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LIPIcs, Volume 370
20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Scandinavian Symposium on Algorithm Theory (SWAT)
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- published at: 2026-06-08
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-421-5
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Complete Volume
LIPIcs, Volume 370, SWAT 2026, Complete Volume
Abstract
LIPIcs, Volume 370, SWAT 2026, Complete Volume
Cite as
20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 1-700, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{fraigniaud:LIPIcs.SWAT.2026,
title = {{LIPIcs, Volume 370, SWAT 2026, Complete Volume}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {1--700},
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},
URN = {urn:nbn:de:0030-drops-262700},
doi = {10.4230/LIPIcs.SWAT.2026},
annote = {Keywords: LIPIcs, Volume 370, SWAT 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fraigniaud:LIPIcs.SWAT.2026.0,
author = {Fraigniaud, Pierre},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {0:i--0:xvi},
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.2026.0},
URN = {urn:nbn:de:0030-drops-262697},
doi = {10.4230/LIPIcs.SWAT.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Classifiers in High Dimensional Hilbert Metrics
Abstract
Classifying points in high-dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address the problem of classifying points in the d-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present an efficient LP-based algorithm in the metric for the large-margin SVM problem. Our algorithm runs in time polynomial in the number of points, the number of bounding facets, and the dimension. This is a significant improvement over previous work, which either provides no theoretical guarantees on runtime or suffers from exponential runtime. We also consider the closely related Funk metric. Finally, we present efficient algorithms for the soft-margin SVM problem and nearest-neighbor-based classification in the Hilbert metric.
Cite as
Aditya Acharya, Auguste H. Gezalyan, and David M. Mount. Classifiers in High Dimensional Hilbert Metrics. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{acharya_et_al:LIPIcs.SWAT.2026.1,
author = {Acharya, Aditya and Gezalyan, Auguste H. and Mount, David M.},
title = {{Classifiers in High Dimensional Hilbert Metrics}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {1:1--1:16},
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.2026.1},
URN = {urn:nbn:de:0030-drops-260376},
doi = {10.4230/LIPIcs.SWAT.2026.1},
annote = {Keywords: Support vector machines, Hilbert geometry, classification, machine learning}
}
Document
On the Fragile Complexity of Geometric Algorithms
Abstract
Surprisingly, the question of bounding the maximum number of operations undergone by each individual element in an algorithm - known as the fragile complexity of the algorithm - has not received much attention. In a foundational paper, Afshani et al. (2019) developed the concept of fragility and explored classic problems such as sorting and selection from this perspective. Motivated by a suggestion for future research by Afshani et al., we initiate a study of fragile complexity in computational geometry. We obtain bounds on several time-honored questions in 2D such as computing the maxima, closest pair, convex hull, triangulation, and approximate Euclidean Minimum Spanning Tree (apx-EMST). Our algorithms for the maxima, convex hull, and triangulation problems are competitive with the classical algorithms in terms of worst-case runtime and guarantee polylogarithmic fragility. We present an O(nlog²n) time algorithm that returns a 1.0125-apx-EMST and achieves O(log² n) fragility, thus matching the best known performance up to polylogarithmic factors.
Cite as
Boris Aronov, Mayank Goswami, John Iacono, and Indu Ramesh. On the Fragile Complexity of Geometric Algorithms. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aronov_et_al:LIPIcs.SWAT.2026.2,
author = {Aronov, Boris and Goswami, Mayank and Iacono, John and Ramesh, Indu},
title = {{On the Fragile Complexity of Geometric Algorithms}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {2:1--2:18},
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.2026.2},
URN = {urn:nbn:de:0030-drops-260386},
doi = {10.4230/LIPIcs.SWAT.2026.2},
annote = {Keywords: Fragile complexity, convex hull, maxima, closest pair, algorithmic complexity}
}
Document
Reachability with Restricted Reactions in Inhibitory Chemical Reaction Networks
Abstract
Chemical Reaction Networks (CRNs) are a well-established model of distributed computing characterized by quantities of molecular species that can transform or change through applications of reactions. A fundamental problem in CRNs is the reachability problem, which asks if an initial configuration of species can transition to a target configuration through an applicable sequence of reactions. It is well-known that the reachability problem in general CRNs was recently proven to be Ackermann-complete. However, if the CRN’s reactions are restricted in both power, such as only deleting species (deletion-only rules) or consuming and producing an equal number of species (volume-preserving rules), and size (unimolecular or bimolecular rules), then reachability falls below Ackermann-completeness, and is even solvable in polynomial time for deletion-only systems.
In this paper, we investigate reachability under this set of restricted unimolecular and bimolecular reactions, but in the Priority-Inhibitory CRN and Inhibitory CRN models. These models extend a traditional CRN by allowing some reactions to be inhibited from firing in a configuration if certain species are present; the exact inhibition behavior varies between the models. We first show that reachability with Priority iCRNs mostly remains in P for deletion-only systems, but becomes NP-complete for one case. We then show that reachability with deletion-only reactions for iCRNs is mostly NP-complete, and PSPACE-complete even for (1,1)-size (general) reactions. We also provide FPT algorithms for solving most of the reachability problems for the iCRN model. Finally, we show reachability for CRNs with states is already NP-hard for the simplest deletion-only systems, and is PSPACE-complete even for (general) (1,1)-size reactions.
Cite as
Divya Bajaj, Bin Fu, Ryan Knobel, Austin Luchsinger, Aiden Massie, Pablo Santos, Ramiro Santos, Robert Schweller, Evan Tomai, and Tim Wylie. Reachability with Restricted Reactions in Inhibitory Chemical Reaction Networks. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bajaj_et_al:LIPIcs.SWAT.2026.3,
author = {Bajaj, Divya and Fu, Bin and Knobel, Ryan and Luchsinger, Austin and Massie, Aiden and Santos, Pablo and Santos, Ramiro and Schweller, Robert and Tomai, Evan and Wylie, Tim},
title = {{Reachability with Restricted Reactions in Inhibitory Chemical Reaction Networks}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {3:1--3:18},
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.2026.3},
URN = {urn:nbn:de:0030-drops-260399},
doi = {10.4230/LIPIcs.SWAT.2026.3},
annote = {Keywords: Chemical Reaction Networks, Vector Addition Systems, Petri-nets, Reachability, Inhibitors, Void Reactions}
}
Document
Improved Bounds for Online TSP on the Half-Line
Abstract
In the open online traveling salesperson problem, requests appear over time at different positions of a metric space. A single agent traveling at unit speed must serve all requests, by visiting their positions, with the objective of minimizing the completion time. We propose online algorithms for this problem for the case that the metric space is the half-line. First, we present a 2-competitive algorithm in which the server always either moves at unit speed or remains stationary, which improves on the previously best-known ratio of 2.04. We further observe that algorithms must sometimes move slower than unit speed to achieve competitive ratios below 2. We introduce a second algorithm that beats the barrier of 2 by carefully modulating its speed, and prove a tight bound of 1.75 on its competitive ratio. Finally, we slightly strengthen a known lower bound on the best-possible competitive ratio on the half-line from 1.627 to 1.646.
Cite as
Júlia Baligács, Yann Disser, and Linda Thelen. Improved Bounds for Online TSP on the Half-Line. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{baligacs_et_al:LIPIcs.SWAT.2026.4,
author = {Balig\'{a}cs, J\'{u}lia and Disser, Yann and Thelen, Linda},
title = {{Improved Bounds for Online TSP on the Half-Line}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {4:1--4:17},
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.2026.4},
URN = {urn:nbn:de:0030-drops-260402},
doi = {10.4230/LIPIcs.SWAT.2026.4},
annote = {Keywords: online algorithms, competitive analysis, server problems, online TSP}
}
Document
Arranging Pairwise Disjoint Shapes to Partition Point Sets
Abstract
We consider the fine-grained complexity of covering a set of n points 𝒫 in the Euclidean plane using a fixed set of geometric objects corresponding to rigid-body translations and, where permitted, rotations of a specified shape Υ. Under the Exponential Time Hypothesis (ETH), and both with and without a pairwise disjointness constraint, we establish that no 2^{o(√n)}-time algorithm can exist for this problem in the following cases: (case 1) translatable unit disks; (case 2) translatable fixed-area axis-aligned squares; or (case 3) translatable and rotatable fixed-area equilateral triangles. Furthermore, by way of establishing the #P-completeness under parsimonious reductions of positive 1-in-3-SAT with a cubic planar 3-connected clause-variable incidence graph - pertinent to hardness reductions for counting tilings {(Moore & Robson; Discrete Comput. Geom. 26(4); 2001), (Pak & Yang; J. Comb. Theory. Ser. A 120(7); 2013)} - we establish in each case that there exists a quadratic time reduction from #SAT to counting the possible coverage-induced partitions of 𝒫. Finally, we consider constraints on the density of the points in 𝒫 that make our coverage problems tractable. In particular, letting Υ be any (not necessarily connected) subregion of a radius 1/2 disk characterized by a semi-algebraic function, and letting 𝒫 be a set of n points, we consider the density requirements that: (constraint 1) every 3 points have a minimum bounding disk of radius greater than 1; or (constraint 2) any 5 points have a minimum bounding disk of radius at least 2. Here, when Υ is part of the input, under both (constraint 1) and (constraint 2), and with and without a pairwise disjointness requirement, we show that finding a minimum cardinality set of translatable and/or rotatable instances of Υ covering all points in 𝒫 is fixed-parameter tractable in the size of the semi-algebraic description of Υ.
