152 Search Results for "Marx, Dániel"

Volume

LIPIcs, Volume 107

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

ICALP 2018, July 9-13, 2018, Prague, Czech Republic

Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella

Document
DOI: 10.4230/LIPIcs.ESA.2024.9
On Connections Between k-Coloring and Euclidean k-Means

Authors: Enver Aman, Karthik C. S., and Sharath Punna

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Euclidean k-means problems we are given as input a set of n points in ℝ^d and the goal is to find a set of k points C ⊆ ℝ^d, so as to minimize the sum of the squared Euclidean distances from each point in P to its closest center in C. In this paper, we formally explore connections between the k-coloring problem on graphs and the Euclidean k-means problem. Our results are as follows: - For all k ≥ 3, we provide a simple reduction from the k-coloring problem on regular graphs to the Euclidean k-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean 2-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. - In the other direction, we mimic the O(1.7297ⁿ) time algorithm of Williams [TCS'05] for the max-cut of problem on n vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in 2ⁿ⋅ poly(n,d) time. - We prove similar results and connections as above for the Euclidean k-min-sum problem.
Cite as

Enver Aman, Karthik C. S., and Sharath Punna. On Connections Between k-Coloring and Euclidean k-Means. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aman_et_al:LIPIcs.ESA.2024.9,
  author =	{Aman, Enver and Karthik C. S. and Punna, Sharath},
  title =	{{On Connections Between k-Coloring and Euclidean k-Means}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.9},
  URN =		{urn:nbn:de:0030-drops-210808},
  doi =		{10.4230/LIPIcs.ESA.2024.9},
  annote =	{Keywords: k-means, k-minsum, Euclidean space, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.10
Sparse Outerstring Graphs Have Logarithmic Treewidth

Authors: Shinwoo An, Eunjin Oh, and Jie Xue

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with n vertices has treewidth O(αlog n), where α denotes the arboricity of the graph, with an almost matching lower bound of Ω(α log (n/α)). As a corollary, we show that a t-biclique-free outerstring graph has treewidth O(t(log t)log n). This leads to polynomial-time algorithms for most of the central NP-complete problems such as Independent Set, Vertex Cover, Dominating Set, Feedback Vertex Set, Coloring for sparse outerstring graphs. Also, we can obtain subexponential-time (exact, parameterized, and approximation) algorithms for various NP-complete problems such as Vertex Cover, Feedback Vertex Set and Cycle Packing for (not necessarily sparse) outerstring graphs.
Cite as

Shinwoo An, Eunjin Oh, and Jie Xue. Sparse Outerstring Graphs Have Logarithmic Treewidth. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{an_et_al:LIPIcs.ESA.2024.10,
  author =	{An, Shinwoo and Oh, Eunjin and Xue, Jie},
  title =	{{Sparse Outerstring Graphs Have Logarithmic Treewidth}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.10},
  URN =		{urn:nbn:de:0030-drops-210816},
  doi =		{10.4230/LIPIcs.ESA.2024.10},
  annote =	{Keywords: Outerstring graphs, geometric intersection graphs, treewidth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.17
Cuts in Graphs with Matroid Constraints

Authors: Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.
Cite as

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Cuts in Graphs with Matroid Constraints. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{banik_et_al:LIPIcs.ESA.2024.17,
  author =	{Banik, Aritra and Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Jana, Satyabrata and Saurabh, Saket},
  title =	{{Cuts in Graphs with Matroid Constraints}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.17},
  URN =		{urn:nbn:de:0030-drops-210887},
  doi =		{10.4230/LIPIcs.ESA.2024.17},
  annote =	{Keywords: s-t-cut, multiway Cut, matroid, odd cycle transversal, feedback vertex set, fixed-parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.32
Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds

