic (k, z)-clustering problem for point sets in metric spaces. These algorithms support adversarial point insertions and deletions under a model with access to pairwise distances. This model differs significantly from the dynamic graph setting, where no oracle access is given to pairwise distances and a single edge update can affect many distances - making these approaches inefficient when applied to graphs. While efficient dynamic k-center approximation algorithms on graphs exist [Cruciani, Forster, Goranci, Nazari, and Skarlatos, SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic (k,z)-clustering problem. As the main result of this paper, we develop a randomized incremental (k, z)-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of Õ(k m^{1+o(1)} + k^{1+1/(λ)} m), where λ ≥ 1 is an arbitrary fixed constant. Our incremental algorithm also achieves an amortized update time of Õ(k n^o(1) + k^{1+1/(λ)}) and consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size Õ(k) with a total update time of m^{1+o(1)} (independent of the parameter k) over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant - a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate (k,z)-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static (k,z)-clustering algorithm.

Cite as

Emilio Cruciani, Sebastian Forster, and Antonis Skarlatos. Incremental (k, z)-Clustering on Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 70:1-70:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cruciani_et_al:LIPIcs.ICALP.2026.70,
  author =	{Cruciani, Emilio and Forster, Sebastian and Skarlatos, Antonis},
  title =	{{Incremental (k, z)-Clustering on Graphs}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{70:1--70:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael 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.2026.70},
  URN =		{urn:nbn:de:0030-drops-264599},
  doi =		{10.4230/LIPIcs.ICALP.2026.70},
  annote =	{Keywords: (k, z)-clustering, k-median, k-means, dynamic graph algorithms}
}

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DOI: 10.4230/LIPIcs.ESA.2023.50

Bootstrapping Dynamic Distance Oracles

Authors: Sebastian Forster, Gramoz Goranci, Yasamin Nazari, and Antonis Skarlatos

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract

Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the O(√n) barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved m^{1/ρ+o(1)} amortized update time with a O(log n)^{3ρ-2} factor in the approximation ratio, for any parameter ρ ≥ 1. In this paper, we give the first constant-stretch fully dynamic distance oracle with small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of O(1/(ρ²))^{4/ρ} in amortized update time Õ(n^{ρ}), and query time Õ(n^{ρ/8}) for any constant parameter 0 < ρ < 1. Our algorithm is randomized and assumes an oblivious adversary. A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution.

Cite as

Sebastian Forster, Gramoz Goranci, Yasamin Nazari, and Antonis Skarlatos. Bootstrapping Dynamic Distance Oracles. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{forster_et_al:LIPIcs.ESA.2023.50,
  author =	{Forster, Sebastian and Goranci, Gramoz and Nazari, Yasamin and Skarlatos, Antonis},
  title =	{{Bootstrapping Dynamic Distance Oracles}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{50:1--50:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.50},
  URN =		{urn:nbn:de:0030-drops-187031},
  doi =		{10.4230/LIPIcs.ESA.2023.50},
  annote =	{Keywords: Dynamic graph algorithms, Distance Oracles, Shortest Paths}
}

Document

DOI: 10.4230/LIPIcs.ESA.2022.9

Computing Smallest Convex Intersecting Polygons

Authors: Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

A polygon C is an intersecting polygon for a set O of objects in ℝ² if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.

Cite as

Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos. Computing Smallest Convex Intersecting Polygons. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{antoniadis_et_al:LIPIcs.ESA.2022.9,
  author =	{Antoniadis, Antonios and de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Skarlatos, Antonis},
  title =	{{Computing Smallest Convex Intersecting Polygons}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{9:1--9:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.9},
  URN =		{urn:nbn:de:0030-drops-169470},
  doi =		{10.4230/LIPIcs.ESA.2022.9},
  annote =	{Keywords: convex hull, imprecise points, computational geometry}
}

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Documents authored by Skarlatos, Antonis

Incremental (k, z)-Clustering on Graphs

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Bootstrapping Dynamic Distance Oracles

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Computing Smallest Convex Intersecting Polygons

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