Cite as
Robert D. Barish and Tetsuo Shibuya. Arranging Pairwise Disjoint Shapes to Partition Point Sets. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{barish_et_al:LIPIcs.SWAT.2026.5,
author = {Barish, Robert D. and Shibuya, Tetsuo},
title = {{Arranging Pairwise Disjoint Shapes to Partition Point Sets}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {5:1--5:18},
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.2026.5},
URN = {urn:nbn:de:0030-drops-260413},
doi = {10.4230/LIPIcs.SWAT.2026.5},
annote = {Keywords: geometric covering, geometric packing, clustering, ply, bounded ply, planar geometry, frequency assignment problem, Exponential Time Hypothesis (ETH), Counting Exponential Time Hypothesis (#ETH)}
}
Document
How Many Slopes Does Polynomial Area Cost?
Abstract
In this work, we study the interplay between the number of slopes, the number of bends per edge, and the area requirements for planar drawings of bounded-degree graphs. Our motivation stems from the fact that, while numerous algorithms produce planar drawings with few slopes for graphs of relatively small degree in polynomial area, existing approaches for higher-degree graphs often require super-polynomial area. We address this gap in the literature by presenting new constructions that yield polynomial-area drawings with few bends per edge while slightly increasing the required number of slopes, thereby providing the first systematic study of slopes, bends and area trade-offs.
Cite as
Michael A. Bekos, Eleni Katsanou, Philipp Kindermann, and Maria Eleni Pavlidi. How Many Slopes Does Polynomial Area Cost?. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bekos_et_al:LIPIcs.SWAT.2026.6,
author = {Bekos, Michael A. and Katsanou, Eleni and Kindermann, Philipp and Pavlidi, Maria Eleni},
title = {{How Many Slopes Does Polynomial Area Cost?}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {6:1--6:18},
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.2026.6},
URN = {urn:nbn:de:0030-drops-260424},
doi = {10.4230/LIPIcs.SWAT.2026.6},
annote = {Keywords: k-bend planar drawings, planar slope number, area requirements}
}
Document
On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons
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
Submodular Max-Min Allocation Under Identical Valuations
Abstract
In the problem of Submodular Max-Min Allocation, we are given a set of items, a set of players, and monotone submodular valuation functions that represent the satisfaction of a player with a certain subset of items. The goal is to find an allocation of the items to the players that maximizes the lowest satisfaction among all players.
We study this problem in the special case where all players have the same valuation function. We devise a greedy algorithm which gives a 0.4-approximation, improving the previously best factor of 10/27 ≈ 0.37 by Uziahu and Feige.
Furthermore, we study the integrality gap of the configuration LP when players have identical valuations. By constructing a variable assignment to the dual from a primal integral solution, we give the first constant upper bound on the integrality gap for submodular valuations. Generalizing the result to the case where players' allocations must be independent in k given matroids, we derive a 𝒪(k)-estimation algorithm for max-min allocation subject to k matroid constraints under identical valuations.
Cite as
Kimon Boehmer. Submodular Max-Min Allocation Under Identical Valuations. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{boehmer:LIPIcs.SWAT.2026.8,
author = {Boehmer, Kimon},
title = {{Submodular Max-Min Allocation Under Identical Valuations}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {8:1--8:17},
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.2026.8},
URN = {urn:nbn:de:0030-drops-260446},
doi = {10.4230/LIPIcs.SWAT.2026.8},
annote = {Keywords: Submodularity, Approximation algorithms, Allocation, Configuration LP}
}
Document
QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes
Abstract
We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (MWIS) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed s and t, we show a QPTAS for MWIS in graphs that exclude sC_t as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs.
This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph H, graphs that exclude H as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs H.
Cite as
Édouard Bonnet, Jadwiga Czyżewska, Tomáš Masařík, Marcin Pilipczuk, and Paweł Rzążewski. QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonnet_et_al:LIPIcs.SWAT.2026.9,
author = {Bonnet, \'{E}douard and Czy\.{z}ewska, Jadwiga and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {9:1--9:14},
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.2026.9},
URN = {urn:nbn:de:0030-drops-260454},
doi = {10.4230/LIPIcs.SWAT.2026.9},
annote = {Keywords: independent set, long holes, QPTAS, induced subgraphs}
}
Document
Robotic Arm Rotation: Standing up Is Harder Than You Think
Abstract
We study motion-planning problems for planar robotic arms that rotate around fixed centers while avoiding collisions. In the SM-RAMP model, each unit-length arm may rotate at most once; the question is whether all arms can be rotated to the vertical position. We resolve an open problem of Bousquet et al. [Bousquet et al., 2026] by proving that SM-RAMP is NP-complete, even in the horizontal-to-vertical setting. Our hardness proof uses a structural analysis of rotation-propagation chains and introduces a combinatorial abstraction of independent interest, the Lighthouse Propagation problem, which we show is itself NP-complete. We then consider the multi-move variant MM-RAMP, where each arm may rotate multiple times among a fixed set of allowed angles (or orientations). We prove that MM-RAMP is PSPACE-complete even when each arm has only a few allowed angles, in sharp contrast with the single-move case. Finally, we give two fixed-parameter tractable algorithms: for MAX-SM-RAMP parameterized by the number k of arms to be made vertical, and for 2A-MM-RAMP (restricted to horizontal and vertical) parameterized by the number 𝓁 of allowed rotations.
Cite as
Nicolas Bousquet, Frank Connor, Remy El Sabeh, Louis-Roy Langevin, Amer E. Mouawad, Naomi Nishimura, and Agnes Totschnig. Robotic Arm Rotation: Standing up Is Harder Than You Think. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bousquet_et_al:LIPIcs.SWAT.2026.10,
author = {Bousquet, Nicolas and Connor, Frank and El Sabeh, Remy and Langevin, Louis-Roy and Mouawad, Amer E. and Nishimura, Naomi and Totschnig, Agnes},
title = {{Robotic Arm Rotation: Standing up Is Harder Than You Think}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {10:1--10:17},
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.2026.10},
URN = {urn:nbn:de:0030-drops-260467},
doi = {10.4230/LIPIcs.SWAT.2026.10},
annote = {Keywords: search, optimization, robotics, robotic arms, parameterized complexity, computational geometry, combinatorial reconfiguration}
}
Document
Improved Approximation of Two Watchmen’s Routes in Simple Polygons
Abstract
We study the watchman route problem for a set of two watchmen for the objective of minimizing the length of the longest route (min-max) inside a simple polygon P having n vertices, which is known to be weakly NP-hard. First, we consider seeing a discrete set of m points in the interior of P. We show that even this problem is weakly NP-hard and present an approximation algorithm with approximation ratio 2+ε that runs in O(m⁵n) time, assuming that a starting point for each of the two routes is given. We generalize the algorithm to see all of the interior of P in O(n⁶) time with approximation ratio 2 + π/2 ≈ 3.571, improving on the previously known best algorithm that has an approximation ratio of ≈ 6.922 and runtime O(n²) [Bengt J. Nilsson and Eli Packer, 2024]. Finally, we describe how to extend this algorithm to the case where no starting points are given, this taking O(n⁸) time, yielding an approximation ratio of 3 + π/2 ≈ 4.571, improving on the previously known best approximation algorithm with ratio ≈ 5.969 also having runtime O(n⁸) [Bengt J. Nilsson and Eli Packer, 2024].