Authors: Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We provide several novel algorithms and lower bounds in central settings of mixed-integer (non-)linear optimization, shedding new light on classic results in the field. This includes an improvement on record running time bounds obtained from a slight extension of Lenstra’s 1983 algorithm [Math. Oper. Res. '83] to optimizing under few constraints with small coefficients. This is important for ubiquitous tasks like knapsack-, subset sum- or scheduling problems [Eisenbrand and Weismantel, SODA'18, Jansen and Rohwedder, ITCS'19]. Further, we extend our algorithm to an intermediate linear optimization problem when the matrix has many rows that exhibit 2-stage stochastic structure, which adds to a prominent line of recent results on this and similarly restricted cases [Jansen et al. ICALP'19, Cslovjecsek et al. SODA'21, Brand et al. AAAI'21, Klein, Reuter SODA'22, Cslovjecsek et al. SODA'24]. We also show that the generalization of two fundamental classes of structured constraints from these works (n-fold and 2-stage stochastic programs) to separable-convex mixed-integer optimization are harder than their mixed-integer, linear counterparts. This counters a widespread belief popularized initially by an influential paper of Hochbaum and Shanthikumar, namely that "convex separable optimization is not much harder than linear optimization" [J. ACM '90]. To obtain our algorithms, we employ the mixed Graver basis introduced by Hemmecke [Math. Prog. '03], and our work is the first to give bounds on the norm of its elements. Importantly, we use these bounds differently from how purely-integer Graver bounds are exploited in related approaches, and prove that, surprisingly, this cannot be avoided.
Cite as

Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak. Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{brand_et_al:LIPIcs.ESA.2024.32,
  author =	{Brand, Cornelius and Kouteck\'{y}, Martin and Lassota, Alexandra and Ordyniak, Sebastian},
  title =	{{Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.32},
  URN =		{urn:nbn:de:0030-drops-211033},
  doi =		{10.4230/LIPIcs.ESA.2024.32},
  annote =	{Keywords: Mixed-Integer Programming, Separable Convex Optimization, Parameterized Algorithms, Parameterized Complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.36
Bicriteria Approximation for Minimum Dilation Graph Augmentation

Authors: Kevin Buchin, Maike Buchin, Joachim Gudmundsson, and Sampson Wong

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget k, which k edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in k. Our main result is a (2 √[r]{2} k^{1/r},2r)-bicriteria approximation that runs in O(n³ log n) time, for all r ≥ 1. In other words, if t^* is the minimum dilation after adding any k edges to a graph, then our algorithm adds O(k^{1+1/r}) edges to the graph to obtain a dilation of 2rt^*. Moreover, our analysis of the algorithm is tight under the Erdős girth conjecture.
Cite as

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, and Sampson Wong. Bicriteria Approximation for Minimum Dilation Graph Augmentation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2024.36,
  author =	{Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Wong, Sampson},
  title =	{{Bicriteria Approximation for Minimum Dilation Graph Augmentation}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{36:1--36:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.36},
  URN =		{urn:nbn:de:0030-drops-211079},
  doi =		{10.4230/LIPIcs.ESA.2024.36},
  annote =	{Keywords: Greedy spanner, Graph augmentation}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.39
List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs

Authors: Barış Can Esmer, Jacob Focke, Dániel Marx, and Paweł Rzążewski

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. Given graphs G, H, and lists L(v) ⊆ V(H) for every v ∈ V(G), a list homomorphism from (G,L) to H is a function f:V(G) → V(H) that preserves the edges (i.e., uv ∈ E(G) implies f(u)f(v) ∈ E(H)) and respects the lists (i.e., f(v) ∈ L(v)). The graph H may have loops. For a fixed H, the input of the optimization problem LHomVD(H) is a graph G with lists L(v), and the task is to find a set X of vertices having minimum size such that (G-X,L) has a list homomorphism to H. We define analogously the edge-deletion variant LHomED(H), where we have to delete as few edges as possible from G to obtain a graph that has a list homomorphism. This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs H that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed H. Second, as our main result, we determine for every graph H for which the problem is NP-hard, the smallest possible constant c_H such that the problem can be solved in time c^t_H⋅ n^{𝒪(1)} if a tree decomposition of G having width t is given in the input. Let i(H) be the maximum size of a set of vertices in H that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD(H), we show that the smallest possible constant is i(H)+1 for every H: - Given a tree decomposition of width t of G, LHomVD(H) can be solved in time (i(H)+1)^t⋅ n^{𝒪(1)}. - For any ε > 0 and H, an (i(H)+1-ε)^t⋅ n^{𝒪(1)} algorithm would violate the Strong Exponential-Time Hypothesis (SETH). The situation is more complex for the edge-deletion version. For every H, one can solve LHomED(H) in time i(H)^t⋅ n^{𝒪(1)} if a tree decomposition of width t is given. However, the existence of a specific type of decomposition of H shows that there are graphs H where LHomED(H) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than i(H). Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed H.
Cite as

