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Volume

LIPIcs, Volume 87

25th Annual European Symposium on Algorithms (ESA 2017)

ESA 2017, September 4-6, 2017, Vienna, Austria

Editors: Kirk Pruhs and Christian Sohler

Document

DOI: 10.4230/LIPIcs.STACS.2026.55

Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion

Authors: Samuel Humeau, Mamadou Moustapha Kanté, Daniel Mock, Timothé Picavet, and Alexandre Vigny

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

Abstract

In property testing, a tester makes queries to (an oracle for) a graph and, on a graph having or being far from having a property P, it decides with high probability whether the graph satisfies P or not. Often, testers are restricted to a constant number of queries. While the graph properties for which there exists such a tester are somewhat well characterized in the dense graph model, it is not the case for sparse graphs. In this area, Czumaj and Sohler (FOCS’19) proved that H-freeness (i.e. the property of excluding the graph H as a subgraph) can be tested with constant queries on planar graphs as well as on graph classes excluding a minor. Using results from the sparsity toolkit, we propose a simpler alternative to the proof of Czumaj and Sohler, for a statement generalized to the broader notion of bounded expansion. That is, we prove that for any class 𝒞 with bounded expansion and any graph H, testing H-freeness can be done with constant query complexity on any graph G in 𝒞, where the constant depends on H and 𝒞, but is independent of G. While classes excluding a minor are prime examples of classes with bounded expansion, so are, for example, cubic graphs, graph classes with bounded maximum degree, or graphs of bounded book thickness. Additionally, random graphs with bounded average degree almost surely have bounded expansion.

Cite as

Samuel Humeau, Mamadou Moustapha Kanté, Daniel Mock, Timothé Picavet, and Alexandre Vigny. Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{humeau_et_al:LIPIcs.STACS.2026.55,
  author =	{Humeau, Samuel and Kant\'{e}, Mamadou Moustapha and Mock, Daniel and Picavet, Timoth\'{e} and Vigny, Alexandre},
  title =	{{Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{55:1--55:18},
  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.2026.55},
  URN =		{urn:nbn:de:0030-drops-255441},
  doi =		{10.4230/LIPIcs.STACS.2026.55},
  annote =	{Keywords: Property testing, Sparsity, Bounded expansion, Treedepth}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.4

Unit Interval Selection in Random Order Streams

Authors: Cezar-Mihail Alexandru, Adithya Diddapur, Magnús M. Halldórsson, Christian Konrad, and Kheeran K. Naidu

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

Abstract

We consider the Unit Interval Selection problem in the one-pass random order streaming model. In this setting, an algorithm is presented with a sequence of n unit-length intervals on the line that arrive in uniform random order, one at a time, and the objective is to output (an approximation of) a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, within these space constraints, a (2/3)-approximation can be achieved in such streams, and this is best possible, in that going beyond such an approximation factor requires space Ω(n) [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, where the expectation is taken over the stream order. More specifically, we give a one-pass streaming algorithm with expected approximation factor 0.7401 that uses space O(|OPT|), where OPT denotes an optimal solution. We also show that random order algorithms with expected approximation factor above 8/9 require space Ω(n), and algorithms that compute a better than 2/3-approximation with probability above 2/3 also require Ω(n) space. On a technical level, we design an algorithm for the restricted domain [0, Δ), for some constant Δ, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain [0, Δ), we run O(Δ) recursive instances of our algorithm, with each instance targeting the situation where a specific interval of an optimal solution arrives first. We establish the interesting property of our algorithm that it performs worst when the input stream consists solely of a set of independent intervals. It then remains to analyse the algorithm on these simple instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds established by [Chakrabarti et al., Theory Comput. 2016].

Cite as

Cezar-Mihail Alexandru, Adithya Diddapur, Magnús M. Halldórsson, Christian Konrad, and Kheeran K. Naidu. Unit Interval Selection in Random Order Streams. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{alexandru_et_al:LIPIcs.STACS.2026.4,
  author =	{Alexandru, Cezar-Mihail and Diddapur, Adithya and Halld\'{o}rsson, Magn\'{u}s M. and Konrad, Christian and Naidu, Kheeran K.},
  title =	{{Unit Interval Selection in Random Order Streams}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{4:1--4:20},
  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.2026.4},
  URN =		{urn:nbn:de:0030-drops-254933},
  doi =		{10.4230/LIPIcs.STACS.2026.4},
  annote =	{Keywords: Random order streaming algorithms, unit interval selection}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.39

Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions

Authors: Keerti Choudhary, Amit Kumar, and Lakshay Saggi

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

Abstract

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

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Keerti Choudhary, Amit Kumar, and Lakshay Saggi. Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{choudhary_et_al:LIPIcs.ITCS.2026.39,
  author =	{Choudhary, Keerti and Kumar, Amit and Saggi, Lakshay},
  title =	{{Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{39:1--39:23},
  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.2026.39},
  URN =		{urn:nbn:de:0030-drops-253267},
  doi =		{10.4230/LIPIcs.ITCS.2026.39},
  annote =	{Keywords: Disjoint paths, Disjoint shortest paths, Algebraic graph algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.82

Dimension Reduction for Clustering: The Curious Case of Discrete Centers

Authors: Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue

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

Abstract

The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of n points into dimension O(log n) with low distortion of their distances. This dimension bound is tight in general, but one can bypass it for specific problems. Indeed, tremendous progress has been made for clustering problems, especially in the continuous setting where centers can be picked from the ambient space ℝ^d. Most notably, for k-median and k-means, the dimension bound was improved to O(log k) [Makarychev, Makarychev and Razenshteyn, STOC 2019]. We explore dimension reduction for clustering in the discrete setting, where centers can only be picked from the dataset, and present two results that are both parameterized by the doubling dimension of the dataset, denoted as ddim. The first result shows that dimension O_{ε}(ddim + log k + log log n) suffices, and is moreover tight, to guarantee that the cost is preserved within factor 1±ε for every set of centers. Our second result eliminates the log log n term in the dimension through a relaxation of the guarantee (namely, preserving the cost only for all approximately-optimal sets of centers), which maintains its usefulness for downstream applications. Overall, we achieve strong dimension reduction in the discrete setting, and find that it differs from the continuous setting not only in the dimension bound, which depends on the doubling dimension, but also in the guarantees beyond preserving the optimal value, such as which clusterings are preserved.

Cite as

Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue. Dimension Reduction for Clustering: The Curious Case of Discrete Centers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 82:1-82:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jiang_et_al:LIPIcs.ITCS.2026.82,
  author =	{Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Sapir, Shay and Silwal, Sandeep and Yue, Di},
  title =	{{Dimension Reduction for Clustering: The Curious Case of Discrete Centers}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{82:1--82:23},
  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.2026.82},
  URN =		{urn:nbn:de:0030-drops-253698},
  doi =		{10.4230/LIPIcs.ITCS.2026.82},
  annote =	{Keywords: dimension reduction, clustering, k-median, k-means, doubling dimension}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.85

Recovering Communities in Structured Random Graphs

Authors: Michael Kapralov, Luca Trevisan, and Weronika Wrzos-Kaminska

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

Abstract

The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the d-dimensional hypercube graph admits d distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability p ∈ (0, 1)? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate p = Clog d/d for a sufficiently large constant C is 1/poly(d)-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all d cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger’s cut counting argument.

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Michael Kapralov, Luca Trevisan, and Weronika Wrzos-Kaminska. Recovering Communities in Structured Random Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 85:1-85:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kapralov_et_al:LIPIcs.ITCS.2026.85,
  author =	{Kapralov, Michael and Trevisan, Luca and Wrzos-Kaminska, Weronika},
  title =	{{Recovering Communities in Structured Random Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{85:1--85:23},
  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.2026.85},
  URN =		{urn:nbn:de:0030-drops-253727},
  doi =		{10.4230/LIPIcs.ITCS.2026.85},
  annote =	{Keywords: Hypercube graphs, Community detection, Fourier analysis of Boolean functions}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.21

