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DOI: 10.4230/LIPIcs.SWAT.2026.7

On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons

Authors: Mark de Berg, Prosenjit Bose, and Leonidas Theocharous

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract

Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any (α,β)-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in (α,β)-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on (α,β)-covered polygons, including an algorithm that computes the closest pair of a set of m points in an (α,β)-covered polygon with n vertices that runs in O(n + mlog n) expected time.

Cite as

Mark de Berg, Prosenjit Bose, and Leonidas Theocharous. On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{deberg_et_al:LIPIcs.SWAT.2026.7,
  author =	{de Berg, Mark and Bose, Prosenjit and Theocharous, Leonidas},
  title =	{{On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.7},
  URN =		{urn:nbn:de:0030-drops-260439},
  doi =		{10.4230/LIPIcs.SWAT.2026.7},
  annote =	{Keywords: Fat polygons, doubling dimension}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.13

Dynamic Light Spanners in Doubling Metrics

Authors: Sujoy Bhore, Jonathan Conroy, and Arnold Filtser

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

A t-spanner of a point set X in a metric space (𝒳, δ) is a graph G with vertex set P such that, for any pair of points u,v ∈ X, the distance between u and v in G is at most t times δ(u,v). We study the problem of maintaining a spanner for a dynamic point set X - that is, when X undergoes a sequence of insertions and deletions - in a metric space of constant doubling dimension. For any constant ε > 0, we maintain a (1+ε)-spanner of P whose total weight remains within a constant factor of the weight of the minimum spanning tree of X. Each update (insertion or deletion) can be performed in poly(log Φ) time, where Φ denotes the aspect ratio of X. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Cite as

Sujoy Bhore, Jonathan Conroy, and Arnold Filtser. Dynamic Light Spanners in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.13,
  author =	{Bhore, Sujoy and Conroy, Jonathan and Filtser, Arnold},
  title =	{{Dynamic Light Spanners in Doubling Metrics}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.13},
  URN =		{urn:nbn:de:0030-drops-258193},
  doi =		{10.4230/LIPIcs.SoCG.2026.13},
  annote =	{Keywords: Dynamic data structures, spanners, light-weight, Euclidean metrics, doubling metrics}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.27

The Squishy Grid Problem

Authors: Zixi Cai, Kuowen Chen, Shengquan Du, Arnold Filtser, Seth Pettie, and Daniel Skora

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment w : E → ℝ_{≥ 0}, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is: For all grid points u,v, dist_w(u,v) = (1± o(1))‖u-v‖₂. We give three methods for solving this problem, each attractive in its own way. - Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway [Charles Radin, 1994; Radin and Sadun, 1996; John H. Conway and Charles Radin, 1998] into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is (1 + 1/Θ(log^ξ log D)), where D is the Euclidean distance and ξ = Θ(1). The pinwheel tiling approach is conceptually simple, but can be improved quantitatively. - Our second construction is based on a hierarchical arrangement of highways. It is simple, achieving stretch (1 + 1/Θ(D^{1/9})), which converges doubly exponentially faster than the pinwheel tiling approach. - The first two methods are deterministic, with rigorous guarantees. An even simpler approach is to sample the edge weights independently and randomly from a common distribution D. Whether there exists a distribution D^* that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when D is a Fisher distribution (which is continuous), grid distances are within 1% of Euclidean distances. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44% and 56%, respectively.

Cite as

Zixi Cai, Kuowen Chen, Shengquan Du, Arnold Filtser, Seth Pettie, and Daniel Skora. The Squishy Grid Problem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai_et_al:LIPIcs.SoCG.2026.27,
  author =	{Cai, Zixi and Chen, Kuowen and Du, Shengquan and Filtser, Arnold and Pettie, Seth and Skora, Daniel},
  title =	{{The Squishy Grid Problem}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{27:1--27:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.27},
  URN =		{urn:nbn:de:0030-drops-258333},
  doi =		{10.4230/LIPIcs.SoCG.2026.27},
  annote =	{Keywords: grid graph, Euclidean distance, metric embedding, first passage percolation}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.36

Upward Book Embeddings of Partitioned Digraphs

Authors: Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph G = (V, ⋃^k_{i=1} E_i), that is, a digraph whose edge set is partitioned into k subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput. 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for k = 1 and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for k ≥ 3. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case k = 2. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when k = 2, thus closing the complexity gap for the problem. Second, we show that, for an n-vertex partitioned digraph with a prescribed planar embedding, the existence of an upward book embedding that respects the given planar embedding can be tested in O(n log³ n) time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial 2-trees.