Barış Can Esmer, Jacob Focke, Dániel Marx, and Paweł Rzążewski. List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{canesmer_et_al:LIPIcs.ESA.2024.39,
  author =	{Can Esmer, Bar{\i}\c{s} and Focke, Jacob and Marx, D\'{a}niel and Rz\k{a}\.{z}ewski, Pawe{\l}},
  title =	{{List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.39},
  URN =		{urn:nbn:de:0030-drops-211103},
  doi =		{10.4230/LIPIcs.ESA.2024.39},
  annote =	{Keywords: Graph Homomorphism, List Homomorphism, Vertex Deletion, Edge Deletion, Multiway Cut, Parameterized Complexity, Tight Bounds, Treewidth, SETH}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.42
From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

Authors: Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. - We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. - We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k).
Cite as

Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chekuri_et_al:LIPIcs.ESA.2024.42,
  author =	{Chekuri, Chandra and Jain, Rhea and Kulkarni, Shubhang and Zheng, Da Wei and Zhu, Weihao},
  title =	{{From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{42:1--42:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.42},
  URN =		{urn:nbn:de:0030-drops-211134},
  doi =		{10.4230/LIPIcs.ESA.2024.42},
  annote =	{Keywords: Directed Planar Graphs, Submodular Functions, Steiner Tree, Network Design}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.43
Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

Authors: Jana Cslovjecsek, Michał Pilipczuk, and Karol Węgrzycki

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are given a weighted set of n axis-parallel rectangles in the plane. The task is to find a subset of pairwise non-overlapping rectangles with the maximum possible total weight. This problem is NP-hard and the best-known polynomial-time approximation algorithm, due to Chalermsook and Walczak [SODA 2021], achieves approximation factor 𝒪(log log n). While in the unweighted setting, constant factor approximation algorithms are known, due to Mitchell [FOCS 2021] and to Gálvez et al. [SODA 2022], it remains open to extend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterized approximation. Grandoni, Kratsch and Wiese [ESA 2019] gave a (1-ε)-approximation algorithm running in k^{𝒪(k/ε⁸)} n^{𝒪(1/ε⁸)} time, where k is the number of rectangles in an optimum solution. Unfortunately, their algorithm works only in the unweighted setting and they left it as an open problem to give a parameterized approximation scheme in the weighted setting. We give a parameterized approximation algorithm for MWISR that given a parameter k ∈ ℕ, finds a set of non-overlapping rectangles of weight at least (1-ε) opt_k in 2^{𝒪(k log(k/ε))} n^{𝒪(1/ε)} time, where opt_k is the maximum weight of a solution of cardinality at most k. We also propose a parameterized approximation scheme with running time 2^{𝒪(k² log(k/ε))} n^{𝒪(1)} that finds a solution with cardinality at most k and total weight at least (1-ε)opt_k for the special case of axis-parallel segments.
Cite as

Jana Cslovjecsek, Michał Pilipczuk, and Karol Węgrzycki. Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cslovjecsek_et_al:LIPIcs.ESA.2024.43,
  author =	{Cslovjecsek, Jana and Pilipczuk, Micha{\l} and W\k{e}grzycki, Karol},
  title =	{{Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.43},
  URN =		{urn:nbn:de:0030-drops-211146},
  doi =		{10.4230/LIPIcs.ESA.2024.43},
  annote =	{Keywords: parameterized approximation, Maximum Weight Independent Set, rectangles, segments}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.49
Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs

Authors: Sally Dong and Guanghao Ye

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We present an algorithm for min-cost flow in graphs with n vertices and m edges, given a tree decomposition of width τ and size S, and polynomially bounded, integral edge capacities and costs, running in Õ(m√{τ} + S) time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver of [Gu and Song, 2022; Dong et al., 2024], which runs in Õ(m τ^{(ω+1)/2}) time, where ω ≈ 2.37 is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). In general graphs where treewidth is trivially bounded by n, the algorithm runs in Õ(m √ n) time, which is the best-known result without using the Lee-Sidford barrier or 𝓁₁ IPM, demonstrating the surprising power of robust interior point methods. As a corollary, we obtain a Õ(tw³ ⋅ m) time algorithm to compute a tree decomposition of width O(tw⋅ log(n)), given a graph with m edges.
Cite as