Overlay Network Construction: Improved Overall and Node-Wise Message Complexity

Authors: Yi-Jun Chang, Yanyu Chen, and Gopinath Mishra

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

Abstract

We consider the problem of constructing distributed overlay networks, where nodes in a reconfigurable system can create or sever connections with nodes whose identifiers they know. Initially, each node knows only its own and its neighbors' identifiers, forming a local channel, while the evolving structure is termed the global channel. The goal is to reconfigure any connected graph into a desired topology, such as a bounded-degree expander graph or a well-formed tree (WFT) with a constant maximum degree and logarithmic diameter, minimizing the total number of rounds and message complexity. This problem mirrors real-world peer-to-peer network construction, where creating robust and efficient systems is desired. We study the overlay reconstruction problem in a network of n nodes in two models: GOSSIP-reply and HYBRID. In the GOSSIP-reply model, each node can send a message and receive a corresponding reply message in one round. In the HYBRID model, a node can send O(1) messages to each neighbor in the local channel and a total of O(log n) messages in the global channel. In both models, we propose protocols for WFT construction with O (n log n) message complexities using messages of O(log n) bits. In the GOSSIP-reply model, our protocol takes O(log n) rounds while in the HYBRID model, our protocol takes O(log² n) rounds. Both protocols use O (n log² n) bits of communication. We obtain improved bounds over prior work: GOSSIP-reply: A recent result by Dufoulon et al. (ITCS 2024) achieved O(log⁵ n) round complexity and O (n log⁵ n) message complexity using messages of at least Ω(log² n) bits in GOSSIP-reply. With messages of size O(log n), our protocol achieves an optimal round complexity of O(log n) and an improved message complexity of O(n log n). HYBRID: Götte et al. (Distributed Computing 2023) showed an optimal O(log n)-round algorithm with O(log² n) global messages per round which incurs a message complexity of Ω(m), where m is the number of edges in the initial topology. At the cost of increasing the round complexity to O(log² n) while using only O(log n) messages globally, our protocol achieves a message complexity that is independent of m. Our approach ensures that the total number of messages for node v, with degree deg(v) in the initial topology, is bounded by O(deg(v) + log n), while the algorithm of Götte et al. requires O(deg(v) + (log⁴ n)/(log log n)) messages per node.

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Yi-Jun Chang, Yanyu Chen, and Gopinath Mishra. Overlay Network Construction: Improved Overall and Node-Wise Message Complexity. 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. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chang_et_al:LIPIcs.FSTTCS.2025.21,
  author =	{Chang, Yi-Jun and Chen, Yanyu and Mishra, Gopinath},
  title =	{{Overlay Network Construction: Improved Overall and Node-Wise Message Complexity}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{21:1--21:21},
  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.2025.21},
  URN =		{urn:nbn:de:0030-drops-251025},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.21},
  annote =	{Keywords: Distributed algorithms, Overlay networks, Expander graphs}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.34

On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem

Authors: Umang Bhaskar, Katharina Eickhoff, Lennart Kauther, Jannik Matuschke, Britta Peis, and Laura Vargas Koch

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

Abstract

In train routing, the headway is the minimum distance that must be maintained between successive trains for safety and robustness. We introduce a model for train routing that requires a fixed headway to be maintained between trains, and study the problem of minimizing the makespan, i.e., the arrival time of the last train, in a single-source single-sink network. For this problem, we first show that there exists an optimal solution where trains move in convoys - that is, the optimal paths for any two trains are either the same or are arc-disjoint. Via this insight, we are able to reduce the approximability of our train routing problem to that of the min-max disjoint paths problem, which asks for a collection of disjoint paths where the maximum length of any path in the collection is as small as possible. While min-max disjoint paths inherits a strong inapproximability result on directed acyclic graphs from the multi-level bottleneck assignment problem, we show that a natural greedy composition approach yields a logarithmic approximation in the number of disjoint paths for series-parallel graphs. We also present an alternative analysis of this approach that yields a guarantee depending on how often the decomposition tree of the series-parallel graph alternates between series and parallel compositions on any root-leaf path.

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Umang Bhaskar, Katharina Eickhoff, Lennart Kauther, Jannik Matuschke, Britta Peis, and Laura Vargas Koch. On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhaskar_et_al:LIPIcs.ESA.2025.34,
  author =	{Bhaskar, Umang and Eickhoff, Katharina and Kauther, Lennart and Matuschke, Jannik and Peis, Britta and Vargas Koch, Laura},
  title =	{{On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{34:1--34:15},
  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.2025.34},
  URN =		{urn:nbn:de:0030-drops-245029},
  doi =		{10.4230/LIPIcs.ESA.2025.34},
  annote =	{Keywords: Train Routing, Scheduling, Approximation Algorithms, Flows over Time, Min-Max Disjoint Paths}
}

@InProceedings{bhaskar_et_al:LIPIcs.ESA.2025.34,
  author =	{Bhaskar, Umang and Eickhoff, Katharina and Kauther, Lennart and Matuschke, Jannik and Peis, Britta and Vargas Koch, Laura},
  title =	{{On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{34:1--34:15},
  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.2025.34},
  URN =		{urn:nbn:de:0030-drops-245029},
  doi =		{10.4230/LIPIcs.ESA.2025.34},
  annote =	{Keywords: Train Routing, Scheduling, Approximation Algorithms, Flows over Time, Min-Max Disjoint Paths}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.24

