Volume
LIPIcs, Volume 294
19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Scandinavian Symposium and Workshops on Algorithm Theory (SWAT)
Event
SWAT 2024, June 12-14, 2024, Helsinki, FinlandEditor
Publication Details
- published at: 2024-05-31
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-318-8
- DBLP: db/conf/swat/swat2024
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Document
Complete Volume
LIPIcs, Volume 294, SWAT 2024, Complete Volume
Abstract
LIPIcs, Volume 294, SWAT 2024, Complete Volume
Cite as
19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1-686, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@Proceedings{bodlaender:LIPIcs.SWAT.2024,
title = {{LIPIcs, Volume 294, SWAT 2024, Complete Volume}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {1--686},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024},
URN = {urn:nbn:de:0030-drops-200397},
doi = {10.4230/LIPIcs.SWAT.2024},
annote = {Keywords: LIPIcs, Volume 294, SWAT 2024, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bodlaender:LIPIcs.SWAT.2024.0,
author = {Bodlaender, Hans L.},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.0},
URN = {urn:nbn:de:0030-drops-200400},
doi = {10.4230/LIPIcs.SWAT.2024.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Eliminating Crossings in Ordered Graphs
Abstract
Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number.
We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.
Cite as
Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2024.1,
author = {Agrawal, Akanksha and Cabello, Sergio and Kaufmann, Michael and Saurabh, Saket and Sharma, Roohani and Uno, Yushi and Wolff, Alexander},
title = {{Eliminating Crossings in Ordered Graphs}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {1:1--1:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.1},
URN = {urn:nbn:de:0030-drops-200417},
doi = {10.4230/LIPIcs.SWAT.2024.1},
annote = {Keywords: Ordered graphs, book embedding, edge deletion, d-planar, hitting set}
}
Document
Local Spanners Revisited
Abstract
For a set P ⊆ ℝ² of points and a family ℱ of regions, a local t-spanner of P is a sparse graph G over P, such that for any region r ∈ ℱ the subgraph restricted to r, denoted by G ∩ r, is a t-spanner for all the points of r ∩ P.
We present algorithms for the construction of local spanners with respect to several families of regions such as homothets of a convex region. Unfortunately, the number of edges in the resulting graph depends logarithmically on the spread of the input point set. We prove that this dependency cannot be removed, thus settling an open problem raised by Abam and Borouny. We also show improved constructions (with no dependency on the spread) of local spanners for fat triangles, and regular k-gons. In particular, this improves over the known construction for axis-parallel squares.
We also study notions of weaker local spanners where one is allowed to shrink the region a "bit". Surprisingly, we show a near linear-size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is multiplicative. Any spanner is a weak local spanner if the shrinking is proportional to the diameter of the region.
Cite as
Stav Ashur and Sariel Har-Peled. Local Spanners Revisited. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ashur_et_al:LIPIcs.SWAT.2024.2,
author = {Ashur, Stav and Har-Peled, Sariel},
title = {{Local Spanners Revisited}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {2:1--2:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.2},
URN = {urn:nbn:de:0030-drops-200420},
doi = {10.4230/LIPIcs.SWAT.2024.2},
annote = {Keywords: Geometric graphs, Fault-tolerant spanners}
}
Document
Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model
Abstract
Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is #P-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length 2 or paths of length 1. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling.
Cite as
Lora Bailey, Heather Smith Blake, Garner Cochran, Nathan Fox, Michael Levet, Reem Mahmoud, Inne Singgih, Grace Stadnyk, and Alexander Wiedemann. Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bailey_et_al:LIPIcs.SWAT.2024.3,
author = {Bailey, Lora and Blake, Heather Smith and Cochran, Garner and Fox, Nathan and Levet, Michael and Mahmoud, Reem and Singgih, Inne and Stadnyk, Grace and Wiedemann, Alexander},
title = {{Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {3:1--3:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.3},
URN = {urn:nbn:de:0030-drops-200436},
doi = {10.4230/LIPIcs.SWAT.2024.3},
annote = {Keywords: Genome Rearrangement, Phylogenetics, Single Cut-and-Join, Computational Complexity}
}
Document
Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage
Abstract
Chordal graphs is a well-studied large graph class that is also a strict super-class of path graphs. Munro and Wu (ISAAC 2018) have given an (n²/4+o(n²))-bit succinct representation for n-vertex unlabeled chordal graphs. A chordal graph G = (V,E) is the intersection graph of sub-trees of a tree T. Based on this characterization, the two parameters of chordal graphs which we consider in this work are leafage, introduced by Lin, McKee and West (Discussiones Mathematicae Graph Theory 1998) and vertex leafage, introduced by Chaplick and Stacho (Discret. Appl. Math. 2014). Leafage is the minimum number of leaves in any possible tree T characterizing G. Let L(u) denote the number of leaves of the sub-tree in T corresponding to u ∈ V and k = max_{u ∈ V} L(u). The smallest k for which there exists a tree T for G is called its vertex leafage.
