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DOI: 10.4230/LIPIcs.SEA.2026.8

General Multiplicative Spanners in Practice

Authors: Fritz Bökler, Markus Chimani, and Henning Jasper

Published in: LIPIcs, Volume 371, 24th International Symposium on Experimental Algorithms (SEA 2026)

Abstract

Given an undirected graph G with edge weights and lengths, a minimum α-spanner is a least-weight subgraph H ⊆ G that preserves distances w.r.t. the lengths between all node pairs up to a factor of α. Literature often takes the simplifying assumption of a single (coupled) edge function for weights and lengths. For such instances, several exact and non-exact algorithms are known and have been thoroughly evaluated in practice. However, many practical instances have decoupled form, as their weights and lengths are generally independent. Due to the increased complexity, only few (and even fewer practical) algorithms are able to guarantee low-weight solutions. This prompts practitioners to force their naturally decoupled instances into a coupled format, forsaking any quality guarantee. We implement several exact, approximative and heuristic algorithms for decoupled α-spanners, and use algorithm engineering to speed them up in practice. Our hypothesis-driven experiments evaluate their performance w.r.t. solution quality and speed. Generally, many practical instances can indeed be solved exactly within reasonable time, while LP-based approximation algorithms are not worthwhile. We find that standard greedy algorithms often yield acceptable results, but there are also practical instances for which they yield arbitrarily poor solutions. Here, augmented greedy variations offer a good compromise between solution quality and speed.

Cite as

Fritz Bökler, Markus Chimani, and Henning Jasper. General Multiplicative Spanners in Practice. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bokler_et_al:LIPIcs.SEA.2026.8,
  author =	{B\"{o}kler, Fritz and Chimani, Markus and Jasper, Henning},
  title =	{{General Multiplicative Spanners in Practice}},
  booktitle =	{24th International Symposium on Experimental Algorithms (SEA 2026)},
  pages =	{8:1--8:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-422-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{371},
  editor =	{Aum\"{u}ller, Martin and Finocchi, Irene},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2026.8},
  URN =		{urn:nbn:de:0030-drops-260120},
  doi =		{10.4230/LIPIcs.SEA.2026.8},
  annote =	{Keywords: Graph spanners, ILP, experimental study, algorithm engineering}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.35

Path-Reporting Distance Oracles for Vertex-Labeled Graphs

Authors: Ofer Neiman and Alon Spector

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

Abstract

Let G = (V,E) be a weighted undirected graph, with n vertices. A distance oracle is a data structure that can quickly answer distance queries, with some stretch factor. A seminal work of [Thorup and Zwick, 2005], given an integer k ≥ 1, provides such an oracle with stretch 2k-1, query time O(k), and size O(k⋅ n^{1+1/k}). Furthermore, this oracle can also report a path in G corresponding to the returned distance. In this paper we focus on vertex-labeled graphs, in which each vertex is given a label from a set L of size 𝓁. A vertex-label distance oracle answers queries of the form (v,λ), where v ∈ V and λ ∈ L, by reporting (an approximation to) the distance from v to the closest vertex of label λ. Following [Danny Hermelin et al., 2011], it was shown in [Chechik, 2012] that for any integer k > 1, there exists a vertex-label distance oracle with stretch 4k-5, query time O(k), and size O(k⋅ n⋅ 𝓁^{1/k}). This state-of-the-art result suffers from two main drawbacks: The stretch is roughly a factor of 2 larger than in [Thorup and Zwick, 2005], and it is not path-reporting. We address these concerns in this work, and provide the following results. - First, we devise a path-reporting vertex-label distance oracle, at the cost of a slight increase in stretch and size. For any constant 0 < ε < 1, our oracle has stretch (4k-5)⋅(1+ε), query time O(k), and size O(n^{1+o(1)}⋅ 𝓁^{1/k}). - Second, we show how to improve the stretch to the optimal 2k-1, at the cost of mildly increasing the query time. Specifically, we devise a vertex-label distance oracle with stretch 2k-1, query time O(𝓁^{1/k}⋅log n), and size O(k⋅ n⋅ 𝓁^{1/k}).

