168 Search Results for "Golovach, Petr A."

Volume

LIPIcs, Volume 214

16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

IPEC 2021, September 8-10, 2021, Lisbon, Portugal

Editors: Petr A. Golovach and Meirav Zehavi

Document
DOI: 10.4230/LIPIcs.WG.2026.18
Polynomial Kernels for Spanning Tree with Diversity Requirements

Authors: Petr A. Golovach, Diptapriyo Majumdar, and Saket Saurabh

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

Abstract
Given a connected undirected graph G, a spanning tree is a subgraph T of G such that V(T) = V(G) and T is a tree. A collection of 𝓁 spanning trees T₁,…,T_{𝓁} is {{pairwise k-diverse}} if for every i ≠ j, |E(T_i) △ E(T_j)| ≥ k. Given a connected undirected graph G and integers p, q, k, 𝓁, {Leaf&Internal-Constrained Diverse Spanning Trees} asks whether there are 𝓁 distinct spanning trees T₁,…,T_{𝓁} of G that are {{pairwise k-diverse}} such that each tree has at least p leaves and at least q internal vertices. Similarly, {Leaf&Non-terminal-Constrained Diverse Spanning Trees} takes a connected undirected graph G, V_NT ⊆ V(G), and three integers p, k, 𝓁, and asks if G has 𝓁 spanning trees that are {{pairwise k-diverse}}, and each has at least p leaves and contains the vertices of V_NT as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for {Leaf&Internal-Constrained Diverse Spanning Trees} and {Leaf&Non-terminal-Constrained Diverse Spanning Trees}, when parameterized by p + q + k + 𝓁 and p + |V_NT| + k + 𝓁, respectively.
Cite as

Petr A. Golovach, Diptapriyo Majumdar, and Saket Saurabh. Polynomial Kernels for Spanning Tree with Diversity Requirements. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{golovach_et_al:LIPIcs.WG.2026.18,
  author =	{Golovach, Petr A. and Majumdar, Diptapriyo and Saurabh, Saket},
  title =	{{Polynomial Kernels for Spanning Tree with Diversity Requirements}},
  booktitle =	{52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-430-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{376},
  editor =	{Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.18},
  URN =		{urn:nbn:de:0030-drops-261840},
  doi =		{10.4230/LIPIcs.WG.2026.18},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Diverse Solutions, Diverse Spanning Trees}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.19
Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study

Authors: Tian Bai, Fedor V. Fomin, Petr A. Golovach, Yash Hiren More, and Simon Wietheger

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

Abstract
Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and median problems, which minimize the maximum and the total distance to the input permutations, respectively. While these problems are well understood under Kendall’s tau and related distances, their parameterized complexity under the Ulam metric, an edit-distance-based metric on permutations, has remained largely unexplored. In this work, we initiate a systematic study of the parameterized complexity of rank aggregation under the Ulam metric. We consider both the center and median problems, as well as their generalizations to the k-center and k-median clustering settings, parameterized by the number of centers k and the distance budget d (corresponding to the maximum distance for center variants and the total distance for median variants). Both problems are known to be NP-hard already for k = 1. We show that the Ulam k-center problem remains NP-hard when d = 1, but is fixed-parameter tractable when parameterized by k + d. Our algorithm is based on a novel local-search framework tailored to the non-local nature of Ulam distances. We complement this by proving that no polynomial kernel exists for the k+d parameterization unless NP ⊆ coNP/poly. For the Ulam k-median problem parameterized by the total distance d, we establish W[1]-hardness and provide an XP algorithm. We also provide a polynomial kernel for the parameter k + d, which in turn yields a fixed-parameter tractable algorithm.
Cite as

Tian Bai, Fedor V. Fomin, Petr A. Golovach, Yash Hiren More, and Simon Wietheger. Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bai_et_al:LIPIcs.ICALP.2026.19,
  author =	{Bai, Tian and Fomin, Fedor V. and Golovach, Petr A. and More, Yash Hiren and Wietheger, Simon},
  title =	{{Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{19:1--19:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.19},
  URN =		{urn:nbn:de:0030-drops-264080},
  doi =		{10.4230/LIPIcs.ICALP.2026.19},
  annote =	{Keywords: parameterized complexity, Ulam distance, rank aggregation, clustering}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.28
The Parameterized Complexity of Coloring Mixed Graphs

Authors: Antonio Lauerbach, Konstanty Junosza-Szaniawski, Marie Diana Sieper, and Alexander Wolff