Cite as
Anna Brötzner, Bengt J. Nilsson, and Christiane Schmidt. Improved Approximation of Two Watchmen’s Routes in Simple Polygons. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brotzner_et_al:LIPIcs.SWAT.2026.11,
author = {Br\"{o}tzner, Anna and Nilsson, Bengt J. and Schmidt, Christiane},
title = {{Improved Approximation of Two Watchmen’s Routes in Simple Polygons}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {11:1--11:16},
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.2026.11},
URN = {urn:nbn:de:0030-drops-260472},
doi = {10.4230/LIPIcs.SWAT.2026.11},
annote = {Keywords: Art gallery problem, watchman route problem, multiple watchmen, path planning, polygons}
}
Document
Strategy Repair in Reachability Games via a Graph Quotientation
Abstract
Reachability Games over graphs (RGs) are a powerful modelling tool for synthesis and planning. The solutions to RGs are strategies that are used in program design. Sometimes, due to model deviation at execution time, specification updates, or simply a bug, the strategy provided as a solution to an RG no longer works. Strategy Repair aims to solve this problem by adjusting strategies with a minimum number of modifications. Such a minimisation requirement is motivated by the costs that one may incur when implementing the new strategy. To minimise implementation costs, one wants to reuse as much of the provided strategy as possible.
In the literature, Strategy Repair has been investigated from both theoretical and practical perspectives. First, it has been shown to be NP-complete. Second, two algorithmic approaches have been proposed to tackle the problem in practice, one provides an optimal solution, the other an approximated solution. Both approaches underutilise the graph-theoretical properties of games, which could significantly improve their performance and accuracy (in the case of approximation algorithms).
This paper provides a graph-theoretic characterisation of Strategy Repair that provably improves every algorithmic approach to solving the problem. It does so by introducing a new notion of quotient graph that allows us to identify and merge those vertices that are equivalent from the perspective of every solution to the problem. This way, solving Strategy Repair can be done in a reduced instance, which we call the quotient game. The approach not only reduces the problem’s input size, but also improves the effectiveness of MustFix, an optimisation condition previously introduced for the problem. Besides the theoretical characterisation, we test our approach empirically by running experiments to demonstrate improvements over the quotient graph approach.
Cite as
Tiziana Calamoneri, Pierre Gaillard, Giacomo Paesani, and Giuseppe Perelli. Strategy Repair in Reachability Games via a Graph Quotientation. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{calamoneri_et_al:LIPIcs.SWAT.2026.12,
author = {Calamoneri, Tiziana and Gaillard, Pierre and Paesani, Giacomo and Perelli, Giuseppe},
title = {{Strategy Repair in Reachability Games via a Graph Quotientation}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {12:1--12:16},
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.2026.12},
URN = {urn:nbn:de:0030-drops-260481},
doi = {10.4230/LIPIcs.SWAT.2026.12},
annote = {Keywords: Reachability Games, Strategy Repair, Strategic Reasoning, Automated Reasoning}
}
Document
Faster Algorithms for Shortest Unique or Absent Substrings
Abstract
We revisit two well-known algorithmic problems on strings: computing a shortest unique substring (SUS) and a shortest absent substring (SAS) in a string S of length n. Both problems admit folklore 𝒪(n)-time solutions using the suffix tree of S. However, for small alphabets, this complexity is not necessarily optimal in the word RAM model, where a string of length n over alphabet [0,σ) can be stored in 𝒪(n log σ/log n) space and read in 𝒪(n log σ/log n) time.
We present an 𝒪(n log σ/√{log n})-time algorithm for computing a SUS in S. This algorithm decomposes the problem according to the length and the period of the sought substring and uses several tools and techniques, such as synchronizing sets, the analysis of runs, and wavelet trees, to reduce the computation of a SUS to a simple geometric problem. Further, we adapt this algorithm and combine it with an efficient construction of de Bruijn sequences in order to obtain an 𝒪(n log σ/√{log n})-time algorithm for computing a SAS in S.
Cite as
Panagiotis Charalampopoulos, Manal Mohamed, Solon P. Pissis, Hilde Verbeek, and Wiktor Zuba. Faster Algorithms for Shortest Unique or Absent Substrings. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{charalampopoulos_et_al:LIPIcs.SWAT.2026.13,
author = {Charalampopoulos, Panagiotis and Mohamed, Manal and Pissis, Solon P. and Verbeek, Hilde and Zuba, Wiktor},
title = {{Faster Algorithms for Shortest Unique or Absent Substrings}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {13:1--13:19},
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.2026.13},
URN = {urn:nbn:de:0030-drops-260493},
doi = {10.4230/LIPIcs.SWAT.2026.13},
annote = {Keywords: string algorithms, unique substrings, absent substrings, absent words}
}
Document
Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds
Abstract
We study a variant of a polygon partition problem, introduced by Chung, Iwama, Liao, and Ahn [ISAAC'25]. Given orthogonal unit vectors 𝐮,𝐯 ∈ ℝ² and a polygon P with n vertices, we partition P into connected pieces by cuts parallel to 𝐯 such that each resulting subpolygon has width at most one in direction 𝐮. We consider the value version, which asks for the minimum number of strips, and the reporting version, which outputs a compact encoding of the cuts in an optimal strip partition.
We give efficient algorithms and lower bounds for both versions on three classes of polygons of increasing generality: convex, simple, and self-overlapping. For convex polygons, we solve the value version in O(log n) time and the reporting version in O(h log (1 + n/h)) time, where h is the width of P in direction 𝐮. We prove matching lower bounds in the decision-tree model, showing that the reporting algorithm is input-sensitive optimal with respect to h. For simple polygons, we present O(n log n)-time, O(n)-space algorithms for both versions and prove an Ω(n) lower bound. For self-overlapping polygons, we extend the approach for simple polygons to obtain O(n log n)-time, O(n)-space algorithms for both versions, and we prove a matching Ω(n log n) lower bound in the algebraic computation-tree model via a reduction from the δ-closeness problem.
Our approach relies on a lattice-theoretic formulation of the problem. We represent strip partitions as antichains of intervals in the Clarke-Cormack-Burkowski lattice, originally developed for minimal-interval semantics in information retrieval. Within this lattice framework, we design a dynamic programming algorithm that uses the lattice operations of meet and join. To the best of our knowledge, this is the first geometric application of the Clarke-Cormack-Burkowski lattice.
Cite as
Jaehoon Chung. Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chung:LIPIcs.SWAT.2026.14,
author = {Chung, Jaehoon},
title = {{Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {14:1--14:17},
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.2026.14},
URN = {urn:nbn:de:0030-drops-260506},
doi = {10.4230/LIPIcs.SWAT.2026.14},
annote = {Keywords: Polygon partitioning, Strip partition, Lattice, Self-overlapping curves}
}
Document
One Color Makes All the Difference in the Tractability of Partial Coloring in Semi-Streaming
Abstract
This paper investigates the semi-streaming complexity of k-partial coloring, a generalization of proper graph coloring. For k ≥ 1, a k-partial coloring requires that each vertex v in an n-node graph is assigned a color such that at least min{k, deg(v)} of its neighbors are assigned colors different from its own. This framework naturally extends classical coloring problems: specifically, k-partial (k+1)-coloring and k-partial k-coloring generalize (Δ+1)-proper coloring and Δ-proper coloring, respectively.
Prior works of Assadi, Chen, and Khanna [SODA 2019] and Assadi, Kumar, and Mittal [TheoretiCS 2023] show that both (Δ+1)-proper coloring and Δ-proper coloring admit one-pass randomized semi-streaming algorithms. We explore whether these efficiency gains extend to their partial coloring generalizations and reveal a sharp computational threshold: while k-partial (k+1)-coloring admits a one-pass randomized semi-streaming algorithm, the k-partial k-coloring remains semi-streaming intractable, effectively demonstrating a "dichotomy of one color" in the streaming model.