Cite as

Ofer Neiman and Alon Spector. Path-Reporting Distance Oracles for Vertex-Labeled Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{neiman_et_al:LIPIcs.SWAT.2026.35,
  author =	{Neiman, Ofer and Spector, Alon},
  title =	{{Path-Reporting Distance Oracles for Vertex-Labeled Graphs}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{35:1--35:16},
  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.2026.35},
  URN =		{urn:nbn:de:0030-drops-260719},
  doi =		{10.4230/LIPIcs.SWAT.2026.35},
  annote =	{Keywords: Graph Algorithms, Shortest Paths, Distance Oracles}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.24

Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support

Authors: Reilly Browne and Hsien-Chih Chang

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

Abstract

Given an unweighted graph G, the minimum r-dominating set problem asks for a subset of vertices S of the smallest cardinality, such that every vertex in G is within radius r to some vertex in S. While the r-dominating set problem on planar graph admits PTAS from Baker’s shifting/layering technique when r is a constant, the problem becomes significantly harder when r can depend on n. In fact, under Exponential-Time Hypothesis, Fox-Epstein ηl [SODA 2019] observed that no efficient PTAS can exist for the unbounded r-dominating set problem on planar graphs. One may consider even harder weighted-variant known as the vertex-weighted metric r-dominating set, where edges are associated with lengths, and every vertex is associated with a positive-valued weight, and the goal is to compute an r-dominating set with minimum total weight. As a result, people resorted to bicriteria algorithms by allowing the returned solution to use radius-(1+ε)r balls instead, in addition to the total weight being a 1+ε approximation to the optimal value. We establish the first single-criteria polynomial-time O(1)-approximation algorithm for the vertex-weighted metric r-dominating set problem on planar graphs when r is part of the input, and can be arbitrarily large compared to n. Our new (single-criteria) O(1)-approximation algorithm uses the quasi-uniformity sampling technique of Chan et al. [SODA 2012] by bounding the shallow cell complexity of the (unbounded) radius-r ball system to be linear in n. To this end we have two technical innovations: 1) The discrete ball system on planar graphs are neither pseudodisks nor have well-defined boundaries for standard union-complexity arguments. We construct a support graph for arbitrary distance ball systems as contractions of Voronoi cells; the sparseness comes as a byproduct. 2) We present an assignment of each depth-(≥3) cell to a unique 3-tuple of ball centers. This allows us to use standard Clarkson-Shor techniques to reduce the counting to cells of depth exactly 3, which we prove to be size O(n) by a novel geometric argument based on our support being a Voronoi contraction.

Cite as

Reilly Browne and Hsien-Chih Chang. Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{browne_et_al:LIPIcs.SoCG.2026.24,
  author =	{Browne, Reilly and Chang, Hsien-Chih},
  title =	{{Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{24:1--24:17},
  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.2026.24},
  URN =		{urn:nbn:de:0030-drops-258300},
  doi =		{10.4230/LIPIcs.SoCG.2026.24},
  annote =	{Keywords: Minimum dominating set, planar graphs, shallow cell complexity}
}

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.2026.48

FPT Approximations for Capacitated Sum of Radii and Diameters

Authors: Arnold Filtser and Ameet Gadekar

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

Abstract

The Capacitated Sum of Radii problem involves partitioning a set of points P, where each point p ∈ P has capacity U_p, into k clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point p is at most U_p. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a ≈5.83-approximation algorithm in FPT time (improving a previous ≈7.61 approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.

Cite as

Arnold Filtser and Ameet Gadekar. FPT Approximations for Capacitated Sum of Radii and Diameters. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{filtser_et_al:LIPIcs.SoCG.2026.48,
  author =	{Filtser, Arnold and Gadekar, Ameet},
  title =	{{FPT Approximations for Capacitated Sum of Radii and Diameters}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{48:1--48:18},
  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.2026.48},
  URN =		{urn:nbn:de:0030-drops-258545},
  doi =		{10.4230/LIPIcs.SoCG.2026.48},
  annote =	{Keywords: clustering, sum of radii, sum of diameter, capacitated clustering, fpt}
}

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.