Cite as

Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter. Upward Book Embeddings of Partitioned Digraphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dalozzo_et_al:LIPIcs.SoCG.2026.36,
  author =	{Da Lozzo, Giordano and Frati, Fabrizio and Rutter, Ignaz},
  title =	{{Upward Book Embeddings of Partitioned Digraphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.36},
  URN =		{urn:nbn:de:0030-drops-258424},
  doi =		{10.4230/LIPIcs.SoCG.2026.36},
  annote =	{Keywords: upward book embeddings, partitioned digraphs, SPQ-trees, 2-trees}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.53

Constructing Doppelgängers of Greedy Geometric Spanners in Practice

Authors: Anirban Ghosh

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

Greedy geometric spanners are considered to be the gold standard for their near-optimal guarantees in terms of sparsity and total weight. However, their inefficient construction poses significant challenges for large-scale geometric networks, especially for low values of stretch factors (< 2). We present Θ-Greedy, a simple and practical parallel algorithm engineered for constructing doppelgängers of greedy geometric spanners that empirically resemble the greedy spanners in key structural and performance metrics, including average degree, degree, and lightness. Unlike approximate greedy spanners, doppelgängers of greedy spanners are almost indistinguishable from the actual greedy spanners in practice. In our experiments, Θ-Greedy consistently produced greedy spanner doppelgängers across a broad range of synthetic and real-world datasets, offering the first practical alternative to the computationally intensive greedy spanners. Θ-Greedy can construct a 1.1-spanner on a 128K-element uniformly distributed point set in well under 5 minutes. In contrast, Bucketing, the most practical greedy spanner algorithm, takes around 3 hours. For million-sized point sets, Θ-Greedy can run to completion in a few hours, making it much faster than Bucketing, which takes days to finish. In extensive experiments on synthetic and real-world datasets, Θ-Greedy delivered speedups of up to 147x over Bucketing while preserving greedy-like sparsity and weight. For broader uses of the algorithm and reproducibility, we share our engineered C++ code.

Cite as

Anirban Ghosh. Constructing Doppelgängers of Greedy Geometric Spanners in Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghosh:LIPIcs.SoCG.2026.53,
  author =	{Ghosh, Anirban},
  title =	{{Constructing Doppelg\"{a}ngers of Greedy Geometric Spanners in Practice}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{53:1--53:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.53},
  URN =		{urn:nbn:de:0030-drops-258599},
  doi =		{10.4230/LIPIcs.SoCG.2026.53},
  annote =	{Keywords: geometric graph, geometric spanners, greedy spanners, algorithm engineering}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.64

Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

Authors: Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

We give an approximation scheme for the TSP in d-dimensional hyperbolic space that has optimal dependence on ε under Gap-ETH. For any fixed dimension d ≥ 2 and for any ε > 0 our randomized algorithm gives a (1+ε)-approximation in time 2^O(1/ε^{d-1}) n^{1+o(1)}. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.

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Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen. Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.64,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Odak, Saeed and Singh, Satyam and van Wordragen, Geert},
  title =	{{Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{64:1--64:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.64},
  URN =		{urn:nbn:de:0030-drops-258710},
  doi =		{10.4230/LIPIcs.SoCG.2026.64},
  annote =	{Keywords: Hyperbolic traveling salesman problem, TSP, Hyperbolic Steiner tree problem, Approximation scheme, Banyan, Hyperbolic geometry}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.87

Robustness of Persistent Topological Features and Minimum Homological Cuts

Authors: Pepijn Roos Hoefgeest and Lucas Slot

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of adversarial robustness, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in 0-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-1 in n-dimensional complexes embedded in ℝⁿ. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.