Sally Dong and Guanghao Ye. Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dong_et_al:LIPIcs.ESA.2024.49,
  author =	{Dong, Sally and Ye, Guanghao},
  title =	{{Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.49},
  URN =		{urn:nbn:de:0030-drops-211207},
  doi =		{10.4230/LIPIcs.ESA.2024.49},
  annote =	{Keywords: Min-cost flow, tree decomposition, interior point method, bounded treewidth graphs}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.51
Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs

Authors: Lech Duraj, Filip Konieczny, and Krzysztof Potępa

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ polylog(n))} time complexity for the graph on n vertices and m edges, where k is the diameter and d is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an 𝒪{(n^{7/4} ⋅ polylog(n))} parameterized diameter algorithm for unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs.
Cite as

Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{duraj_et_al:LIPIcs.ESA.2024.51,
  author =	{Duraj, Lech and Konieczny, Filip and Pot\k{e}pa, Krzysztof},
  title =	{{Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.51},
  URN =		{urn:nbn:de:0030-drops-211229},
  doi =		{10.4230/LIPIcs.ESA.2024.51},
  annote =	{Keywords: Graph Diameter, Geometric Intersection Graphs, Vapnik-Chervonenkis Dimension}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.55
Hitting Meets Packing: How Hard Can It Be?

Authors: Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of 𝒳-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing 𝓁 vertex-disjoint objects of type 𝒳? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination 𝒳-HitPack can be significantly harder than these two base problems. Already for one particular choice of 𝒳, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain at the intersection of the areas of hitting and packing problems. At a high level, we present two case studies: (1) 𝒳 being all cycles, and (2) 𝒳 being all copies of a fixed graph H. In each, we explore the classical complexity as well as the parameterized complexity with the natural parameters k+𝓁 and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph on at least 3 vertices, the problem is Σ₂^𝖯-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For 𝒳 being all cycles, we establish a 2^{poly(k+𝓁)}⋅ n^{𝒪(1)} algorithm using an involved branching method, for example. Also, for 𝒳 being all edges (i.e., H = K₂; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^{poly(tw)}⋅ n^{𝒪(1)} on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.
Cite as

Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki. Hitting Meets Packing: How Hard Can It Be?. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 55:1-55:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{focke_et_al:LIPIcs.ESA.2024.55,
  author =	{Focke, Jacob and Frei, Fabian and Li, Shaohua and Marx, D\'{a}niel and Schepper, Philipp and Sharma, Roohani and W\k{e}grzycki, Karol},
  title =	{{Hitting Meets Packing: How Hard Can It Be?}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{55:1--55:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.55},
  URN =		{urn:nbn:de:0030-drops-211261},
  doi =		{10.4230/LIPIcs.ESA.2024.55},
  annote =	{Keywords: Hitting, Packing, Covering, Parameterized Algorithms, Lower Bounds, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.57
Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices

Authors: Dvir Fried, Tsvi Kopelowitz, and Ely Porat

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We revisit the problem of multiplying two square matrices over the (min, +) semi-ring, where all entries are integers from a bounded range [-M : M] ∪ {∞}. The current state of the art for this problem is a simple O(M n^{ω} log M) time algorithm by Alon, Galil and Margalit [JCSS'97], where ω is the exponent in the runtime of the fastest matrix multiplication (FMM) algorithm. We design a new simple algorithm whose runtime is O(M n^ω + M n² log M), thereby removing the logM factor in the runtime if ω > 2 or if n^ω = Ω (n²log n).
Cite as