Cite as
Avinandan Das. One Color Makes All the Difference in the Tractability of Partial Coloring in Semi-Streaming. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{das:LIPIcs.SWAT.2026.15,
author = {Das, Avinandan},
title = {{One Color Makes All the Difference in the Tractability of Partial Coloring in Semi-Streaming}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {15:1--15:16},
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.2026.15},
URN = {urn:nbn:de:0030-drops-260515},
doi = {10.4230/LIPIcs.SWAT.2026.15},
annote = {Keywords: Graph Coloring, Semi-streaming algorithms, Lower bounds}
}
Document
Online Hitting Set for Axis-Aligned Squares
Abstract
Given a set P of n points in the plane and a sequence of axis-aligned squares that arrive in an online fashion, the online hitting set problem consists of maintaining, by adding new points from P if necessary, a hitting set H ⊆ P, which contains at least one point in every input square that has already arrived. We present an O(log n)-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first O(log n)-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., unbounded scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with k ≥ 3 vertices in the plane and provide an O(k²log n)-competitive algorithm.
Cite as
Minati De, Satyam Singh, and Csaba D. Tóth. Online Hitting Set for Axis-Aligned Squares. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{de_et_al:LIPIcs.SWAT.2026.16,
author = {De, Minati and Singh, Satyam and T\'{o}th, Csaba D.},
title = {{Online Hitting Set for Axis-Aligned Squares}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {16:1--16:18},
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.2026.16},
URN = {urn:nbn:de:0030-drops-260528},
doi = {10.4230/LIPIcs.SWAT.2026.16},
annote = {Keywords: axis-aligned squares, hitting set, homothets of a polygon, online algorithm}
}
Document
Incremental Strongly Connected Components with Predictions
Abstract
Algorithms with predictions is a growing area that aims to leverage machine-learned predictions to design faster beyond-worst-case algorithms. In this paper, we use this framework to design a learned data structure for the incremental strongly connected components (SCC) problem. In this problem, the n vertices of a graph are known a priori and the m directed edges arrive over time. The goal is to efficiently maintain the strongly connected components of the graph after each insert. Our algorithm receives a possibly erroneous prediction of the edge sequence and uses it to precompute partial solutions to support fast inserts. We show that our algorithm achieves nearly optimal bounds with good predictions and its performance smoothly degrades with the prediction error. We also implement our data structure and perform experiments on real datasets. Our empirical results show that the theory is predictive of practical runtime improvements.
Cite as
Ronald Deng, Samuel McCauley, Aidin Niaparast, Helia Niaparast, Bennett Ptak, Shirel Quintanilla, Shikha Singh, and Nathan Vosburg. Incremental Strongly Connected Components with Predictions. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deng_et_al:LIPIcs.SWAT.2026.17,
author = {Deng, Ronald and McCauley, Samuel and Niaparast, Aidin and Niaparast, Helia and Ptak, Bennett and Quintanilla, Shirel and Singh, Shikha and Vosburg, Nathan},
title = {{Incremental Strongly Connected Components with Predictions}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {17:1--17:16},
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.2026.17},
URN = {urn:nbn:de:0030-drops-260530},
doi = {10.4230/LIPIcs.SWAT.2026.17},
annote = {Keywords: algorithms with predictions, learning augmented algorithms, incremental graph algorithms, strongly connected components, data structures}
}
Document
On Fréchet Traveling Salesmen Problems
Abstract
The Fréchet distance is a well-studied distance measure between two curves. In this work, we demonstrate that the merit of Fréchet distance extends beyond evaluating similarity, and introduce a new setting in which it proves useful. Consider a situation where two agents are required to visit a given set of sites, while staying close to each other throughout their traversal. In this paper, we study problems where the goal is to construct two curves whose vertices are from a given set of points, under the constraint that the Fréchet distance between the curves is kept as small as possible. This problem can be viewed as a variant of the Traveling Salesman Problem (TSP), and thus may be of interest in routing, network planning and more. We present a near-linear algorithm for this problem under the discrete Fréchet distance, and explore several variants of the problem, including minimizing the lengths of the curves and balancing the number of sites assigned to each agent. Lastly, we prove that the problem is NP-hard under the continuous Fréchet Distance.
Cite as
Omrit Filtser, Tzalik Maimon, and Michal Moiseev. On Fréchet Traveling Salesmen Problems. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{filtser_et_al:LIPIcs.SWAT.2026.18,
author = {Filtser, Omrit and Maimon, Tzalik and Moiseev, Michal},
title = {{On Fr\'{e}chet Traveling Salesmen Problems}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {18:1--18:17},
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.2026.18},
URN = {urn:nbn:de:0030-drops-260545},
doi = {10.4230/LIPIcs.SWAT.2026.18},
annote = {Keywords: Fr\'{e}chet distance, traveling salesman problem}
}
Document
Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means
Abstract
We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute.
Dickerson, Esmaeili, Morgenstern, and Pena [John P. Dickerson et al., 2023] denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen [Matthew Jones et al., 2020]. For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.
Cite as
Nicole Funk, Annika Hennes, Johanna Hillebrand, and Sarah Sturm. Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{funk_et_al:LIPIcs.SWAT.2026.19,
author = {Funk, Nicole and Hennes, Annika and Hillebrand, Johanna and Sturm, Sarah},
title = {{Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {19:1--19:19},
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.2026.19},
URN = {urn:nbn:de:0030-drops-260551},
doi = {10.4230/LIPIcs.SWAT.2026.19},
annote = {Keywords: Clustering, Fairness, Approximation Algorithms, k-center, k-median, k-means}
}
Document
Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs
Abstract
A disk graph is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time O(n^{7/3+ o(1)}), where n denotes the number of disks [Jared Espenant et al., 2023; J. Mark Keil and Debajyoti Mondal, 2025]. Moreover, for the case of disk graphs with t distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time O^*(n^{2t}) [J. Mark Keil and Debajyoti Mondal, 2025]. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: [(i)]
1) for unit disk graphs, we give an algorithm that, with constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(n/ε²); and
2) for disk graphs with t distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(f(t)⋅ (1/ε)^{O(t)} ⋅ n), for some (exponential) function f(t).
Cite as
Jie Gao, Paweł Gawrychowski, Panos Giannopoulos, Wolfgang Mulzer, Satyam Singh, Frank Staals, and Meirav Zehavi. Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gao_et_al:LIPIcs.SWAT.2026.20,
author = {Gao, Jie and Gawrychowski, Pawe{\l} and Giannopoulos, Panos and Mulzer, Wolfgang and Singh, Satyam and Staals, Frank and Zehavi, Meirav},
title = {{Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {20:1--20:17},
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.2026.20},
URN = {urn:nbn:de:0030-drops-260563},
doi = {10.4230/LIPIcs.SWAT.2026.20},
annote = {Keywords: Maximum Clique, Disk Graphs, Unit Disk Graphs, FPT Approximation}
}
Document
Dynamic MIS Revisited: Incremental, Fault Tolerant and Fully Dynamic
Abstract
Given a dynamic graph G, we aim to maintain a maximal independent set (MIS). This problem admits a deterministic solution in O(m^{2/3}) time [SOSA21], while randomized algorithms for oblivious adversary take O(polylog n) [FOCS19] time. Recently, Bernstein et al. [SODA26] proved a conditional lower bound of Ω(n^{1-o(1)}) amortized update time even for incremental MIS with adaptive adversary. In this paper, we establish similar deterministic bounds through the lens of incremental and fault tolerant algorithms for MIS, and show the following:
1) Incremental: MIS can be maintained under edge insertions in O(√m) amortized update time.
2) Fault Tolerant: Using O(m) preprocessing time, the MIS can be reported after k edge deletions in O(k+ ̅{n}²) time, where ̅{n} is the number of vertices on which deleted edges are incident.
3) Fully Dynamic: MIS can be maintained under fully dynamic edge updates (insertions and deletions) in O(m^{2/3}) amortized update time.
Our incremental result is thus polynomially optimal even on allowing randomization with an adaptive adversary. Our fully dynamic result matches the state-of-the-art [SOSA21], and uses our incremental and fault tolerant algorithms as a black box. Although our fully dynamic algorithm does not improve the existing bounds, it provides additional insights into the harder edge-insertion case. We believe these insights may potentially be useful for improving the bounds for fully dynamic MIS in the future to match the incremental MIS bounds.