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Pepijn Roos Hoefgeest and Lucas Slot. Robustness of Persistent Topological Features and Minimum Homological Cuts. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rooshoefgeest_et_al:LIPIcs.SoCG.2026.87,
  author =	{Roos Hoefgeest, Pepijn and Slot, Lucas},
  title =	{{Robustness of Persistent Topological Features and Minimum Homological Cuts}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{87:1--87:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.87},
  URN =		{urn:nbn:de:0030-drops-258636},
  doi =		{10.4230/LIPIcs.SoCG.2026.87},
  annote =	{Keywords: Topological Data Analysis, Persistent Homology, Min-cut Max-flow, Robustness, Vietoris-Rips Filtration}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.50

A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs

Authors: Thekla Hamm, Sukanya Pandey, and Krisztina Szilágyi

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

Given a planar graph, a subset of its vertices called terminals, and k ∈ ℕ, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most k faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel [Valentin Garnero et al., 2017]. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.

Cite as

Thekla Hamm, Sukanya Pandey, and Krisztina Szilágyi. A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hamm_et_al:LIPIcs.STACS.2026.50,
  author =	{Hamm, Thekla and Pandey, Sukanya and Szil\'{a}gyi, Krisztina},
  title =	{{A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{50:1--50:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.50},
  URN =		{urn:nbn:de:0030-drops-255392},
  doi =		{10.4230/LIPIcs.STACS.2026.50},
  annote =	{Keywords: Kernelization, Planar Graphs, SPQR-tree}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.74

Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications

Authors: Paweł Gawrychowski, Egor Gorbachev, and Tomasz Kociumaka

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Min-plus matrix multiplication is a fundamental tool for designing algorithms operating on distances in graphs and different problems solvable by dynamic programming. We know that, assuming the APSP hypothesis, no subcubic-time algorithm exists for the case of general matrices. However, in many applications the matrices admit certain structural properties that can be used to design faster algorithms. For example, when considering a planar graph, one often works with a Monge matrix A, meaning that the density matrix A^◻ has non-negative entries, that is, A^◻_{i,j} := A_{i+1,j} + A_{i,j+1} - A_{i,j} -A_{i+1,j+1} ≥ 0. The min-plus product of two n×n Monge matrices can be computed in 𝒪(n²) time using the famous SMAWK algorithm. In applications such as longest common subsequence, edit distance, and longest increasing subsequence, the matrices are even more structured, as observed by Tiskin [J. Discrete Algorithms, 2008]: they are (or can be converted to) simple unit-Monge matrices, meaning that the density matrix is a permutation matrix and, furthermore, the first column and the last row of the matrix consist of only zeroes. Such matrices admit an implicit representation of size 𝒪(n) and, as shown by Tiskin [SODA 2010 & Algorithmica, 2015], their min-plus product can be computed in 𝒪(nlog n) time. Russo [SPIRE 2010 & Theor. Comput. Sci., 2012] identified a general structural property of matrices that admit such efficient representation and min-plus multiplication algorithms: the core size δ, defined as the number of non-zero entries in the density matrices of the input and output matrices. He provided an adaptive implementation of the SMAWK algorithm that runs in 𝒪((n+δ)log³ n) or 𝒪((n+δ)log² n) time (depending on the representation of the input matrices). In this work, we further investigate the core size as the parameter that enables efficient min-plus matrix multiplication. On the combinatorial side, we provide a (linear) bound on the core size of the product matrix in terms of the core sizes of the input matrices. On the algorithmic side, we generalize Tiskin’s algorithm (but, arguably, with a more elementary analysis) to solve the core-sparse Monge matrix multiplication problem in 𝒪(n+δlog δ) ⊆ 𝒪(n + δ log n) time, matching the complexity for simple unit-Monge matrices. As witnessed by the recent work of Gorbachev and Kociumaka [STOC'25] for edit distance with integer weights, our generalization opens up the possibility of speed-ups for weighted sequence alignment problems. Furthermore, our multiplication algorithm is also capable of producing an efficient data structure for recovering the witness for any given entry of the output matrix. This allows us, for example, to preprocess an integer array of size n in Õ(n) time so that the longest increasing subsequence of any sub-array can be reconstructed in Õ(𝓁) time, where 𝓁 is the length of the reported subsequence. In comparison, Karthik C. S. and Rahul [arXiv, 2024] recently achieved 𝒪(𝓁+n^{1/2}polylog n)-time reporting after 𝒪(n^{3/2}polylog n)-time preprocessing.