In this work, we improve the worst-case information theoretic lower bound of Munro and Wu (ISAAC 2018) for n-vertex unlabeled chordal graphs when vertex leafage is bounded and leafage is unbounded. The class of unlabeled k-vertex leafage chordal graphs that consists of all chordal graphs with vertex leafage at most k and unbounded leafage, denoted 𝒢_k, is introduced for the first time. For k > 0 in o(n^c), c > 0, we obtain a lower bound of ((k-1)n log n -kn log k - O(log n))-bits on the size of any data structure that encodes a graph in 𝒢_k. Further, for every k-vertex leafage chordal graph G and k > 1 in o(n^c), c > 0, we present a ((k-1)n log n + o(kn log n))-bit succinct data structure, constructed using the succinct data structure for path graphs with (k-1)n vertices. Our data structure supports adjacency query in O(k log n) time and using additional 2n log n bits, an O(k² d_v log n + log² n) time neighbourhood query where d_v is degree of v ∈ V.
Cite as
Girish Balakrishnan, Sankardeep Chakraborty, N. S. Narayanaswamy, and Kunihiko Sadakane. Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{balakrishnan_et_al:LIPIcs.SWAT.2024.4,
author = {Balakrishnan, Girish and Chakraborty, Sankardeep and Narayanaswamy, N. S. and Sadakane, Kunihiko},
title = {{Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {4:1--4:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.4},
URN = {urn:nbn:de:0030-drops-200446},
doi = {10.4230/LIPIcs.SWAT.2024.4},
annote = {Keywords: succinct data structure, chordal graphs, leafage, vertex leafage, path graphs}
}
Document
Recognition and Proper Coloring of Unit Segment Intersection Graphs
Abstract
In this work, we concern ourselves with the fine-grained complexity of recognition and proper coloring problems on highly restricted classes of geometric intersection graphs of "thin" objects (i.e., objects with unbounded aspect ratios). As a point of motivation, we remark that there has been significant interest in finding algorithmic lower bounds for classic decision and optimization problems on these types of graphs, as they appear to escape the net of known planar or geometric separator theorems for "fat" objects (i.e., objects with bounded aspect ratios). In particular, letting n be the order of a geometric intersection graph, and assuming a geometric ply bound, per what is known as the "square root phenomenon", these separator theorems often imply the existence of 𝒪(2^√n) algorithms for problems ranging from finding proper colorings to finding Hamiltonian cycles. However, in contrast, it is known for instance that no 2^o(n) time algorithm can exist under the Exponential Time Hypothesis (ETH) for proper 6-coloring intersection graphs of line segments embedded in the plane (Biró et. al.; J. Comput. Geom. 9(2); pp. 47-80; 2018).
We begin by establishing algorithmic lower bounds for proper k-coloring and recognition problems of intersection graphs of line segments embedded in the plane under the most stringent constraints possible that allow either problem to be non-trivial. In particular, we consider the class UNIT-PURE-k-DIR of unit segment geometric intersection graphs, in which segments are constrained to lie in at most k directions in the plane, and no two parallel segments are permitted to intersect. Here, under the ETH, we show for every k ≥ 3 that no 2^o(√{n/k}) time algorithm can exist for either recognizing or proper k-coloring UNIT-PURE-k-DIR graphs of order n. In addition, for every k ≥ 4, we establish the same algorithmic lower bound under the ETH for the problem of proper (k-1)-coloring UNIT-PURE-k-DIR graphs when provided a list of segment coordinates specified using 𝒪(n ⋅ k) bits witnessing graph class membership. As a consequence of our approach, we are also able to show that the problem of properly 3-coloring an arbitrary graph on m edges can be reduced in 𝒪(m²) time to the problem of properly (k-1)-coloring a UNIT-PURE-k-DIR graph. Finally, we consider a slightly less constrained class of geometric intersection graphs of lines (of unbounded length) in which line-line intersections must occur on any one of (r = 3) parallel planes in ℝ³. In this context, for every k ≥ 3, we show that no 2^o(n/k) time algorithm can exist for proper k-coloring these graphs unless the ETH is false.
Cite as
Robert D. Barish and Tetsuo Shibuya. Recognition and Proper Coloring of Unit Segment Intersection Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{barish_et_al:LIPIcs.SWAT.2024.5,
author = {Barish, Robert D. and Shibuya, Tetsuo},
title = {{Recognition and Proper Coloring of Unit Segment Intersection Graphs}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {5:1--5:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.5},
URN = {urn:nbn:de:0030-drops-200452},
doi = {10.4230/LIPIcs.SWAT.2024.5},
annote = {Keywords: graph class recognition, proper coloring, geometric intersection graph, segment intersection graph, fine-grained complexity, Exponential Time Hypothesis}
}
Document
Destroying Densest Subgraphs Is Hard
Abstract
We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph G, a budget k and a target density τ_ρ, are there k edges (k vertices) whose removal from G results in a graph where the densest subgraph has density at most τ_ρ? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that Bounded-Density Edge Deletion is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.