Cite as
Manoj Gupta, Shahbaz Khan, and Madhu Surendra. Dynamic MIS Revisited: Incremental, Fault Tolerant and Fully Dynamic. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 21:1-21:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gupta_et_al:LIPIcs.SWAT.2026.21,
author = {Gupta, Manoj and Khan, Shahbaz and Surendra, Madhu},
title = {{Dynamic MIS Revisited: Incremental, Fault Tolerant and Fully Dynamic}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {21:1--21:11},
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.2026.21},
URN = {urn:nbn:de:0030-drops-260577},
doi = {10.4230/LIPIcs.SWAT.2026.21},
annote = {Keywords: Maximal Independent Set, MIS, Incremental, Fault Tolerant, Fully dynamic}
}
Document
Search-Space Reduction for Boolean MinCSPs via Essential Constraints
Abstract
For a fixed set ℱ of Boolean constraint types, a MinCSP(ℱ)-instance consists of a formula F that applies m constraints from ℱ to a set of n Boolean variables. The goal is to remove a minimum subset of constraint applications from F to make the remaining formula satisfiable. Previous work characterized how the choice of ℱ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula F as c-essential if it is contained in all c-approximate solutions to F. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets ℱ for which c_ℱ-essential constraint applications can be detected efficiently for some c_{ℱ} ∈ 𝒪(1), from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set ℱ of bijunctive constraints, there is a polynomial-time algorithm that detects 𝒪(1)-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP(ℱ)-problem is intractable under the Unique Games Conjecture.
Cite as
Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction for Boolean MinCSPs via Essential Constraints. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jansen_et_al:LIPIcs.SWAT.2026.22,
author = {Jansen, Bart M. P. and Verhaegh, Ruben F. A.},
title = {{Search-Space Reduction for Boolean MinCSPs via Essential Constraints}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {22:1--22:17},
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.2026.22},
URN = {urn:nbn:de:0030-drops-260586},
doi = {10.4230/LIPIcs.SWAT.2026.22},
annote = {Keywords: fixed-parameter tractability, constraint satisfaction problems}
}
Document
Indexing Range Maximum-Sum Segment Queries with Offsets
Abstract
Given an array of n real numbers, the maximum segment sum (MSS) problem is to find a contiguous subarray that has the largest sum. While the MSS problem can be solved optimally with Kadane’s algorithm in O(n) time, the study of its indexing version spawned new extensions such as (a) retrieving the MSS after subtracting a query offset parameter for all array entries or (b) retrieving the MSS for arbitrary query ranges. We here study the combination of both problems (a) and (b), which requires retrieving the MSS for arbitrary query ranges after subtracting a query offset parameter for all array entries. For that, we present an index whose query time is only slower than the best known for (a) by a factor of O(log n). In detail, our index uses O(n log n) space, supports queries in O(log² n) time, and can be constructed in O(n log³ n) time. As side results, we study our combined problem in the context of run-length compressed input, and also deduce a solution for (a) that works in run-length compressed space and time. Finally we give supportive lower bounds for our query problem, showing that there is only a polylogarithmic gap of improvement left.
Cite as
Seungbum Jo and Dominik Köppl. Indexing Range Maximum-Sum Segment Queries with Offsets. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jo_et_al:LIPIcs.SWAT.2026.23,
author = {Jo, Seungbum and K\"{o}ppl, Dominik},
title = {{Indexing Range Maximum-Sum Segment Queries with Offsets}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {23:1--23:16},
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.2026.23},
URN = {urn:nbn:de:0030-drops-260597},
doi = {10.4230/LIPIcs.SWAT.2026.23},
annote = {Keywords: maximum segment sum, data structure, range query}
}
Document
Cutwidth Versus BFS-Width with Applications to Graph Reconstruction from Distance Queries
Abstract
Eppstein, Goodrich, and Liu [ESA 2025] introduced a new graph parameter, called BFS-width, and gave polylogarithmic bounds on it for bounded bandwidth graphs. Their bounds naturally imply several applications, e.g. in graph reconstruction via shortest path distance queries, graph drawing, and matrix reordering.
We study this parameter for a broader class of graphs, namely bounded cutwidth graphs. We prove a sublinear upper bound on the BFS-width of bounded cutwidth graphs and show that our bounds are asymptotically tight. Our upper bound implies the first deterministic algorithm for reconstructing a bounded cutwidth graph with a subquadratic number of shortest path distance queries.
Cite as
Chirag Kaudan and Amir Nayyeri. Cutwidth Versus BFS-Width with Applications to Graph Reconstruction from Distance Queries. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaudan_et_al:LIPIcs.SWAT.2026.24,
author = {Kaudan, Chirag and Nayyeri, Amir},
title = {{Cutwidth Versus BFS-Width with Applications to Graph Reconstruction from Distance Queries}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {24:1--24:13},
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.2026.24},
URN = {urn:nbn:de:0030-drops-260600},
doi = {10.4230/LIPIcs.SWAT.2026.24},
annote = {Keywords: Graph algorithms, graph theory, cutwidth, pathwidth, BFS-width}
}
Document
Parameterized Critical Node Cut Revisited
Abstract
We study how to sparsify connectivity in graphs under a tight deletion budget. Given a graph G and integers k,x ≥ 0, Critical Node Cut (CNC) asks whether we can delete at most k vertices so that the number of remaining unordered pairs of connected vertices is at most x. CNC generalizes Vertex Cover (the case x = 0) and models tasks in network design, epidemiology, and social network analysis. We comprehensively map the structural parameterized complexity landscape for Critical Node Cut. First, we prove W[1]-hardness for the combined parameter k + fes + Δ + pw, where fes is the feedback edge set number, Δ the maximum degree, and pw the pathwidth of the input graph, respectively. This significantly improves over the known W[1]-hardness for k+tw, where tw denotes the treewidth, and is tight in that tree-depth together with maximum degree trivially yields FPT. Second, we give new positive results. Specifically, we identify three structural parameters-max-leaf number, vertex integrity, and modular-width-that render the problem fixed-parameter tractable, and develop a polynomial-time algorithm for graphs of constant clique-width. Third, leveraging a technique introduced by Lampis [ICALP '14], we develop an FPT approximation scheme that, for any ε > 0, computes a (1+ε)-approximate solution in time (tw / ε)^{𝒪(tw)} n^{𝒪(1)}. Finally, we show that CNC admits no polynomial kernel when parameterized by vertex cover number, unless standard assumptions fail. Together, these results substantially sharpen the known complexity landscape for CNC.
Cite as
Dušan Knop, Nikolaos Melissinos, and Manolis Vasilakis. Parameterized Critical Node Cut Revisited. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{knop_et_al:LIPIcs.SWAT.2026.25,
author = {Knop, Du\v{s}an and Melissinos, Nikolaos and Vasilakis, Manolis},
title = {{Parameterized Critical Node Cut Revisited}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {25:1--25:17},
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.2026.25},
URN = {urn:nbn:de:0030-drops-260617},
doi = {10.4230/LIPIcs.SWAT.2026.25},
annote = {Keywords: Critical Node Cut, Parameterized Complexity, Treewidth}
}
Document
On the Parameterized Complexity of Min-Sum-Radii
Abstract
In the Min-Sum-Radii (MSR) clustering problem, we are given a finite set X of n points in a metric space. The objective is to find at most k clusters centered at a subset of these points such that every point of X is assigned to one of the clusters, minimizing the sum of the radii of the clusters. The problem is known to be NP-hard even on metrics induced by weighted planar graphs and metrics with constant doubling dimension, as shown by Gibson et al. (SWAT 2008). In this work, we investigate the parameterized complexity of MSR on metrics induced by undirected graphs. We distinguish between weighted graph metrics (with positive edge weights) and unweighted graph metrics (where all edges have unit weight).
Weighted Graph Metrics. We show that MSR is W[1]-hard on metrics induced by weighted bipartite graphs, when parameterized by the combined parameter k the number of clusters and Δ the cost of the clustering. We then investigate the structural parameterized complexity of the problem. Drexler et al. [doi:10.48550/arXiv.2310.02130] showed that the MSR problem admits an XP algorithm on metrics induced by weighted graphs when parameterized by treewidth, and asked whether this can be improved to fixed-parameter tractability. We first answer their question in the negative, and more strongly show that MSR stays W[1]-hard on metrics induced by undirected weighted bipartite graphs when parameterized by the vertex cover number plus k. We then turn our attention to parameters for dense graphs and show that MSR remains W[1]-hard when parameterized by k+Δ even on cliques and complete bipartite graphs.