Cite as

Paweł Gawrychowski, Egor Gorbachev, and Tomasz Kociumaka. Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2025.74,
  author =	{Gawrychowski, Pawe{\l} and Gorbachev, Egor and Kociumaka, Tomasz},
  title =	{{Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{74:1--74:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.74},
  URN =		{urn:nbn:de:0030-drops-245427},
  doi =		{10.4230/LIPIcs.ESA.2025.74},
  annote =	{Keywords: Min-plus matrix multiplication, Monge matrix, longest increasing subsequence}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.13

Tight Bounds for Some Classical Problems Parameterized by Cutwidth

Authors: Narek Bojikian, Vera Chekan, and Stefan Kratsch

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Cutwidth is a widely studied parameter and it quantifies how well a graph can be decomposed along small edge-cuts. It complements pathwidth, which captures decomposition by small vertex separators, and it is well-known that cutwidth upper-bounds pathwidth. The SETH-tight parameterized complexity of problems on graphs of bounded pathwidth (and treewidth) has been actively studied over the past decade while for cutwidth the complexity of many classical problems remained open. For Hamiltonian Cycle, it is known that a (2+√2)^{pw} n^𝒪(1) algorithm is optimal for pathwidth under SETH [Cygan et al. JACM 2018]. Van Geffen et al. [J. Graph Algorithms Appl. 2020] and Bojikian et al. [STACS 2023] asked which running time is optimal for this problem parameterized by cutwidth. We answer this question with (1+√2)^{ctw} n^𝒪(1) by providing matching upper and lower bounds. Second, as our main technical contribution, we close the gap left by van Heck [2018] for Partition Into Triangles (and Triangle Packing) by improving both upper and lower bound and getting a tight bound of ∛{3}^{ctw} n^𝒪(1), which to our knowledge exhibits the only known tight non-integral basis apart from Hamiltonian Cycle [Cygan et al. JACM 2018] and C₄-Hitting Set [SODA 2025]. We show that the cuts inducing a disjoint union of paths of length three (unions of so-called Z-cuts) lie at the core of the complexity of the problem - usually lower-bound constructions use simpler cuts inducing either a matching or a disjoint union of bicliques. Finally, we determine the optimal running times for Max Cut (2^{ctw} n^𝒪(1)) and Induced Matching (3^{ctw} n^𝒪(1)) by providing matching lower bounds for the existing algorithms - the latter result also answers an open question for treewidth by Chaudhary and Zehavi [WG 2023].

Cite as

Narek Bojikian, Vera Chekan, and Stefan Kratsch. Tight Bounds for Some Classical Problems Parameterized by Cutwidth. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bojikian_et_al:LIPIcs.ESA.2025.13,
  author =	{Bojikian, Narek and Chekan, Vera and Kratsch, Stefan},
  title =	{{Tight Bounds for Some Classical Problems Parameterized by Cutwidth}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.13},
  URN =		{urn:nbn:de:0030-drops-244815},
  doi =		{10.4230/LIPIcs.ESA.2025.13},
  annote =	{Keywords: Parameterized complexity, cutwidth, Hamiltonian cycle, triangle packing, max cut, induced matching}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2025.65