Cite as

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.2026.65

Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces

Authors: Sándor Kisfaludi-Bak and Geert van Wordragen

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

Abstract

We consider Steiner spanners in Euclidean and non-Euclidean geometries. In the Euclidean setting, a recent line of work initiated by Le and Solomon [FOCS'19] and further improved by Chang et al. [SoCG'24] obtained Steiner (1+ε)-spanners of size O_d(ε^{(1-d)/2} log(1/ε) n), nearly matching the lower bound Ω_d(ε^{(1-d)/2} n) of Bhore and Tóth [SIDMA'22]. We obtain Steiner (1+ε)-spanners of size O_d(ε^{(1-d)/2} log(1/ε)n) not only in d-dimensional Euclidean space, but also in d-dimensional spherical and hyperbolic space. For any fixed dimension d, the obtained edge count is optimal up to an O(log(1/ε)) factor in each of these spaces. Unlike earlier constructions, our Steiner spanners are based on simple quadtrees, and they can be dynamically maintained, leading to efficient data structures for dynamic approximate nearest neighbours and bichromatic closest pair. In the hyperbolic setting, we also show that 2-spanners in the hyperbolic plane must have Ω(nlog n) edges, and we obtain a 2-spanner of size O_d(nlog n) in d-dimensional hyperbolic space, matching our lower bound for any constant d. Finally, we give a Steiner spanner with additive error ε in hyperbolic space with O_d(ε^{(1-d)/2} log(α(n)/ε)n) edges, where α(n) is the inverse Ackermann function. Our techniques generalize to closed orientable surfaces of constant curvature as well as to some other quotient spaces.

Cite as

Sándor Kisfaludi-Bak and Geert van Wordragen. Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.65,
  author =	{Kisfaludi-Bak, S\'{a}ndor and van Wordragen, Geert},
  title =	{{Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{65:1--65:17},
  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.2026.65},
  URN =		{urn:nbn:de:0030-drops-258728},
  doi =		{10.4230/LIPIcs.SoCG.2026.65},
  annote =	{Keywords: hyperbolic geometry, Steiner spanner, dynamic approximate nearest neighbours}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.68

Optimal Bounds for Spanners and Tree Covers in Doubling Metrics

Authors: An La, Hung Le, Shay Solomon, Cuong Than, Vinayak, Shuang Yang, and Tianyi Zhang

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

Abstract

It is known that any n-point set in the d-dimensional Euclidean space ℝ^d, for d = O(1), admits: 1) A (1+ε)-spanner with maximum degree Õ(ε^{-d+1}) and with lightness Õ(ε^{-d}), for any ε > 0. 2) A (1+ε)-tree cover with Õ(n ⋅ ε^{-d+1}) trees and maximum degree of O(1) in each tree. Moreover, all the parameters in these constructions are optimal: For any 2 ≤ d = O(1), there exists an n-point set in ℝ^d, for which any (1+ε)-spanner has Ω̃(n⋅ε^{-d+1}) edges and lightness Ω̃(ε^{-d}). The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in ℝ^d, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has Õ(n⋅ε^{-d}) edges, and it could be transformed into a spanner of maximum degree Õ(ε^{-d}) and lightness Õ(n⋅ε^{-(d+1)}) by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a (1+ε)-tree cover with Õ(ε^{-d}) trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) A surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters. 2) A new construction of (1+ε)-tree covers with Õ(n⋅ε^{-d}) trees, with maximum degree O(1) in each tree. This construction is optimal with respect to the number of trees and maximum degree.

Cite as

An La, Hung Le, Shay Solomon, Cuong Than, Vinayak, Shuang Yang, and Tianyi Zhang. Optimal Bounds for Spanners and Tree Covers in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 68:1-68:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{la_et_al:LIPIcs.SoCG.2026.68,
  author =	{La, An and Le, Hung and Solomon, Shay and Than, Cuong and Vinayak and Yang, Shuang and Zhang, Tianyi},
  title =	{{Optimal Bounds for Spanners and Tree Covers in Doubling Metrics}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{68:1--68:16},
  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.2026.68},
  URN =		{urn:nbn:de:0030-drops-258756},
  doi =		{10.4230/LIPIcs.SoCG.2026.68},
  annote =	{Keywords: doubling metrics, doubling spanners, Euclidean spanners, tree cover}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.70

Tree-Like Shortcuttings of Trees

Authors: Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than

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

Abstract

Sparse shortcuttings of trees - equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter - have been studied extensively (under different names and settings), since the pioneering works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for n-node trees with sparsity O(log^* n). Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity Ω(log n)), which is a significant drawback for many applications. We initiate a systematic study of constant-hop tree shortcuttings that are "tree-like". We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold. - New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to O(log log n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((log log n)²) for all values of hop-diameter, resolving an open question of [Arnold Filtser and Hung Le, 2022; H. Le, 2023]. - Applications of these bounds, focusing on low-dimensional Euclidean and doubling metrics. A seminal work of Arya et al. [S. Arya et al., 1995] presented a (1+ε)-spanner with constant hop-diameter and sparsity O(log^* n), but with large arboricity. We show that constant hop-diameter is sufficient to achieve arboricity O(log^*{n}). Furthermore, we present a (1+ε)-stretch routing scheme in the fixed-port model with 3 hops and a local memory of O(log²n / log log n) bits, resolving an open question of [Omri Kahalon et al., 2022].