Cite as
Cristina Bazgan, André Nichterlein, and Sofia Vazquez Alferez. Destroying Densest Subgraphs Is Hard. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bazgan_et_al:LIPIcs.SWAT.2024.6,
author = {Bazgan, Cristina and Nichterlein, Andr\'{e} and Vazquez Alferez, Sofia},
title = {{Destroying Densest Subgraphs Is Hard}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.6},
URN = {urn:nbn:de:0030-drops-200461},
doi = {10.4230/LIPIcs.SWAT.2024.6},
annote = {Keywords: Graph modification problems, NP-hardness, fixed-parameter tractability, W-hardness, special graph classes}
}
Document
The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs
Abstract
We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined as the smallest number d of labels such that the graph admits a d-simultaneous interval representation, that is, an assignment of intervals and label sets to the vertices such that two vertices are adjacent if and only if the corresponding intervals, as well as their label sets, intersect. We show that this parameter is NP-hard to compute and give several bounds for the parameter, showing in particular that it is sandwiched between pathwidth and linear mim-width. For classes of graphs with bounded parameter values, assuming that the graph is equipped with a simultaneous interval representation with a constant number of labels, we give FPT algorithms for the clique, independent set, and dominating set problems, and hardness results for the independent dominating set and coloring problems. The FPT results for independent set and dominating set are for the simultaneous interval number plus solution size. In contrast, both problems are known to be 𝖶[1]-hard for linear mim-width plus solution size.
Cite as
Jesse Beisegel, Nina Chiarelli, Ekkehard Köhler, Martin Milanič, Peter Muršič, and Robert Scheffler. The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{beisegel_et_al:LIPIcs.SWAT.2024.7,
author = {Beisegel, Jesse and Chiarelli, Nina and K\"{o}hler, Ekkehard and Milani\v{c}, Martin and Mur\v{s}i\v{c}, Peter and Scheffler, Robert},
title = {{The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {7:1--7:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.7},
URN = {urn:nbn:de:0030-drops-200470},
doi = {10.4230/LIPIcs.SWAT.2024.7},
annote = {Keywords: Interval graph, simultaneous representation, width parameter, algorithm, parameterized complexity}
}
Document
Correlation Clustering with Vertex Splitting
Abstract
We explore CLUSTER EDITING and its generalization CORRELATION CLUSTERING with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both problems are NP-hard, yet they exhibit significant differences in terms of parameterized complexity and approximability. For CLUSTER EDITING WITH PERMISSIVE VERTEX SPLITTING, we show a polynomial kernel when parameterized by the solution size and develop a polynomial-time 7-approximation. In the case of CORRELATION CLUSTERING, we establish para-NP-hardness when parameterized by the solution size and demonstrate that computing an n^{1-ε}-approximation is NP-hard for any constant ε > 0. Additionally, we extend an established link between CORRELATION CLUSTERING and MULTICUT to the setting with permissive vertex splits.
Cite as
Matthias Bentert, Alex Crane, Pål Grønås Drange, Felix Reidl, and Blair D. Sullivan. Correlation Clustering with Vertex Splitting. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bentert_et_al:LIPIcs.SWAT.2024.8,
author = {Bentert, Matthias and Crane, Alex and Drange, P\r{a}l Gr{\o}n\r{a}s and Reidl, Felix and Sullivan, Blair D.},
title = {{Correlation Clustering with Vertex Splitting}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {8:1--8:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.8},
URN = {urn:nbn:de:0030-drops-200483},
doi = {10.4230/LIPIcs.SWAT.2024.8},
annote = {Keywords: graph modification, cluster editing, overlapping clustering, approximation, parameterized complexity}
}
Document
Daisy Bloom Filters
Abstract
A filter is a widely used data structure for storing an approximation of a given set S of elements from some universe 𝒰 (a countable set). It represents a superset S' ⊇ S that is "close to S" in the sense that for x ∉ S, the probability that x ∈ S' is bounded by some ε > 0. The advantage of using a Bloom filter, when some false positives are acceptable, is that the space usage becomes smaller than what is required to store S exactly.
Though filters are well-understood from a worst-case perspective, it is clear that state-of-the-art constructions may not be close to optimal for particular distributions of data and queries. Suppose, for instance, that some elements are in S with probability close to 1. Then it would make sense to always include them in S', saving space by not having to represent these elements in the filter. Questions like this have been raised in the context of Weighted Bloom filters (Bruck, Gao and Jiang, ISIT 2006) and Bloom filter implementations that make use of access to learned components (Vaidya, Knorr, Mitzenmacher, and Krask, ICLR 2021).
In this paper, we present a lower bound for the expected space that such a filter requires. We also show that the lower bound is asymptotically tight by exhibiting a filter construction that executes queries and insertions in worst-case constant time, and has a false positive rate at most ε with high probability over input sets drawn from a product distribution. We also present a Bloom filter alternative, which we call the Daisy Bloom filter, that executes operations faster and uses significantly less space than the standard Bloom filter.