Parameterized Spanning Tree Congestion

Authors: Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

In this paper we study the Spanning Tree Congestion problem, where we are given an undirected graph G = (V,E) and are asked to find a spanning tree T of minimum maximum congestion. Here, the congestion of an edge e ∈ T is the number of edges uv ∈ E such that the (unique) path from u to v in T traverses e. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results: - We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form "vertex-deletion distance to class 𝒞", thus obtaining W[1]-hardness for more restricted parameters, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width 4. - Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. - Complementing the problem’s W[1]-hardness for treewidth, we formulate an algorithm that runs in time roughly {(k+w)}^{𝒪(w)}, where k is the desired congestion and w the treewidth, improving a previous argument for parameter k+w that was based on Courcelle’s theorem. This explicit algorithm pays off in two ways: it allows us to obtain an FPT approximation scheme for parameter treewidth, that is, a (1+ε)-approximation running in time roughly {(w/ε)}^{𝒪(w)}; and it leads to an exact FPT algorithm for parameter clique-width+k via a Win/Win argument. - Finally, motivated by the problem’s hardness for most standard structural parameters, we present FPT algorithms for several more restricted cases, namely, for the parameters vertex-deletion distance to clique; vertex integrity; and feedback edge set, in the latter case also achieving a single-exponential running time dependence on the parameter.

Cite as

Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz. Parameterized Spanning Tree Congestion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lampis_et_al:LIPIcs.MFCS.2025.65,
  author =	{Lampis, Michael and Mitsou, Valia and Nemery, Edouard and Otachi, Yota and Vasilakis, Manolis and Vaz, Daniel},
  title =	{{Parameterized Spanning Tree Congestion}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{65:1--65:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.65},
  URN =		{urn:nbn:de:0030-drops-241724},
  doi =		{10.4230/LIPIcs.MFCS.2025.65},
  annote =	{Keywords: Parameterized Complexity, Treewidth, Graph Width Parameters}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.109

Faster Diameter Computation in Graphs of Bounded Euler Genus

Authors: Kacper Kluk, Marcin Pilipczuk, Michał Pilipczuk, and Giannos Stamoulis

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

Abstract

We show that for any fixed integer k ⩾ 0, there exists an algorithm that computes the diameter and the eccentricies of all vertices of an input unweighted, undirected n-vertex graph of Euler genus at most k in time 𝒪_k(n^{2-1/25}). Furthermore, for the more general class of graphs that can be constructed by clique-sums from graphs that are of Euler genus at most k after deletion of at most k vertices, we show an algorithm for the same task that achieves the running time bound 𝒪_k(n^{2-1/356} log^{6k} n). Up to today, the only known subquadratic algorithms for computing the diameter in those graph classes are that of [Ducoffe, Habib, Viennot; SICOMP 2022], [Le, Wulff-Nilsen; SODA 2024], and [Duraj, Konieczny, Potępa; ESA 2024]. These algorithms work in the more general setting of K_h-minor-free graphs, but the running time bound is 𝒪_h(n^{2-c_h}) for some constant c_h > 0 depending on h. That is, our savings in the exponent of the polynomial function of n, as compared to the naive quadratic algorithm, are independent of the parameter k. The main technical ingredient of our work is an improved bound on the number of distance profiles, as defined in [Le, Wulff-Nilsen; SODA 2024], in graphs of bounded Euler genus.

Cite as

Kacper Kluk, Marcin Pilipczuk, Michał Pilipczuk, and Giannos Stamoulis. Faster Diameter Computation in Graphs of Bounded Euler Genus. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 109:1-109:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kluk_et_al:LIPIcs.ICALP.2025.109,
  author =	{Kluk, Kacper and Pilipczuk, Marcin and Pilipczuk, Micha{\l} and Stamoulis, Giannos},
  title =	{{Faster Diameter Computation in Graphs of Bounded Euler Genus}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{109:1--109:19},
  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.2025.109},
  URN =		{urn:nbn:de:0030-drops-234869},
  doi =		{10.4230/LIPIcs.ICALP.2025.109},
  annote =	{Keywords: Diameter, eccentricity, subquadratic algorithms, surface-embeddable graphs}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.30