Fréchet Distance in Unweighted Planar Graphs

Authors: Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf

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

Abstract

The Fréchet distance is a distance measure between trajectories in ℝ^d or walks in a graph G. Given constant-time shortest path queries, the Discrete Fréchet distance D_G(P, Q) between two walks P and Q can be computed in O(|P|⋅|Q|) time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a 1.01-approximation of D_G(P, Q) unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fréchet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist (1+ε)-approximations, both in weighted graphs and in ℝ^d, that take near-linear time for c-packed or κ-straight walks in the graph. In ℝ^d there also exists a near-linear time algorithm to compute the Fréchet distance whenever all input edges are long compared to the distance. We consider computing the Fréchet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic 1.25-approximations of the discrete Fréchet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an Õ((|P|+|Q|)^{1.5})-time algorithm to compute D_G(P, Q). Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate (1+ε)-approximations of their discrete Fréchet distance in subquadratic time.

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Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf. Fréchet Distance in Unweighted Planar Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2025.24,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Rotenberg, Eva and Wulf, Lasse},
  title =	{{Fr\'{e}chet Distance in Unweighted Planar Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{24:1--24:16},
  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.2025.24},
  URN =		{urn:nbn:de:0030-drops-244924},
  doi =		{10.4230/LIPIcs.ESA.2025.24},
  annote =	{Keywords: Fr\'{e}chet distance, planar graphs, lower bounds, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.78

Testing Depth First Search Numbering

Authors: Artur Czumaj, Christian Sohler, and Stefan Walzer

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

Abstract

Property Testing is a formal framework to study the computational power and complexity of sampling from combinatorial objects. A central goal in standard graph property testing is to understand which graph properties are testable with sublinear query complexity. Here, a graph property P is testable with a sublinear query complexity if there is an algorithm that makes a sublinear number of queries to the input graph and accepts with probability at least 2/3, if the graph has property P, and rejects with probability at least 2/3 if it is ε-far from every graph that has property P. In this paper, we introduce a new variant of the bounded degree graph model. In this variant, in addition to the standard representation of a bounded degree graph, we assume that every vertex v has a unique label num(v) from {1, … , |V|}, and in addition to the standard queries in the bounded degree graph model, we also allow a property testing algorithm to query for the label of a vertex (but not for a vertex with a given label). Our new model is motivated by certain graph processes such as a DFS traversal, which assign consecutive numbers (labels) to the vertices of the graph. We want to study which of these numberings can be tested in sublinear time. As a first step in understanding such a model, we develop a property testing algorithm for discovery times of a DFS traversal with query complexity O(n^{1/3}/ε) and for constant ε > 0 we give a matching lower bound.

Cite as

Artur Czumaj, Christian Sohler, and Stefan Walzer. Testing Depth First Search Numbering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{czumaj_et_al:LIPIcs.ESA.2025.78,
  author =	{Czumaj, Artur and Sohler, Christian and Walzer, Stefan},
  title =	{{Testing Depth First Search Numbering}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{78:1--78:16},
  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.2025.78},
  URN =		{urn:nbn:de:0030-drops-245466},
  doi =		{10.4230/LIPIcs.ESA.2025.78},
  annote =	{Keywords: Randomized Algorithms, Graph Algorithms, Property Testing}
}

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}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.60

Fast and Memory-Efficient BWT Construction of Repetitive Texts Using Lyndon Grammars

Authors: Jannik Olbrich

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

Abstract

The Burrows-Wheeler Transform (BWT) serves as the basis for many important sequence indexes. On very large datasets (e.g. genomic databases), classical BWT construction algorithms are often infeasible because they usually need to have the entire dataset in main memory. Fortunately, such large datasets are often highly repetitive. It can thus be beneficial to compute the BWT from a compressed representation. We propose an algorithm for computing the BWT via the Lyndon straight-line program, a grammar based on the standard factorization of Lyndon words. Our algorithm can also be used to compute the extended BWT (eBWT) of a multiset of sequences. We empirically evaluate our implementation and find that we can compute the BWT and eBWT of very large datasets faster and/or with less memory than competing methods.