Cite as
Ioana O. Bercea, Jakob Bæk Tejs Houen, and Rasmus Pagh. Daisy Bloom Filters. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bercea_et_al:LIPIcs.SWAT.2024.9,
author = {Bercea, Ioana O. and Houen, Jakob B{\ae}k Tejs and Pagh, Rasmus},
title = {{Daisy Bloom Filters}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {9:1--9:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.9},
URN = {urn:nbn:de:0030-drops-200491},
doi = {10.4230/LIPIcs.SWAT.2024.9},
annote = {Keywords: Bloom filters, input distribution, learned data structures}
}
Document
Online Bin Covering with Frequency Predictions
Abstract
We study the bin covering problem where a multiset of items from a fixed set S ⊆ (0,1] must be split into disjoint subsets while maximizing the number of subsets whose contents sum to at least 1. We focus on the online discrete variant, where S is finite, and items arrive sequentially. In the purely online setting, we show that the competitive ratios of best deterministic (and randomized) algorithms converge to 1/2 for large S, similar to the continuous setting. Therefore, we consider the problem under the prediction setting, where algorithms may access a vector of frequencies predicting the frequency of items of each size in the instance. In this setting, we introduce a family of online algorithms that perform near-optimally when the predictions are correct. Further, we introduce a second family of more robust algorithms that presents a tradeoff between the performance guarantees when the predictions are perfect and when predictions are adversarial. Finally, we consider a stochastic setting where items are drawn independently from any fixed but unknown distribution of S. Using results from the PAC-learnability of probabilities in discrete distributions, we introduce a purely online algorithm whose average-case performance is near-optimal with high probability for all finite sets S and all distributions of S.
Cite as
Magnus Berg and Shahin Kamali. Online Bin Covering with Frequency Predictions. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{berg_et_al:LIPIcs.SWAT.2024.10,
author = {Berg, Magnus and Kamali, Shahin},
title = {{Online Bin Covering with Frequency Predictions}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.10},
URN = {urn:nbn:de:0030-drops-200504},
doi = {10.4230/LIPIcs.SWAT.2024.10},
annote = {Keywords: Bin Covering, Online Algorithms with Predictions, PAC Learning, Learning-Augmented Algorithms}
}
Document
Subexponential Algorithms in Geometric Graphs via the Subquadratic Grid Minor Property: The Role of Local Radius
Abstract
We investigate the existence in geometric graph classes of subexponential parameterized algorithms for cycle-hitting problems like Triangle Hitting (TH), Feedback Vertex Set (FVS) or Odd Cycle Transversal (OCT). These problems respectively ask for the existence in a graph G of a set X of at most k vertices such that G-X is triangle-free, acyclic, or bipartite. It is know that subexponential FPT algorithms of the form 2^o(k)n^𝒪(1) exist in planar and even H-minor free graphs from bidimensionality theory [Demaine et al. 2005], and there is a recent line of work lifting these results to geometric graph classes consisting of intersection of similarly sized "fat" objects ([Fomin et al. 2012], [Grigoriev et al. 2014], or disk graphs [Lokshtanov et al. 2022], [An et al. 2023]).
In this paper we first identify sufficient conditions, for any graph class 𝒞 included in string graphs, to admit subexponential FPT algorithms for any problem in 𝒫, a family of bidimensional problems where one has to find a set of size at most k hitting a fixed family of graphs, containing in particular FVS. Informally, these conditions boil down to the fact that for any G ∈ 𝒞, the local radius of G (a new parameter introduced in [Lokshtanov et al. 2023]) is polynomial in the clique number of G and in the maximum matching in the neighborhood of a vertex. To demonstrate the applicability of this generic result, we bound the local radius for two special classes: intersection graphs of axis-parallel squares and of contact graphs of segments in the plane. This implies that any problem Π ∈ 𝒫 (in particular, FVS) can be solved in:
- 2^𝒪(k^{3/4}log k) n^𝒪(1)-time in contact segment graphs,
- 2^𝒪(k^{9/10}log k) n^𝒪(1) in intersection graphs of axis-parallel squares On the positive side, we also provide positive results for TH by solving it in:
- 2^𝒪(k^{3/4}log k) n^𝒪(1)-time in contact segment graphs,
- 2^𝒪(√dt²(log t)k^{2/3}log k) n^𝒪(1)-time in K_{t,t}-free d-DIR graphs (intersection of segments with d slopes)
On the negative side, assuming the ETH we rule out the existence of algorithms solving:
- TH and OCT in time 2^o(n) in 2-DIR graphs and more generally in time 2^o(√{Δn}) in 2-DIR graphs with maximum degree Δ, and
- TH, FVS, and OCT in time 2^o(√n) in K_{2,2}-free contact-2-DIR graphs of maximum degree 6.
Observe that together, these results show that the absence of large K_{t,t} is a necessary and sufficient condition for the existence of subexponential FPT algorithms for TH in 2-DIR.