On the positive side, we employ the known XP algorithm parameterized by treewidth, to show that the MSR problem is FPT when parameterized by the parameter treewidth plus Δ. Together, these results provide a complete picture of the parameterized complexity of MSR with respect to any combination of parameters k, Δ, as well as structural parameters for sparse graphs above vertex cover and known parameters for dense graphs (such as neighborhood diversity and modular width).
Unweighted Graph Metrics. The story is rather different for unweighted graphs, since it is a long standing open question whether MSR on metrics induced by undirected graphs is solvable in polynomial-time. Although we cannot answer this question, we provide classical and parameterized hardness results for two very closely related problems, namely Exact-MSR (MSR and one wants to find exactly k clusters) and Allowed-Centers-MSR (MSR with an additional set of allowed cluster centers). We also show that MSR as well as these two problems are fixed-parameter tractable parameterized by the treedepth of the input graph.
Cite as
Pankaj Kumar, Haiko Müller, Sebastian Ordyniak, and Melanie Schmidt. On the Parameterized Complexity of Min-Sum-Radii. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kumar_et_al:LIPIcs.SWAT.2026.26,
author = {Kumar, Pankaj and M\"{u}ller, Haiko and Ordyniak, Sebastian and Schmidt, Melanie},
title = {{On the Parameterized Complexity of Min-Sum-Radii}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {26:1--26:18},
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.2026.26},
URN = {urn:nbn:de:0030-drops-260623},
doi = {10.4230/LIPIcs.SWAT.2026.26},
annote = {Keywords: Parameterized complexity, Min-Sum-Radii clustering}
}
Document
New Results on Three-Sided Skyline Range Counting and Reporting
Abstract
In the orthogonal skyline range counting (resp. reporting) problem we store the set of points P in a data structure so that for any query range Q the number of points (resp. the list of all points) on the skyline of Q∩ P can be found efficiently. In this paper we study two-dimensional range counting and reporting problems in the case when the query range is bounded on three sides.
We describe a linear-space data structure that answers top-open three-sided skyline counting queries in O(log log N) time, where N is the number of points stored in the data structure. We also show that bottom-open three-sided skyline counting queries are as difficult as general four-sided queries and any data structure that uses O(Nlog^c N) space for a constant c requires Ω(log N/log log N) time to answer such queries.
Next, we turn to skyline color range queries. In this variant of the problem each point in P is assigned a color and we must count (resp. report) the distinct colors of point on the skyline of Q∩ P. We describe an O(N)-space data structure that answers top-open three-sided color counting queries in O(log N/log log N) time. Finally, we study top-open three-sided skyline color reporting in the EM model and describe a data structure that uses linear space and answers queries in O(k/B+1) I/Os where k is the number of colors on the skyline. This is the first external-memory data structure with optimal query cost and space usage for this problem.
Cite as
Suruchi Kushwaha and Yakov Nekrich. New Results on Three-Sided Skyline Range Counting and Reporting. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kushwaha_et_al:LIPIcs.SWAT.2026.27,
author = {Kushwaha, Suruchi and Nekrich, Yakov},
title = {{New Results on Three-Sided Skyline Range Counting and Reporting}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {27:1--27:14},
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.2026.27},
URN = {urn:nbn:de:0030-drops-260631},
doi = {10.4230/LIPIcs.SWAT.2026.27},
annote = {Keywords: Data Structures, Range Searching, Skyline Queries}
}
Document
The Parameterized Complexity of Coloring Mixed Graphs
Abstract
A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring c of a mixed graph G assigns a positive integer to each vertex such that c(u)≠c(v) for every edge {u,v} and c(u)<c(v) for every arc (u,v) of G. As in classical coloring, the objective is to minimize the number of colors. Thus, mixed (graph) coloring generalizes classical coloring of undirected graphs and allows for more general applications, such as scheduling with precedence constraints, modeling metabolic pathways, and process management in operating systems; see a survey by Sotskov [Mathematics, 2020].
We initiate the systematic study of the parameterized complexity of mixed coloring. We focus on structural graph parameters that lie between cliquewidth and vertex cover, primarily with respect to the underlying undirected graph. Unlike classical coloring, which is fixed-parameter tractable (FPT) parameterized by treewidth or neighborhood diversity, we show that mixed coloring is W[1]-hard for treewidth and even paraNP-hard for neighborhood diversity. To utilize the directedness of arcs, we introduce and analyze natural generalizations of neighborhood diversity and cliquewidth to mixed graphs, and show that mixed coloring becomes FPT when parameterized by (the generalized) mixed neighborhood diversity. Further, we investigate how these parameters are affected if we add transitive arcs, which do not affect colorings. Finally, we provide tight bounds on the chromatic number of mixed graphs, generalizing known bounds on mixed interval graphs.
Cite as
Antonio Lauerbach, Konstanty Junosza-Szaniawski, Marie Diana Sieper, and Alexander Wolff. The Parameterized Complexity of Coloring Mixed Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lauerbach_et_al:LIPIcs.SWAT.2026.28,
author = {Lauerbach, Antonio and Junosza-Szaniawski, Konstanty and Sieper, Marie Diana and Wolff, Alexander},
title = {{The Parameterized Complexity of Coloring Mixed Graphs}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {28:1--28:18},
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.2026.28},
URN = {urn:nbn:de:0030-drops-260644},
doi = {10.4230/LIPIcs.SWAT.2026.28},
annote = {Keywords: Mixed Graphs, Coloring, Parameterized Complexity, Structural Graph Parameters}
}
Document
Bichromatic Classifications of Points Using Strips
Abstract
Given a set of n points in the plane, each colored either blue or red, we study the problem of finding a strip that separates the blue points from the red points. Specifically, we consider the following two variants: (1) locating a strip that contains no red points while maximizing the number of blue points within the strip, and (2) locating a strip that contains all blue points while minimizing the number of red points within the strip. For variant (1), we present an O(n²)-time algorithm, improving upon the previously best O(n²log n)-time result. We also show that this running time is optimal under the standard 3SUM conjecture. We also give an output-sensitive algorithm with running time O(k_{opt} n log n) that returns a strip, where k_{opt} is the number of blue points not contained within the strip in an optimal solution. We extend our results to the case of up to t parallel strips, obtaining an O(n²log n)-time algorithm. For variant (2), an optimal Θ(nlog n)-time algorithm is known for t = 1. We show 3SUM-hardness for t = 2 and give an O(n²)-time algorithm. For any t ≥ 3, we present an O(n²log n)-time algorithm.
Cite as
Jaegun Lee, Chaeyoon Chung, and Hee-Kap Ahn. Bichromatic Classifications of Points Using Strips. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lee_et_al:LIPIcs.SWAT.2026.29,
author = {Lee, Jaegun and Chung, Chaeyoon and Ahn, Hee-Kap},
title = {{Bichromatic Classifications of Points Using Strips}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {29:1--29:17},
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.2026.29},
URN = {urn:nbn:de:0030-drops-260659},
doi = {10.4230/LIPIcs.SWAT.2026.29},
annote = {Keywords: Bichromatic Classification, Separation, Strip, Duality}
}
Document
Parallel Algorithms for Group Isomorphism via Code Equivalence
Abstract
In this paper, we exhibit AC³ isomorphism tests for coprime extensions H ⋉ N where H is elementary Abelian and N is Abelian; and groups where Rad(G) = Z(G) is elementary Abelian and G = Soc^{*}(G). The fact that isomorphism testing for these families is in P was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017).
The polynomial-time isomorphism tests for both of these families crucially leveraged small (size O(log |G|)) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that G is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in AC³.
As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using AC circuits of depth O(log³ n) and size n^{O(log log n)}. This improves upon the previous bound of n^{O(log log n)}-time due to Grochow and Qiao (ibid.).