Light Spanners with Small Hop-Diameter

Authors: Sujoy Bhore and Lazar Milenković

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

Abstract

Lightness, sparsity, and hop-diameter are the fundamental parameters of geometric spanners. Arya et al. [STOC'95] showed in their seminal work that there exists a construction of Euclidean (1+ε)-spanners with hop-diameter O(log n) and lightness O(log n). They also gave a general tradeoff of hop-diameter k and sparsity O(α_k(n)), where α_k is a very slowly growing inverse of an Ackermann-style function. The former combination of logarithmic hop-diameter and lightness is optimal due to the lower bound by Dinitz et al. [FOCS'08]. Later, Elkin and Solomon [STOC'13] generalized the light spanner construction to doubling metrics and extended the tradeoff for more values of hop-diameter k. In a recent line of work [SoCG'22, SoCG'23], Le et al. proved that the aforementioned tradeoff between the hop-diameter and sparsity is tight for every choice of hop-diameter k. A fundamental question remains: What is the optimal tradeoff between the hop-diameter and lightness for every value of k? In this paper, we present a general framework for constructing light spanners with small hop-diameter. Our framework is based on tree covers. In particular, we show that if a metric admits a tree cover with γ trees, stretch t, and lightness L, then it also admits a t-spanner with hop-diameter k and lightness O(kn^{2/k}⋅ γ L). Further, we note that the tradeoff for trees is tight due to a construction in uniform line metric, which is perhaps the simplest tree metric. As a direct consequence of this framework, we obtain new tradeoffs between lightness and hop-diameter for doubling metrics.

Cite as

Sujoy Bhore and Lazar Milenković. Light Spanners with Small Hop-Diameter. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhore_et_al:LIPIcs.ICALP.2025.30,
  author =	{Bhore, Sujoy and Milenkovi\'{c}, Lazar},
  title =	{{Light Spanners with Small Hop-Diameter}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{30:1--30:16},
  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.2025.30},
  URN =		{urn:nbn:de:0030-drops-234075},
  doi =		{10.4230/LIPIcs.ICALP.2025.30},
  annote =	{Keywords: Geometric Spanners, Lightness, Hop-Diameter, Recurrences, Lower Bounds}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.33

Faster Construction of a Planar Distance Oracle with Õ(1) Query Time

Authors: Itai Boneh, Shay Golan, Shay Mozes, Daniel Prigan, and Oren Weimann

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

Abstract

We show how to preprocess a weighted undirected n-vertex planar graph in Õ(n^{4/3}) time, such that the distance between any pair of vertices can then be reported in Õ(1) time. This improves the previous Õ(n^{3/2}) preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of additively weighted Voronoi diagrams in undirected planar graphs. Namely, given a planar graph G and a face f, we show that one can preprocess G in Õ(n) time such that given any weight assignment to the vertices of f one can construct the additively weighted Voronoi diagram of f in near optimal Õ(|f|) time. This improves the Õ(√{n|f|}) construction time of [JACM'23].

Cite as

Itai Boneh, Shay Golan, Shay Mozes, Daniel Prigan, and Oren Weimann. Faster Construction of a Planar Distance Oracle with Õ(1) Query Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{boneh_et_al:LIPIcs.ICALP.2025.33,
  author =	{Boneh, Itai and Golan, Shay and Mozes, Shay and Prigan, Daniel and Weimann, Oren},
  title =	{{Faster Construction of a Planar Distance Oracle with \~{O}(1) Query Time}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{33:1--33:21},
  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.2025.33},
  URN =		{urn:nbn:de:0030-drops-234106},
  doi =		{10.4230/LIPIcs.ICALP.2025.33},
  annote =	{Keywords: Distance Oracle, Planar Graph, Construction Time}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2025.66

Lipschitz Decompositions of Finite 𝓁_{p} Metrics

Authors: Robert Krauthgamer and Nir Petruschka

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for n-point subsets of 𝓁_p, for p > 2, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound β = O(log^{1-1/p} n). Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for 1 < p < 2, due to Filtser and Neiman [Algorithmica 2022].

Cite as

Robert Krauthgamer and Nir Petruschka. Lipschitz Decompositions of Finite 𝓁_{p} Metrics. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{krauthgamer_et_al:LIPIcs.SoCG.2025.66,
  author =	{Krauthgamer, Robert and Petruschka, Nir},
  title =	{{Lipschitz Decompositions of Finite 𝓁\underline\{p\} Metrics}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.66},
  URN =		{urn:nbn:de:0030-drops-232182},
  doi =		{10.4230/LIPIcs.SoCG.2025.66},
  annote =	{Keywords: Lipschitz decompositions, metric embeddings, geometric spanners}
}

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