Cite as
Gaétan Berthe, Marin Bougeret, Daniel Gonçalves, and Jean-Florent Raymond. Subexponential Algorithms in Geometric Graphs via the Subquadratic Grid Minor Property: The Role of Local Radius. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{berthe_et_al:LIPIcs.SWAT.2024.11,
author = {Berthe, Ga\'{e}tan and Bougeret, Marin and Gon\c{c}alves, Daniel and Raymond, Jean-Florent},
title = {{Subexponential Algorithms in Geometric Graphs via the Subquadratic Grid Minor Property: The Role of Local Radius}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.11},
URN = {urn:nbn:de:0030-drops-200519},
doi = {10.4230/LIPIcs.SWAT.2024.11},
annote = {Keywords: geometric intersection graphs, subexponential FPT algorithms, cycle-hitting problems, bidimensionality}
}
Document
Arboricity-Dependent Algorithms for Edge Coloring
Abstract
The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph.
Currently, it is not known whether it is possible to maintain a (Δ + O(Δ^(1-μ)))-edge coloring in Õ(1) update time, for any constant μ > 0, where Δ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a (Δ + O(α))-edge coloring in Õ(1) amortized update time, where α is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.
Cite as
Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Arboricity-Dependent Algorithms for Edge Coloring. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bhattacharya_et_al:LIPIcs.SWAT.2024.12,
author = {Bhattacharya, Sayan and Costa, Mart{\'\i}n and Panski, Nadav and Solomon, Shay},
title = {{Arboricity-Dependent Algorithms for Edge Coloring}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {12:1--12:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.12},
URN = {urn:nbn:de:0030-drops-200524},
doi = {10.4230/LIPIcs.SWAT.2024.12},
annote = {Keywords: Dynamic Algorithms, Graph Algorithms, Edge Coloring, Arboricity}
}
Document
On the Independence Number of 1-Planar Graphs
Abstract
An independent set in a graph is a set of vertices where no two vertices are adjacent to each other. A maximum independent set is the largest possible independent set that can be formed within a given graph G. The cardinality of this set is referred to as the independence number of G. This paper investigates the independence number of 1-planar graphs, a subclass of graphs defined by drawings in the Euclidean plane where each edge can have at most one crossing point. Borodin establishes a tight upper bound of six for the chromatic number of every 1-planar graph G, leading to a corresponding lower bound of n/6 for the independence number, where n is the number of vertices of G. In contrast, the upper bound for the independence number in 1-planar graphs is less studied. This paper addresses this gap by presenting upper bounds based on the minimum degree δ. A comprehensive table summarizes these upper bounds for various δ values, providing insights into achievable independence numbers under different conditions.
Cite as
Therese Biedl, Prosenjit Bose, and Babak Miraftab. On the Independence Number of 1-Planar Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{biedl_et_al:LIPIcs.SWAT.2024.13,
author = {Biedl, Therese and Bose, Prosenjit and Miraftab, Babak},
title = {{On the Independence Number of 1-Planar Graphs}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.13},
URN = {urn:nbn:de:0030-drops-200537},
doi = {10.4230/LIPIcs.SWAT.2024.13},
annote = {Keywords: 1-planar graph, independent set, minimum degree}
}
Document
Size-Constrained Weighted Ancestors with Applications
Abstract
The weighted ancestor problem on a rooted node-weighted tree T is a generalization of the classic predecessor problem: construct a data structure for a set of integers that supports fast predecessor queries. Both problems are known to require Ω(log log n) time for queries provided 𝒪(n poly log n) space is available, where n is the input size. The weighted ancestor problem has attracted a lot of attention by the combinatorial pattern matching community due to its direct application to suffix trees. In this formulation of the problem, the nodes are weighted by string depth. This research has culminated in a data structure for weighted ancestors in suffix trees with 𝒪(1) query time and an 𝒪(n)-time construction algorithm [Belazzougui et al., CPM 2021].
In this paper, we consider a different version of the weighted ancestor problem, where the nodes are weighted by any function weight that maps each node of T to a positive integer, such that weight(u) ≤ size(u) for any node u and weight(u₁) ≤ weight(u₂) if node u₁ is a descendant of node u₂, where size(u) is the number of nodes in the subtree rooted at u. In the size-constrained weighted ancestor (SWA) problem, for any node u of T and any integer k, we are asked to return the lowest ancestor w of u with weight at least k. We show that for any rooted tree with n nodes, we can locate node w in 𝒪(1) time after 𝒪(n)-time preprocessing. In particular, this implies a data structure for the SWA problem in suffix trees with 𝒪(1) query time and 𝒪(n)-time preprocessing, when the nodes are weighted by weight. We also show several string-processing applications of this result.