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

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
DOI: 10.4230/LIPIcs.CPM.2026.12
Maximizing Diversity in (Near-)Median String Selection

Authors: Diptarka Chakraborty, Rudrayan Kundu, Nidhi Purohit, and Aravinda Kanchana Ruwanpathirana

Published in: LIPIcs, Volume 369, 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)

Abstract
Given a set of strings over a specified alphabet, identifying a median or consensus string that minimizes the total distance to all input strings is a fundamental data aggregation problem. When the Hamming distance is considered as the underlying metric, this problem has extensive applications, ranging from bioinformatics to pattern recognition. However, modern applications often require the generation of multiple (near-)optimal yet diverse median strings to enhance flexibility and robustness in decision-making. In this study, we address this need by focusing on two prominent diversity measures: sum dispersion and min dispersion. We first introduce an exact algorithm for the diameter variant of the problem, which identifies pairs of near-optimal medians that are maximally diverse. Subsequently, we propose a (1-ε)-approximation algorithm (for any ε > 0) for sum dispersion, as well as a bi-criteria approximation algorithm for the more challenging min dispersion case, allowing the generation of multiple (more than two) diverse near-optimal Hamming medians. Our approach primarily leverages structural insights into the Hamming median space and also draws on techniques from error-correcting code construction to establish these results.
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Diptarka Chakraborty, Rudrayan Kundu, Nidhi Purohit, and Aravinda Kanchana Ruwanpathirana. Maximizing Diversity in (Near-)Median String Selection. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chakraborty_et_al:LIPIcs.CPM.2026.12,
  author =	{Chakraborty, Diptarka and Kundu, Rudrayan and Purohit, Nidhi and Ruwanpathirana, Aravinda Kanchana},
  title =	{{Maximizing Diversity in (Near-)Median String Selection}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-420-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{369},
  editor =	{Bille, Philip and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.12},
  URN =		{urn:nbn:de:0030-drops-259382},
  doi =		{10.4230/LIPIcs.CPM.2026.12},
  annote =	{Keywords: Diversity maximization, Hamming median, diameter, dispersion, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.25
Parameterized Critical Node Cut Revisited

Authors: Dušan Knop, Nikolaos Melissinos, and Manolis Vasilakis

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

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
DOI: 10.4230/LIPIcs.FORC.2026.10
Packing Compact Subgraphs with Applications to Districting

Authors: Ho-Lin Chen, Po-Yu Chou, Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract
Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements. In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs [Prathamesh Dharangutte et al., 2025]. In particular, we improve the approximation factor from O(log n) to O(1) for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an O(1) approximation for packing radius-k districts (with a constant k) in planar and apex-minor-free graphs. In order to get a (1+ε) approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to 1), for bounded radius-k districts on planar and apex-minor-free graphs. We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types.
Cite as

Ho-Lin Chen, Po-Yu Chou, Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu. Packing Compact Subgraphs with Applications to Districting. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 10:1-10:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.FORC.2026.10,
  author =	{Chen, Ho-Lin and Chou, Po-Yu and Dharangutte, Prathamesh and Gao, Jie and Huang, Shang-En and Yu, Fang-Yi},
  title =	{{Packing Compact Subgraphs with Applications to Districting}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{10:1--10:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.10},
  URN =		{urn:nbn:de:0030-drops-259820},
  doi =		{10.4230/LIPIcs.FORC.2026.10},
  annote =	{Keywords: Approximation algorithms, algorithmic fairness}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.33
Near-Optimal Bounds for Parameterized Euclidean k-Means

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

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

Abstract
The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1). In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.33,
  author =	{Cohen-Addad, Vincent and C. S., Karthik and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Near-Optimal Bounds for Parameterized Euclidean k-Means}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.33},
  URN =		{urn:nbn:de:0030-drops-258391},
  doi =		{10.4230/LIPIcs.SoCG.2026.33},
  annote =	{Keywords: k-means clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.34
Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

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

Abstract
The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
  author =	{Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.34},
  URN =		{urn:nbn:de:0030-drops-258404},
  doi =		{10.4230/LIPIcs.SoCG.2026.34},
  annote =	{Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.49
Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

Authors: Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, and Saket Saurabh

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

Abstract
In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: 1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on 𝒪(1) rows (for fixed values of the parameters). 2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. 3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.
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Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, and Saket Saurabh. Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2026.49,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.49},
  URN =		{urn:nbn:de:0030-drops-258552},
  doi =		{10.4230/LIPIcs.SoCG.2026.49},
  annote =	{Keywords: Parameterized Complexity, Euclidean Embedding, Polynomial Compression}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.77
Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs

Authors: Malory Marin, Jean-Florent Raymond, and Rémi Watrigant

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

Abstract
We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in ℝ^d. In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a λ-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph. We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time 2^O(n^{1-1/d}) on intersection graphs of similarly sized fat objects in ℝ^d. This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms. We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension d. More precisely, we obtain a randomized robust algorithm running in time 2^O(k^{1-1/d} log² k) n^O(1) on intersection graphs of similarly sized fat objects in ℝ^d, where k is the natural parameter. Besides λ-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017]. This structural result may be of independent interest.
Cite as

Malory Marin, Jean-Florent Raymond, and Rémi Watrigant. Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{marin_et_al:LIPIcs.SoCG.2026.77,
  author =	{Marin, Malory and Raymond, Jean-Florent and Watrigant, R\'{e}mi},
  title =	{{Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.77},
  URN =		{urn:nbn:de:0030-drops-258842},
  doi =		{10.4230/LIPIcs.SoCG.2026.77},
  annote =	{Keywords: Robust algorithms, geometric intersection graphs, subexponential FPT algorithms}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.7
Nemesis, an Escape Game in Graphs

Authors: Pierre Bergé, Antoine Dailly, and Yan Gerard

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
We define a new escape game in graphs that we call Nemesis. The game is played on a graph having a subset of vertices labeled as exits and the goal of one of the two players, called the fugitive, is to reach one of these exit vertices. The second player, i.e. the fugitive adversary, is called the Nemesis. Her goal is to trap the fugitive in a connected component which does not contain any exit. At each round of the game, the fugitive moves from one vertex to an adjacent vertex. Then the Nemesis deletes one edge anywhere in the graph. The game ends when either the fugitive reached an exit or when he is in a connected component that does not contain any exit. In trees and graphs of maximum degree bounded by 3, Nemesis can be solved in linear time. For arbitrary graphs, we show that Nemesis is PSPACE-complete, and that it is NP-hard on planar multigraphs. We extend our results to the related Cat Herding problem, proving its PSPACE-completeness.
Cite as

Pierre Bergé, Antoine Dailly, and Yan Gerard. Nemesis, an Escape Game in Graphs. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{berge_et_al:LIPIcs.FUN.2026.7,
  author =	{Berg\'{e}, Pierre and Dailly, Antoine and Gerard, Yan},
  title =	{{Nemesis, an Escape Game in Graphs}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{7:1--7:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.7},
  URN =		{urn:nbn:de:0030-drops-257261},
  doi =		{10.4230/LIPIcs.FUN.2026.7},
  annote =	{Keywords: Graphs, Evasion and Pursuit Games, PSPACE-completeness, Quantified SAT, Canadian Traveler Problem, Cat Herding Problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.13
Line Cover and Related Problems

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana

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

Abstract
We study several extensions of the classic Line Cover problem of covering a set of n points in the plane with k lines. Line Cover is known to be NP-hard and our focus is on two natural generalizations: (1) Line Clustering, where the objective is to find k lines in the plane that minimize the sum of squares of distances of a given set of input points to the closest line, and (2) Hyperplane Cover, where the goal is to cover n points in ℝ^d by k hyperplanes. We also consider the more general Projective Clustering problem, which unifies both of these and has numerous applications in machine learning, data mining, and computational geometry. In this problem one seeks k affine subspaces of dimension r minimizing the sum of squares of distances of a given set of n points in ℝ^d to the closest point within one of the k affine subspaces. Our main contributions reveal interesting differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable parameterized by the number k of lines in the solution, we show that Line Clustering is W[1]-hard when parameterized by k and rule out algorithms of running time n^{o(k)} under the Exponential Time Hypothesis. Hyperplane Cover is known to be NP-hard even when d = 2 and by the work of Langerman and Morin [Discrete & Computational Geometry, 2005], it is FPT parameterized by k and d. We complement this result by establishing that Hyperplane Cover is W[2]-hard when parameterized by only k. We complement our hardness results by presenting an algorithm for Projective Clustering. We show that this problem is solvable in n^{𝒪(dk(r+1))} time. Not only does this yield an upper bound for Line Clustering that asymptotically matches our lower bound, but it also significantly extends the seminal work on k-Means Clustering (the special case r = 0) by Inaba, Katoh, and Imai [SoCG 1994].
Cite as