Cite as

Jannik Olbrich. Fast and Memory-Efficient BWT Construction of Repetitive Texts Using Lyndon Grammars. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 60:1-60:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{olbrich:LIPIcs.ESA.2025.60,
  author =	{Olbrich, Jannik},
  title =	{{Fast and Memory-Efficient BWT Construction of Repetitive Texts Using Lyndon Grammars}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{60:1--60:19},
  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.2025.60},
  URN =		{urn:nbn:de:0030-drops-245286},
  doi =		{10.4230/LIPIcs.ESA.2025.60},
  annote =	{Keywords: Burrows-Wheeler Transform, Grammar compression}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.63

Connected k-Median with Disjoint and Non-Disjoint Clusters

Authors: Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, and Melanie Schmidt

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

Abstract

The connected k-median problem is a constrained clustering problem that combines distance-based k-clustering with connectivity information. The problem allows to input a metric space and an unweighted undirected connectivity graph that is completely unrelated to the metric space. The goal is to compute k centers and corresponding clusters such that each cluster forms a connected subgraph of G, and such that the k-median cost is minimized. The problem has applications in very different fields like geodesy (particularly districting), social network analysis (especially community detection), or bioinformatics. We study a version with overlapping clusters where points can be part of multiple clusters which is natural for the use case of community detection. This problem variant is Ω(log n)-hard to approximate, and our main result is an 𝒪(k² log n)-approximation algorithm for the problem. We complement it with an Ω(n^{1-ε})-hardness result for the case of disjoint clusters without overlap with general connectivity graphs, as well as an exact algorithm in this setting if the connectivity graph is a tree.

Cite as

Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, and Melanie Schmidt. Connected k-Median with Disjoint and Non-Disjoint Clusters. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eube_et_al:LIPIcs.ESA.2025.63,
  author =	{Eube, Jan and Luo, Kelin and Reineccius, Dorian and R\"{o}glin, Heiko and Schmidt, Melanie},
  title =	{{Connected k-Median with Disjoint and Non-Disjoint Clusters}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{63:1--63:14},
  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.2025.63},
  URN =		{urn:nbn:de:0030-drops-245317},
  doi =		{10.4230/LIPIcs.ESA.2025.63},
  annote =	{Keywords: Clustering, Connectivity constraints, Approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.58

Streaming Diameter of High-Dimensional Points

Authors: Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý

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

Abstract

We improve the space bound for streaming approximation of Diameter but also of Farthest Neighbor queries, Minimum Enclosing Ball and its Coreset, in high-dimensional Euclidean spaces. In particular, our deterministic streaming algorithms store 𝒪(ε^{-2}log(1/(ε))) points. This improves by a factor of ε^{-1} the previous space bound of Agarwal and Sharathkumar (SODA 2010), while retaining the state-of-the-art approximation guarantees, such as √2+ε for Diameter or Farthest Neighbor queries, and also offering a simpler and more complete argument. Moreover, we show that storing Ω(ε^{-1}) points is necessary for a streaming (√2+ε)-approximation of Farthest Pair and Farthest Neighbor queries.

Cite as

Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý. Streaming Diameter of High-Dimensional Points. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 58:1-58:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{halldorsson_et_al:LIPIcs.ESA.2025.58,
  author =	{Halld\'{o}rsson, Magn\'{u}s M. and Matsakis, Nicolaos and Vesel\'{y}, Pavel},
  title =	{{Streaming Diameter of High-Dimensional Points}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{58:1--58:10},
  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.2025.58},
  URN =		{urn:nbn:de:0030-drops-245263},
  doi =		{10.4230/LIPIcs.ESA.2025.58},
  annote =	{Keywords: streaming algorithm, farthest pair, diameter, minimum enclosing ball, coreset}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.39

Going Beyond Surfaces in Diameter Approximation

Authors: Michał Włodarczyk

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

Abstract

Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known (1+ε)-approximation algorithms with running time poly(1/ε, log n)⋅ n. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving (1+ε)-approximation algorithms with the following running times: - 𝒪_h((1/ε)^𝒪(h) ⋅ nlog² n)-time algorithm for graphs excluding an apex graph of size h as a minor, - 𝒪_d((1/ε)^𝒪(d) ⋅ nlog² n)-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+ε)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time 𝒪_h((1/ε)⁸⋅ nlog nlog W) and query time 𝒪_h((1/ε)²⋅log n log W), where W is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.

Cite as

Michał Włodarczyk. Going Beyond Surfaces in Diameter Approximation. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wlodarczyk:LIPIcs.ESA.2025.39,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Going Beyond Surfaces in Diameter Approximation}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{39:1--39:19},
  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.2025.39},
  URN =		{urn:nbn:de:0030-drops-245076},
  doi =		{10.4230/LIPIcs.ESA.2025.39},
  annote =	{Keywords: diameter, approximation, distance oracles, graph minors, treewidth}
}

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