Cite as
Philip Bille, Yakov Nekrich, and Solon P. Pissis. Size-Constrained Weighted Ancestors with Applications. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bille_et_al:LIPIcs.SWAT.2024.14,
author = {Bille, Philip and Nekrich, Yakov and Pissis, Solon P.},
title = {{Size-Constrained Weighted Ancestors with Applications}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {14:1--14:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.14},
URN = {urn:nbn:de:0030-drops-200544},
doi = {10.4230/LIPIcs.SWAT.2024.14},
annote = {Keywords: weighted ancestors, string indexing, data structures}
}
Document
Range Reporting for Time Series via Rectangle Stabbing
Abstract
We study the Fréchet queries problem. It is a data structure problem for range reporting, where we are given a set S of n polygonal curves and a distance threshold ρ. The data structure should support queries with a polygonal curve q for the elements of S, for which the continuous Fréchet distance to q is at most ρ. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves of constant complexity and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point q should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with size in 𝒪(n log^{t-1} n) and query time in 𝒪(log^{t-1} n+k), where k denotes the output size and t can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. Note that we omit factors depending on the complexity of the curves that do not depend on n. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter d = ⌊t/2⌋ can be applied to our problem via reduction.
Cite as
Lotte Blank and Anne Driemel. Range Reporting for Time Series via Rectangle Stabbing. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{blank_et_al:LIPIcs.SWAT.2024.15,
author = {Blank, Lotte and Driemel, Anne},
title = {{Range Reporting for Time Series via Rectangle Stabbing}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {15:1--15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.15},
URN = {urn:nbn:de:0030-drops-200559},
doi = {10.4230/LIPIcs.SWAT.2024.15},
annote = {Keywords: Data Structures, Fr\'{e}chet distance, Rectangle Stabbing, Orthogonal Range Searching}
}
Document
On the Online Weighted Non-Crossing Matching Problem
Abstract
We introduce and study the weighted version of an online matching problem in the Euclidean plane with non-crossing constraints: 2n points with non-negative weights arrive online, and an algorithm can match an arriving point to one of the unmatched previously arrived points. In the vanilla model, the decision on how to match (if at all) a newly arriving point is irrevocable. The goal is to maximize the total weight of matched points under the constraint that straight-line segments corresponding to the edges of the matching do not intersect. The unweighted version of the problem was introduced in the offline setting by Atallah in 1985, and this problem became a subject of study in the online setting with and without advice in several recent papers.
We observe that deterministic online algorithms cannot guarantee a non-trivial competitive ratio for the weighted problem. We study various regimes of the problem which permit non-trivial online algorithms. In particular, when weights are restricted to the interval [1, U] we give a deterministic algorithm achieving competitive ratio Ω(2^{-2√{log U}}). We also prove that deterministic online algorithms cannot achieve competitive ratio better than O (2^{-√{log U}}). Interestingly, we establish that randomization alone suffices to achieve competitive ratio 1/3 even when there are no restrictions on the weights. Additionally, if one allows an online algorithm to revoke acceptances, then one can achieve a competitive ratio ≈ 0.2862 deterministically for arbitrary weights. We also establish a lower bound on the competitive ratio of randomized algorithms in the unweighted setting, and improve the best-known bound on advice complexity to achieve a perfect matching.
Cite as
Joan Boyar, Shahin Kamali, Kim S. Larsen, Ali Mohammad Lavasani, Yaqiao Li, and Denis Pankratov. On the Online Weighted Non-Crossing Matching Problem. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{boyar_et_al:LIPIcs.SWAT.2024.16,
author = {Boyar, Joan and Kamali, Shahin and Larsen, Kim S. and Lavasani, Ali Mohammad and Li, Yaqiao and Pankratov, Denis},
title = {{On the Online Weighted Non-Crossing Matching Problem}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {16:1--16:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.16},
URN = {urn:nbn:de:0030-drops-200567},
doi = {10.4230/LIPIcs.SWAT.2024.16},
annote = {Keywords: Online algorithms, weighted matching problem, Euclidean plane, non-crossing constraints, competitive analysis, randomized online algorithms, online algorithms with advice, online algorithms with revoking}
}
Document
Deterministic Cache-Oblivious Funnelselect
Abstract
In the multiple-selection problem one is given an unsorted array S of N elements and an array of q query ranks r_1 < ⋯ < r_q, and the task is to return, in sorted order, the q elements in S of rank r_1, …, r_q, respectively. The asymptotic deterministic comparison complexity of the problem was settled by Dobkin and Munro [JACM 1981]. In the I/O model an optimal I/O complexity was achieved by Hu et al. [SPAA 2014]. Recently [ESA 2023], we presented a cache-oblivious algorithm with matching I/O complexity, named funnelselect, since it heavily borrows ideas from the cache-oblivious sorting algorithm funnelsort from the seminal paper by Frigo, Leiserson, Prokop and Ramachandran [FOCS 1999]. Funnelselect is inherently randomized as it relies on sampling for cheaply finding many good pivots. In this paper we present deterministic funnelselect, achieving the same optimal I/O complexity cache-obliviously without randomization. Our new algorithm essentially replaces a single (in expectation) reversed-funnel computation using random pivots by a recursive algorithm using multiple reversed-funnel computations. To meet the I/O bound, this requires a carefully chosen subproblem size based on the entropy of the sequence of query ranks; deterministic funnelselect thus raises distinct technical challenges not met by randomized funnelselect. The resulting worst-case I/O bound is O(∑_{i = 1}^{q+1} Δ_i/B ⋅ log_{M/B} N/Δ_i + N/B), where B is the external memory block size, M ≥ B^{1+ε} is the internal memory size, for some constant ε > 0, and Δ_i = r_i - r_{i-1} (assuming r_0 = 0 and r_{q+1} = N + 1).