Cite as
Michael Levet. Parallel Algorithms for Group Isomorphism via Code Equivalence. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{levet:LIPIcs.SWAT.2026.30,
author = {Levet, Michael},
title = {{Parallel Algorithms for Group Isomorphism via Code Equivalence}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {30:1--30:19},
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.2026.30},
URN = {urn:nbn:de:0030-drops-260660},
doi = {10.4230/LIPIcs.SWAT.2026.30},
annote = {Keywords: Group Isomorphism, Circuit Complexity, Code Equivalence}
}
Document
Improved and Parameterized Algorithms for Online Multi-Level Aggregation
Abstract
We study the online multi-level aggregation problem with deadlines (MLAP-D) introduced by Bienkowski, Böhm, Byrka, Chrobak, Dürr, Folwarczný, Jeż, Sgall, Thang, and Veselý (ESA 2016, OR 2020). In this problem, requests arrive over time at the vertices of a given vertex-weighted tree, and each request has a deadline that it must be served by. The cost of serving a request equals the cost of a path from the root to the vertex where the request resides. Instead of serving each request individually, requests can be aggregated and served by transmitting a subtree from the root that spans the vertices on which the requests reside, to potentially be more cost-effective. The aggregated cost is the weight of the transmission subtree. The goal of MLAP-D is to find an aggregation solution that minimizes the total cost while serving all requests. MLAP-D generalizes some well-studied problems including the TCP acknowledgment problem and the joint replenishment problem, and arises in natural scenarios such as multi-casting, sensor networks, and supply chain management.
We present improved and parameterized algorithms for MLAP-D. Our result is twofold. First, we present an e(D+1)-competitive algorithm where D is the depth of the tree. Second, we present an e(4H+2)-competitive algorithm where H is the caterpillar dimension of the tree. Here, H ≤ D and H ≤ log₂ |V| where |V| is the number of vertices in the given tree. The caterpillar dimension remains constant for rich but simple classes of trees, such as line graphs (H = 1), caterpillar graphs (H = 2), and lobster graphs (H = 3). To the best of our knowledge, this is the first online algorithm parameterized on a measure better than depth. The state-of-the-art online algorithms are 6(D+1)-competitive by Buchbinder, Feldman, Naor, and Talmon (SODA 2017) and O(log |V|)-competitive by Azar and Touitou (FOCS 2020). Our framework outperforms the state-of-the-art ratios when H = o(min{D,log₂ |V|}). Our memory-based algorithms extend transmission subtrees with a cost comparable to transmission subtrees used to serve previous requests. Our simple framework directly applies to trees with any structure and differs from the previous frameworks that reduce the problem to trees with specific structures.
Cite as
Young-San Lin and Alex Turoczy. Improved and Parameterized Algorithms for Online Multi-Level Aggregation. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lin_et_al:LIPIcs.SWAT.2026.31,
author = {Lin, Young-San and Turoczy, Alex},
title = {{Improved and Parameterized Algorithms for Online Multi-Level Aggregation}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {31:1--31:17},
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.2026.31},
URN = {urn:nbn:de:0030-drops-260673},
doi = {10.4230/LIPIcs.SWAT.2026.31},
annote = {Keywords: Online Algorithms, Approximation Algorithms, Graph Problems}
}
Document
Semirandom Planted Bipartite Subgraphs
Abstract
There have been many recent works studying planted subgraphs problems. The semirandom planted bipartite subgraph problem is defined as follows. Starting with a vertex set V, an arbitrary subset S ⊂ V of size k is chosen, then an arbitrary bipartite graph is added on S. After this between each pair of vertices in S × (V ⧵ S) an edge is added independently with probability p, then an arbitrary graph is added on V⧵ S. The analogous semirandom planted clique problem, where S forms a clique, has been studied starting with the work of Fiege and Kilian [Uriel Feige and Joe Kilian, 2001]; recent work by [Blasiok et al., 2024; Venkatesan Guruswami and Hsin-Po Wang, 2025] gave an algorithm for this problem when k = Ω(√{n log n}). We give an algorithm for semirandom planted bipartite subgraph problem when k = Ω(√{n log n}) and the two color classes are roughly balanced.
Our algorithms are essentially the same as the elegant greedy algorithm of [Blasiok et al., 2024]. We generalize their idea to our setting. Handling the arbitrary nature of the bipartite graph requires some new technical ideas and is our main technical contribution.
Cite as
Anand Louis and Kirtan Vora. Semirandom Planted Bipartite Subgraphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{louis_et_al:LIPIcs.SWAT.2026.32,
author = {Louis, Anand and Vora, Kirtan},
title = {{Semirandom Planted Bipartite Subgraphs}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {32:1--32:13},
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.2026.32},
URN = {urn:nbn:de:0030-drops-260681},
doi = {10.4230/LIPIcs.SWAT.2026.32},
annote = {Keywords: Semirandom Models, Spectral Algorithms, Planted Subgraphs, Random Graphs, Approximate Recovery Algorithms}
}
Document
Polychromatic 2-Colorings with Bounded Discrepancy for Triangulations
Abstract
A polychromatic 2-coloring of a triangulation is a 2-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic 2-coloring with discrepancy at most (5n-16)/9, and that there exists a class of triangulations for which every polychromatic 2-coloring has discrepancy at least n/3 - 2, where n is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic 2-coloring with discrepancy at most (3n-16)/7 and such a 2-coloring can be computed in quadratic time. We also show a discrepancy of at most n-4M/3 for triangulations with a matching of size M. This implies, for example, that Delaunay triangulations admit a discrepancy of at most n/3. We provide a linear-time algorithm to compute a 2-coloring whose discrepancy is at most (5n-24)/7.
Cite as
Alma Arevalo Loyola, Ahmad Biniaz, Prosenjit Bose, and Thomas Shermer. Polychromatic 2-Colorings with Bounded Discrepancy for Triangulations. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 33:1-33:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{loyola_et_al:LIPIcs.SWAT.2026.33,
author = {Loyola, Alma Arevalo and Biniaz, Ahmad and Bose, Prosenjit and Shermer, Thomas},
title = {{Polychromatic 2-Colorings with Bounded Discrepancy for Triangulations}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {33:1--33:13},
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.2026.33},
URN = {urn:nbn:de:0030-drops-260691},
doi = {10.4230/LIPIcs.SWAT.2026.33},
annote = {Keywords: polychromatic coloring, triangulation, balanced coloring, matching}
}
Document
Linear-Time Exact Computation of Influence Spread on Bounded-Pathwidth Graphs
Abstract
Given a network and a set of vertices called seeds to initially inject information, influence spread is the expected number of vertices that eventually receive the information under a certain stochastic model of information propagation. Under the commonly used independent cascade model, influence spread is equivalent to the expected number of vertices reachable from the seeds on a directed uncertain graph, and the exact evaluation of influence spread offers many applications, e.g., influence maximization. Although its evaluation is a #P-hard task, there is an algorithm that can precisely compute the influence spread in O(mnω_p²⋅ 2^{ω_p²}) time, where ω_p is the pathwidth of the graph. We improve this by developing an algorithm that computes the influence spread in O((m+n)ω_p²⋅ 2^{ω_p²}) time. This is achieved by identifying the similarities in the repetitive computations in the existing algorithm and sharing them to reduce computation. Although similar refinements have been considered for the probability computation on undirected uncertain graphs, a greater number of similarities must be leveraged for directed graphs to achieve linear time complexity.
Cite as
Kengo Nakamura and Masaaki Nishino. Linear-Time Exact Computation of Influence Spread on Bounded-Pathwidth Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nakamura_et_al:LIPIcs.SWAT.2026.34,
author = {Nakamura, Kengo and Nishino, Masaaki},
title = {{Linear-Time Exact Computation of Influence Spread on Bounded-Pathwidth Graphs}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {34:1--34:17},
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.2026.34},
URN = {urn:nbn:de:0030-drops-260704},
doi = {10.4230/LIPIcs.SWAT.2026.34},
annote = {Keywords: Influence spread, bounded pathwidth, network reliability, linear time algorithm}
}
Document
Path-Reporting Distance Oracles for Vertex-Labeled Graphs
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
Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates
Abstract
In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets R,B ⊂ [Δ]² with |R|+|B| = n, the goal is to select a set of edges between R and B so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that R and B are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes Õ(n²) time. We present an exact Õ(n^{1.5} log Δ) time algorithm for point sets in [Δ]². To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.