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana. Line Cover and Related Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bentert_et_al:LIPIcs.STACS.2026.13,
  author =	{Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Saha, Souvik and Seetharaman, Sanjay and Upasana, Anannya},
  title =	{{Line Cover and Related Problems}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.13},
  URN =		{urn:nbn:de:0030-drops-255023},
  doi =		{10.4230/LIPIcs.STACS.2026.13},
  annote =	{Keywords: Point Line Cover, Projective Clustering, W-hardness, XP algorithm}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.15
The Complexity of Resilience for Digraph Queries

Authors: Manuel Bodirsky and Žaneta Semanišinová

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

Abstract
We prove a complexity dichotomy for the resilience problem for unions of conjunctive digraph queries (i.e., for existential positive sentences over the signature {R} of directed graphs). Specifically, for every union μ of conjunctive digraph queries, the following problem is in P or NP-complete: given a directed multigraph G and a natural number u, can we remove u edges from G so that G ⊧ ¬ μ? In fact, we verify a more general dichotomy conjecture from [Bodirsky et al., 2024] for all resilience problems in the special case of directed graphs, and show that for such unions of queries μ there exists a countably infinite (`dual') valued structure Δ_μ which either primitively positively constructs 1-in-3-3-SAT, and hence the resilience problem for μ is NP-complete by general principles, or has a pseudo cyclic canonical fractional polymorphism, and the resilience problem for μ is in P.
Cite as

Manuel Bodirsky and Žaneta Semanišinová. The Complexity of Resilience for Digraph Queries. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bodirsky_et_al:LIPIcs.STACS.2026.15,
  author =	{Bodirsky, Manuel and Semani\v{s}inov\'{a}, \v{Z}aneta},
  title =	{{The Complexity of Resilience for Digraph Queries}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{15:1--15:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.15},
  URN =		{urn:nbn:de:0030-drops-255045},
  doi =		{10.4230/LIPIcs.STACS.2026.15},
  annote =	{Keywords: valued constraints, unions of conjunctive queries, resilience, computational complexity, pp-constructions}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.73
Colouring Probe H-Free Graphs

Authors: Daniël Paulusma, Johannes Rauch, and Erik Jan van Leeuwen

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

Abstract
The NP-complete problems Colouring and k-Colouring (k ≥ 3) are well studied on H-free graphs, i.e., graphs that do not contain some fixed graph H as an induced subgraph. We research to what extent the known polynomial-time algorithms for H-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph H, a partitioned probe H-free graph (G,P,N) consists of a graph G = (V,E), together with a set P ⊆ V of probes and an independent set N = V ⧵ P of non-probes, such that G+F is H-free for some edge set F ⊆ binom(N,2). We show the following: - We fully classify Colouring on partitioned probe H-free graphs and show that the obtained complexity dichotomy differs from the known dichotomy of Colouring for H-free graphs. - We fully classify 3-Colouring on partitioned probe P_t-free graphs: we prove polynomial-time solvability for t ≤ 5 and NP-completeness for t ≥ 6. In contrast, 3-Colouring on P_t-free graphs is known to be polynomial-time solvable for t ≤ 7 and quasi-polynomial-time solvable for t ≥ 8. Our main result is our polynomial-time algorithm for 3-Colouring on partitioned P₅-free graphs. For this result, and also for all our other polynomial-time results, we do not need to know the edge set F; we only need to know its existence. Moreover, the class of probe P₅-free graphs includes not only paths of arbitrary length but even all bipartite graphs and is much richer than the class of P₅-free graphs. The latter is also evidenced by the fact that there exist graph problems, such as Matching Cut, that are known to be polynomial-time solvable for P₅-free graphs but NP-complete for partitioned probe P₅-free graphs. In particular, unlike the class of 3-colourable P₅-free graphs, the class of 3-colourable probe P₅-free graphs has unbounded mim-width. Hence, our polynomial-time result for 3-Colouring for probe P₅-free graphs suggests that there may be another, deeper overarching reason why 3-Colouring is polynomial-time solvable for P₅-free graphs.
Cite as

Daniël Paulusma, Johannes Rauch, and Erik Jan van Leeuwen. Colouring Probe H-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{paulusma_et_al:LIPIcs.STACS.2026.73,
  author =	{Paulusma, Dani\"{e}l and Rauch, Johannes and van Leeuwen, Erik Jan},
  title =	{{Colouring Probe H-Free Graphs}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{73:1--73:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.73},
  URN =		{urn:nbn:de:0030-drops-255621},
  doi =		{10.4230/LIPIcs.STACS.2026.73},
  annote =	{Keywords: colouring, probe graph, forbidden induced subgraph, complexity dichotomy}
}
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