Cite as

Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Tree-Like Shortcuttings of Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{le_et_al:LIPIcs.SoCG.2026.70,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
  title =	{{Tree-Like Shortcuttings of Trees}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{70:1--70:15},
  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.2026.70},
  URN =		{urn:nbn:de:0030-drops-258776},
  doi =		{10.4230/LIPIcs.SoCG.2026.70},
  annote =	{Keywords: spanner, tree shortcutting, arboricity, treewidth}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.15

Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

Authors: Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong

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

Abstract

A Euclidean noncrossing Steiner (1+ε)-spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε)-spanner with O(n/ε^{3/2}) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O(n/ε⁴) obtained nearly three decades ago. We also establish an almost matching lower bound: There exist n points in the plane for which any Euclidean noncrossing Steiner (1+ε)-spanner has Ω_μ(n/ε^{3/2-μ}) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

Cite as

Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.15,
  author =	{Bhore, Sujoy and Kisfaludi‑Bak, S\'{a}ndor and Milenkovi\'{c}, Lazar and T\'{o}th, Csaba D. and W\k{e}grzycki, Karol and Wong, Sampson},
  title =	{{Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{15:1--15:18},
  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.2026.15},
  URN =		{urn:nbn:de:0030-drops-258210},
  doi =		{10.4230/LIPIcs.SoCG.2026.15},
  annote =	{Keywords: geometric network design, spanners, crossing number, incidences}
}

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.2026.17

Fréchet Distance in the Imbalanced Case

Authors: Lotte Blank

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

Abstract

Given two polygonal curves P and Q defined by n and m vertices with m ≤ n, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of 2-ε in 𝒪((nm)^{1-δ}) time for any ε, δ > 0 unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to 1+√2-ε (resp. 3-ε) if the curves lie in the Euclidean space (resp. in the L_∞-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where m = n^α for α ∈ (0,1) and increases the approximation factor of 1.001 by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any L_p space, we present a (3+ε)-approximation algorithm for the continuous and discrete Fréchet distance using 𝒪((n+m²)log n) time, which almost matches the approximation factor of the lower bound for the L_∞ metric.

Cite as

Lotte Blank. Fréchet Distance in the Imbalanced Case. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{blank:LIPIcs.SoCG.2026.17,
  author =	{Blank, Lotte},
  title =	{{Fr\'{e}chet Distance in the Imbalanced Case}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{17:1--17:17},
  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.2026.17},
  URN =		{urn:nbn:de:0030-drops-258232},
  doi =		{10.4230/LIPIcs.SoCG.2026.17},
  annote =	{Keywords: Fr\'{e}chet distance, SETH, Orthogonal Vectors, Lower Bounds, distance oracle, data structures}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.94

Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion

Authors: Yingxi Li, Ellen Vitercik, and Mingwei Yang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

In the online metric matching problem, n servers and n requests lie in a metric space. Servers are available upfront, and requests arrive sequentially. An arriving request must be matched immediately and irrevocably to an available server, incurring a cost equal to their distance. The goal is to minimize the total matching cost. We study this problem in [0, 1]^d with the Euclidean metric, when servers are adversarial and requests are independently drawn from distinct distributions that satisfy a mild smoothness condition. Our main result is an O(1)-competitive algorithm for d ≠ 2 that requires no distributional knowledge, relying only on a single sample from each request distribution. To our knowledge, this is the first algorithm to achieve an o(log n) competitive ratio for non-trivial metrics beyond the i.i.d. setting. Our approach bypasses the Ω(log n) barrier introduced by probabilistic metric embeddings: instead of analyzing the embedding distortion and the algorithm separately, we directly bound the cost of the algorithm on the target metric space of a simple deterministic embedding. We then combine this analysis with lower bounds on the offline optimum for Euclidean metrics, derived via majorization arguments, to obtain our guarantees.