Cite as
Seongbin Park and Eunjin Oh. Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{park_et_al:LIPIcs.SWAT.2026.36,
author = {Park, Seongbin and Oh, Eunjin},
title = {{Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {36:1--36:16},
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.2026.36},
URN = {urn:nbn:de:0030-drops-260728},
doi = {10.4230/LIPIcs.SWAT.2026.36},
annote = {Keywords: Edge cover, many-to-many matching, similarity, geometric matching}
}
Document
Faster Linear-Space Data Structures for Path Frequency Queries
Abstract
We present linear-space data structures for several frequency queries on trees, namely: path mode, path least frequent element, and path α-minority queries. We present the first linear-space data structures, requiring O(n √{nw}) preprocessing time, that can answer path mode and path least frequent element queries in O(√{n/w}) time. This improves upon the best previously known bound of O(log log n √{n/w}) achieved by Durocher et al. [Durocher et al., 2016] in 2016.
For the path α-minority problem, where α is specified at query time, we reduce the query time of the linear-space data structure of Durocher et al. [Durocher et al., 2016] from O(α^{-1}log log n) down to O(α^{-1}) by employing a simple randomized algorithm with a success probability ≥ 1/2.
We also present the first linear-space data structure supporting "Path Maximum g-value Color" queries in O(√{n/w}) time, requiring O(n √{nw}) preprocessing time. This general framework encapsulates both path mode and path least frequent element queries. For our data structures, we consider the word-RAM model with w ∈ Ω(log n), where w is the word size in bits.
Cite as
Ovidiu Rața. Faster Linear-Space Data Structures for Path Frequency Queries. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rata:LIPIcs.SWAT.2026.37,
author = {Rața, Ovidiu},
title = {{Faster Linear-Space Data Structures for Path Frequency Queries}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {37:1--37:17},
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.2026.37},
URN = {urn:nbn:de:0030-drops-260732},
doi = {10.4230/LIPIcs.SWAT.2026.37},
annote = {Keywords: Data structure, Range query, Mode, Minority, Least frequent element, Trees, Linear-space, Path query}
}
Document
New Algorithms for Girth and Cycle Detection
Abstract
Let G = (V,E) be an unweighted undirected graph with n vertices and m edges. Let g be the girth of G, that is, the length of a shortest cycle in G. We present a randomized algorithm with a running time of Õ(𝓁 ⋅ n^{1 + 1/(𝓁-ε)}) that returns a cycle of length at most 2𝓁 ⌈g/2⌉ - 2 ⌊ε⌈g/2⌉⌋, where 𝓁 ≥ 2 is an integer and ε ∈ [0,1], for every graph with g = polylog(n).
Our algorithm generalizes an algorithm of Kadria et al. [SODA'22] that computes a cycle of length at most 4 ⌈g/2⌉ - 2 ⌊ε⌈g/2⌉⌋ in Õ(n^{1 + 1/(2 - ε)}) time. Kadria et al. presented also an algorithm that finds a cycle of length at most 2𝓁 ⌈g/2⌉ in Õ(n^{1 + 1/(𝓁)}) time, where 𝓁 must be an integer. Our algorithm generalizes this algorithm, as well, by replacing the integer parameter 𝓁 in the running time exponent with a real-valued parameter 𝓁 - ε, thereby offering greater flexibility in parameter selection and enabling a broader spectrum of combinations between running times and cycle lengths.
We also show that for sparse graphs a better tradeoff is possible, by presenting an Õ(𝓁⋅ m^{1+ 1/(𝓁-ε)}) time randomized algorithm that returns a cycle of length at most 2𝓁(⌊(g-1)/2⌋) - 2(⌊ε⌊(g-1)/2⌋⌋+1), where 𝓁 ≥ 3 is an integer and ε ∈ [0,1), for every graph with g = polylog(n).
To obtain our algorithms we develop several techniques and introduce a formal definition of hybrid cycle detection algorithms. Both may prove useful in broader contexts, including other cycle detection and approximation problems. Among our techniques is a new cycle searching technique, in which we search for a cycle from a given vertex and possibly all its neighbors in linear time. Using this technique together with more ideas we develop two hybrid algorithms. The first allows us to obtain an Õ(m^{2-2/(⌈g/2⌉+1))-time, (+1)-approximation of g. The second is used to obtain our Õ(𝓁⋅ n^{1+ 1/(𝓁-ε)})-time and Õ(𝓁⋅ m^{1+ 1/(𝓁-ε)})-time approximation algorithms.
Cite as
Liam Roditty and Plia Trabelsi. New Algorithms for Girth and Cycle Detection. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{roditty_et_al:LIPIcs.SWAT.2026.38,
author = {Roditty, Liam and Trabelsi, Plia},
title = {{New Algorithms for Girth and Cycle Detection}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {38:1--38:17},
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.2026.38},
URN = {urn:nbn:de:0030-drops-260742},
doi = {10.4230/LIPIcs.SWAT.2026.38},
annote = {Keywords: Graph algorithms, All pairs shortest path, Girth, Cycle approximation}
}
Document
Faster Approximate Linear Matroid Intersection
Abstract
We consider a fast approximation algorithm for the linear matroid intersection problem. In this problem, we are given two r × n matrices M₁ and M₂, and the objective is to find a largest set of columns that are linearly independent in both M₁ and M₂. We design a (1 - ε)-approximation algorithm with time complexity Õ_{ε}(nnz(M₁) + nnz(M₂) + r_{*}^{ω}), where nnz(M_i) denotes the number of nonzero entries in M_i for i = 1, 2, r_{*} denotes the maximum size of a common independent set, and ω < 2.372 denotes the matrix multiplication exponent. Our approximation algorithm is faster than the exact algorithm by Harvey [FOCS'06 & SICOMP'09] and Cheung-Kwok-Lau [STOC'12 & JACM'13], which runs in Õ(nnz(M₁) + nnz(M₂) + n r_{*}^{ω - 1}) time.
We also develop a fast (1 - ε)-approximation algorithm for the weighted version of the linear matroid intersection problem. In fact, we design a (1 - ε)-approximation algorithm for weighted linear matroid intersection with time complexity Õ_{ε}(nnz(M₁) + nnz(M₂) + r_{*}^{ω}). Our algorithm improves upon the (1 - ε)-approximation algorithm by Huang-Kakimura-Kamiyama [SODA'16 & Math. Program.'19], which runs in Õ_{ε}(nnz(M₁) + nnz(M₂) + nr_{*}^{ω - 1}) time.
To obtain these results, we combine Quanrud’s adaptive sparsification framework [ICALP'24] with a simple yet effective method for efficiently checking whether a given vector lies in the linear span of a subset of vectors, which is of independent interest.
Cite as
Tatsuya Terao. Faster Approximate Linear Matroid Intersection. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{terao:LIPIcs.SWAT.2026.39,
author = {Terao, Tatsuya},
title = {{Faster Approximate Linear Matroid Intersection}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {39:1--39:19},
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.2026.39},
URN = {urn:nbn:de:0030-drops-260756},
doi = {10.4230/LIPIcs.SWAT.2026.39},
annote = {Keywords: Linear matroid intersection, fast approximation algorithm}
}
Document
Maximum Independent Sets in Disk Graphs with Disks in Convex Position
Abstract
For a set 𝒟 of disks in the plane, its disk graph G(𝒟) is the graph with vertex set 𝒟, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set 𝒟 of n weighted disks, computing a maximum independent set of G(𝒟) is NP-hard. In this paper, we present an O(n³log n)-time algorithm for this problem in a special setting in which the disks are in convex position, meaning that every disk appears on the convex hull of 𝒟. This setting has been studied previously for disks of equal radius, for which an O(n^{37/11})-time algorithm was known. Our algorithm also works in the weighted case where disks have weights and the goal is to compute a maximum-weight independent set. As an application of our result, we obtain an O(n³log² n)-time algorithm for the dispersion problem on a set of n disks in convex position: given an integer k, compute a subset of k disks that maximizes the minimum pairwise distance among all disks in the subset.
Cite as
Anastasiia Tkachenko and Haitao Wang. Maximum Independent Sets in Disk Graphs with Disks in Convex Position. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tkachenko_et_al:LIPIcs.SWAT.2026.40,
author = {Tkachenko, Anastasiia and Wang, Haitao},
title = {{Maximum Independent Sets in Disk Graphs with Disks in Convex Position}},
booktitle = {20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
pages = {40:1--40:18},
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.2026.40},
URN = {urn:nbn:de:0030-drops-260766},
doi = {10.4230/LIPIcs.SWAT.2026.40},
annote = {Keywords: disk graphs, independent sets, convex position, dispersion}
}