Dvir Fried, Tsvi Kopelowitz, and Ely Porat. Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 57:1-57:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fried_et_al:LIPIcs.ESA.2024.57,
  author =	{Fried, Dvir and Kopelowitz, Tsvi and Porat, Ely},
  title =	{{Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{57:1--57:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.57},
  URN =		{urn:nbn:de:0030-drops-211283},
  doi =		{10.4230/LIPIcs.ESA.2024.57},
  annote =	{Keywords: FMM, (min , +)-product, FFT}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.58
Random-Order Online Independent Set of Intervals and Hyperrectangles

Authors: Mohit Garg, Debajyoti Kar, and Arindam Khan

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Maximum Independent Set of Hyperrectangles problem, we are given a set of n (possibly overlapping) d-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For d = 1, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for d-dimensional hyperrectangles, polynomial time (log n)^{O(d)}-approximation algorithms are known [Chalermsook and Chuzhoy, 2009]. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an Ω(n) lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis (see the survey by Gupta and Singla [Anupam Gupta and Sahil Singla, 2020]). Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple (log n)^{O(d)}-competitive algorithm for d-dimensional hyperrectangles in this model, which runs in O_d̃(n) time. Our approach also yields (log n)^{O(d)}-competitive algorithms in the random-order model for more general objects such as d-dimensional fat objects and ellipsoids. Furthermore, all our competitiveness guarantees hold with high probability, and not just in expectation.
Cite as

Mohit Garg, Debajyoti Kar, and Arindam Khan. Random-Order Online Independent Set of Intervals and Hyperrectangles. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{garg_et_al:LIPIcs.ESA.2024.58,
  author =	{Garg, Mohit and Kar, Debajyoti and Khan, Arindam},
  title =	{{Random-Order Online Independent Set of Intervals and Hyperrectangles}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{58:1--58:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.58},
  URN =		{urn:nbn:de:0030-drops-211298},
  doi =		{10.4230/LIPIcs.ESA.2024.58},
  annote =	{Keywords: Online Algorithms, Random-Order Model, Maximum Independent Set of Rectangles, Hyperrectangles, Fat Objects, Interval Scheduling}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.76
Steiner Tree Parameterized by Multiway Cut and Even Less

Authors: Bart M.P. Jansen and Céline M.F. Swennenhuis

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Steiner Tree problem we are given an undirected edge-weighted graph as input, along with a set K of vertices called terminals. The task is to output a minimum-weight connected subgraph that spans all the terminals. The famous Dreyfus-Wagner algorithm running in 3^{|K|}poly(n) time shows that the problem is fixed-parameter tractable parameterized by the number of terminals. We present fixed-parameter tractable algorithms for Steiner Tree using structurally smaller parameterizations. Our first result concerns the parameterization by a multiway cut S of the terminals, which is a vertex set S (possibly containing terminals) such that each connected component of G-S contains at most one terminal. We show that Steiner Tree can be solved in 2^{𝒪(|S|log|S|)}poly(n) time and polynomial space, where S is a minimum multiway cut for K. The algorithm is based on the insight that, after guessing how an optimal Steiner tree interacts with a multiway cut S, computing a minimum-cost solution of this type can be formulated as minimum-cost bipartite matching. Our second result concerns a new hybrid parameterization called K-free treewidth that simultaneously refines the number of terminals |K| and the treewidth of the input graph. By utilizing recent work on ℋ-Treewidth in order to find a corresponding decomposition of the graph, we give an algorithm that solves Steiner Tree in time 2^{𝒪(k)} poly(n), where k denotes the K-free treewidth of the input graph. To obtain this running time, we show how the rank-based approach for solving Steiner Tree parameterized by treewidth can be extended to work in the setting of K-free treewidth, by exploiting existing algorithms parameterized by |K| to compute the table entries of leaf bags of a tree K-free decomposition.
Cite as

Bart M.P. Jansen and Céline M.F. Swennenhuis. Steiner Tree Parameterized by Multiway Cut and Even Less. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2024.76,
  author =	{Jansen, Bart M.P. and Swennenhuis, C\'{e}line M.F.},
  title =	{{Steiner Tree Parameterized by Multiway Cut and Even Less}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{76:1--76:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.76},
  URN =		{urn:nbn:de:0030-drops-211471},
  doi =		{10.4230/LIPIcs.ESA.2024.76},
  annote =	{Keywords: fixed-parameter tractability, Steiner Tree, structural parameterization, H-treewidth}
}
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