Cite as
Gerth Stølting Brodal and Sebastian Wild. Deterministic Cache-Oblivious Funnelselect. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{brodal_et_al:LIPIcs.SWAT.2024.17,
author = {Brodal, Gerth St{\o}lting and Wild, Sebastian},
title = {{Deterministic Cache-Oblivious Funnelselect}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {17:1--17:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.17},
URN = {urn:nbn:de:0030-drops-200576},
doi = {10.4230/LIPIcs.SWAT.2024.17},
annote = {Keywords: Multiple selection, cache-oblivious algorithm, entropy bounds}
}
Document
Dynamic L-Budget Clustering of Curves
Abstract
A key goal of clustering is data reduction. In center-based clustering of complex objects therefore not only the number of clusters but also the complexity of the centers plays a crucial role. We propose L-Budget Clustering as unifying perspective on this task, optimizing the clustering under the constraint that the summed complexity of all centers is at most L. We present algorithms for clustering planar curves under the Fréchet distance, but note that our algorithms more generally apply to objects in metric spaces if a notion of simplification of objects is applicable. A scenario in which data reduction is of particular importance is when the space is limited. Our main result is an efficient (8 + ε)-approximation algorithm with a (1 + ε)-resource augmentation that maintains an L-budget clustering under insertion of curves using only O(Lε^{-1}) space and O^*(L³log(L) + L²log(r^*/r₀)) time where O^* hides factors of ε^{-1}.
Cite as
Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Lukas Plätz, Lea Thiel, and Sampson Wong. Dynamic L-Budget Clustering of Curves. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{buchin_et_al:LIPIcs.SWAT.2024.18,
author = {Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Pl\"{a}tz, Lukas and Thiel, Lea and Wong, Sampson},
title = {{Dynamic L-Budget Clustering of Curves}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {18:1--18:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.18},
URN = {urn:nbn:de:0030-drops-200588},
doi = {10.4230/LIPIcs.SWAT.2024.18},
annote = {Keywords: clustering, streaming algorithm, polygonal curves, Fr\'{e}chet distance, storage efficiency, simplification, approximation algorithms}
}
Document
Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond
Abstract
For a positive real γ ≥ 1, a γ-certified algorithm for a vertex-weighted graph optimization problem is an algorithm that, given a weighted graph (G,w), outputs a re-weighting of the graph obtained by scaling each weight individually with a factor between 1 and γ, along with a solution which is optimal for the perturbed weight function. Here we provide (1+ε)-certified algorithms for Dominating Set and H-Subgraph-Free-Deletion which, for any ε > 0, run in time f(1/ε)⋅n^𝒪(1) on minor-closed classes of graphs of bounded local tree-width with polynomially-bounded weights. We obtain our algorithms as corollaries of a more general result establishing FPT-time certified algorithms for problems admitting, at an intuitive level, certain "local solution-improvement properties". These results improve - in terms of generality, running time and parameter dependence - on Angelidakis, Awasthi, Blum, Chatziafratis and Dan’s XP-time (1+ε)-certified algorithm for Independent Set on planar graphs (ESA2019). Furthermore, our methods are also conceptually simpler: our algorithm is based on elementary local re-optimizations inspired by Baker’s technique, as opposed to the heavy machinery of the Sherali-Adams hierarchy required in previous work.
Cite as
Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jaime Venne. Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bumpus_et_al:LIPIcs.SWAT.2024.19,
author = {Bumpus, Benjamin Merlin and Jansen, Bart M. P. and Venne, Jaime},
title = {{Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {19:1--19:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.19},
URN = {urn:nbn:de:0030-drops-200595},
doi = {10.4230/LIPIcs.SWAT.2024.19},
annote = {Keywords: fixed-parameter tractability, certified algorithms}
}
Document
Sparsity-Parameterised Dynamic Edge Colouring
Abstract
We study the edge-colouring problem, and give efficient algorithms where the number of colours is parameterised by the graph’s arboricity, α. In a dynamic graph, subject to insertions and deletions, we give a deterministic algorithm that updates a proper Δ + O(α) edge colouring in poly(log n) amortized time. Our algorithm is fully adaptive to the current value of the maximum degree and arboricity.
In this fully-dynamic setting, the state-of-the-art edge-colouring algorithms are either a randomised algorithm using (1 + ε)Δ colours in poly(log n, ε^{-1}) time per update, or the naive greedy algorithm which is a deterministic 2Δ -1 edge colouring with log(Δ) update time.