Cite as

Yingxi Li, Ellen Vitercik, and Mingwei Yang. Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 94:1-94:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{li_et_al:LIPIcs.ITCS.2026.94,
  author =	{Li, Yingxi and Vitercik, Ellen and Yang, Mingwei},
  title =	{{Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{94:1--94:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.94},
  URN =		{urn:nbn:de:0030-drops-253815},
  doi =		{10.4230/LIPIcs.ITCS.2026.94},
  annote =	{Keywords: Online algorithm, Metric matching, Competitive analysis, Smoothed analysis}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.40

Improved Upper Bounds on Multiflow-Multicut Gaps in Cactus Graphs

Authors: Sina Kalantarzadeh and Nikhil Kumar

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the largest total amount of flow that can be feasibly routed between them. The minimum multicut problem, which is dual to multiflow, seeks the lowest-cost set of edges whose removal disconnects all source-sink pairs. It is straightforward to see that the value of a minimum multicut is at least that of the corresponding maximum multiflow. The ratio between the two is known as the multiflow-multicut gap. The classical max-flow min-cut theorem tells us that this gap is exactly one when there is only a single source-sink pair. However, for multiple source-sink pairs, the gap can be arbitrarily large. In this work, we investigate the multiflow-multicut gap in cactus graphs, and establish the following results (i) tight upper bound of 1.5 for cycle (ii) an upper bound of 2 + 2/(ln 2) < 3.45 for general cactus graph (iii) tight upper bound of 2 for unicyclic graphs, where the graph contains exactly one cycle (iv) tight upper bound of 2 for path cactus graphs, where cycles are arranged along a single path. We develop novel generalizations of the classical rounding algorithm to establish our results.

Cite as

Sina Kalantarzadeh and Nikhil Kumar. Improved Upper Bounds on Multiflow-Multicut Gaps in Cactus Graphs. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 40:1-40:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kalantarzadeh_et_al:LIPIcs.FSTTCS.2025.40,
  author =	{Kalantarzadeh, Sina and Kumar, Nikhil},
  title =	{{Improved Upper Bounds on Multiflow-Multicut Gaps in Cactus Graphs}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{40:1--40:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.40},
  URN =		{urn:nbn:de:0030-drops-251205},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.40},
  annote =	{Keywords: Approximation Algorithms, Randomized Algorithms, Linear Programming, Graph Algorithms, Multicut, Multicommodity flow}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.62

Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering

Authors: Sina Bagheri Nezhad, Sayan Bandyapadhyay, and Tianzhi Chen

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

Abstract

In a seminal work, Chierichetti et al. [Chierichetti et al., 2017] introduced the (t,k)-fair clustering problem: Given a set of red points and a set of blue points in a metric space, a clustering is called fair if the number of red points in each cluster is at most t times and at least 1/t times the number of blue points in that cluster. The goal is to compute a fair clustering with at most k clusters that optimizes certain objective function. Considering this problem, they designed a polynomial-time O(1)- and O(t)-approximation for the k-center and the k-median objective, respectively. Recently, Carta et al. [Carta et al., 2024] studied this problem with the sum-of-radii objective and obtained a (6+ε)-approximation with running time O((k log_{1+ε}(k/ε))^k n^O(1)), i.e., fixed-parameter tractable in k. Here n is the input size. In this work, we design the first polynomial-time O(1)-approximation for (t,k)-fair clustering with the sum-of-radii objective, improving the result of Carta et al. Our result places sum-of-radii in the same group of objectives as k-center, that admit polynomial-time O(1)-approximations. This result also implies a polynomial-time O(1)-approximation for the Euclidean version of the problem, for which an f(k)⋅n^O(1)-time (1+ε)-approximation was known due to Drexler et al. [Drexler et al., 2023]. Here f is an exponential function of k. We are also able to extend our result to any arbitrary 𝓁 ≥ 2 number of colors when t = 1. This matches known results for the k-center and k-median objectives in this case. The significant disparity of sum-of-radii compared to k-center and k-median presents several complex challenges, all of which we successfully overcome in our work. Our main contribution is a novel cluster-merging-based analysis technique for sum-of-radii that helps us achieve the constant-approximation bounds.

Cite as

Sina Bagheri Nezhad, Sayan Bandyapadhyay, and Tianzhi Chen. Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bagherinezhad_et_al:LIPIcs.ESA.2025.62,
  author =	{Bagheri Nezhad, Sina and Bandyapadhyay, Sayan and Chen, Tianzhi},
  title =	{{Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{62:1--62:16},
  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.2025.62},
  URN =		{urn:nbn:de:0030-drops-245309},
  doi =		{10.4230/LIPIcs.ESA.2025.62},
  annote =	{Keywords: fair clustering, sum-of-radii clustering, approximation algorithms}
}

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