Compared to the (1+ε)Δ algorithm, our algorithm is deterministic and asymptotically faster, and when α is sufficiently small compared to Δ, it even uses fewer colours. In particular, ours is the first Δ+O(1) edge-colouring algorithm for dynamic forests, and dynamic planar graphs, with polylogarithmic update time.
Additionally, in the static setting, we show that we can find a proper edge colouring with Δ + 2α colours in O(mlog n) time. Moreover, the colouring returned by our algorithm has the following local property: every edge uv is coloured with a colour in {1, max{deg(u), deg(v)} + 2α}. The time bound matches that of the greedy algorithm that computes a 2Δ-1 colouring of the graph’s edges, and improves the number of colours when α is sufficiently small compared to Δ.
Cite as
Aleksander B. G. Christiansen, Eva Rotenberg, and Juliette Vlieghe. Sparsity-Parameterised Dynamic Edge Colouring. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{christiansen_et_al:LIPIcs.SWAT.2024.20,
author = {Christiansen, Aleksander B. G. and Rotenberg, Eva and Vlieghe, Juliette},
title = {{Sparsity-Parameterised Dynamic Edge Colouring}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.20},
URN = {urn:nbn:de:0030-drops-200608},
doi = {10.4230/LIPIcs.SWAT.2024.20},
annote = {Keywords: edge colouring, arboricity, hierarchical partition, dynamic algorithms, amortized analysis}
}
Document
Approximating Minimum Sum Coloring with Bundles
Abstract
In the Minimum Sum Coloring with Bundles problem, we are given an undirected graph G = (V,E) and (not necessarily disjoint) bundles V_1, V_2, …, V_p ⊆ V with associated weights w_1, …, w_p ≥ 0. The goal is to give a proper coloring of G using positive integers to minimize the weighted average/total completion time of all bundles, where the completion time of a bundle is the maximum integer assigned to one of its nodes. This is a common generalization of the classic Minimum Sum Coloring problem, i.e. when all bundles are singleton nodes, and the classic Chromatic Number problem, i.e. the only bundle is all of V.
Despite its generality as an extension of Minimum Sum Coloring, only very special cases have been studied with the most common being the line graph L(H) of a graph H (also known as Coflow Scheduling). We provide the first constant-factor approximation in perfect graphs and, more generally, graphs whose chromatic number is within a constant factor of the maximum clique size in any induced subgraph. For example, we obtain constant-factor approximations for graphs that are well-studied in minimum sum coloring such as interval graphs and unit disk graphs.
Next, we extend our results to get constant-factor approximations for a general model where the bundles are disjoint (i.e. can be thought of as jobs brought by the corresponding client) and we are only permitted to color/schedule a bounded number of jobs from each bundle at any given time. Specifically, we get constant-factor approximations for this model if the nodes of graph G have an ordering v_1, v_2, …, v_n such that the left-neighborhood N_𝓁(v_i) : = {v_j : j < i, v_iv_j ∈ E} can be covered by O(1) cliques. For example, this applies to chordal graphs, unit disc graphs, and circular arc graphs.
Cite as
Seyed Parsa Darbouy and Zachary Friggstad. Approximating Minimum Sum Coloring with Bundles. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{darbouy_et_al:LIPIcs.SWAT.2024.21,
author = {Darbouy, Seyed Parsa and Friggstad, Zachary},
title = {{Approximating Minimum Sum Coloring with Bundles}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.21},
URN = {urn:nbn:de:0030-drops-200611},
doi = {10.4230/LIPIcs.SWAT.2024.21},
annote = {Keywords: Approximation Algorithms, Scheduling, Coloring}
}
Document
Stability in Graphs with Matroid Constraints
Abstract
We study the following INDEPENDENT STABLE SET problem. Let G be an undirected graph and ℳ = (V(G), ℐ) be a matroid whose elements are the vertices of G. For an integer k ≥ 1, the task is to decide whether G contains a set S ⊆ V(G) of size at least k which is independent (stable) in G and independent in ℳ. This problem generalizes several well-studied algorithmic problems, including RAINBOW INDEPENDENT SET, RAIBOW MATCHING, and BIPARTITE MATCHING WITH SEPARATION. We show that
- When the matroid ℳ is represented by the independence oracle, then for any computable function f, no algorithm can solve INDEPENDENT STABLE SET using f(k)⋅n^o(k) calls to the oracle.
- On the other hand, when the graph G is of degeneracy d, then the problem is solvable in time 𝒪((d+1)^k ⋅ n), and hence is FPT parameterized by d+k. Moreover, when the degeneracy d is a constant (which is not a part of the input), the problem admits a kernel polynomial in k. More precisely, we prove that for every integer d ≥ 0, the problem admits a kernelization algorithm that in time n^𝒪(d) outputs an equivalent framework with a graph on dk^{𝒪(d)} vertices. A lower bound complements this when d is part of the input: INDEPENDENT STABLE SET does not admit a polynomial kernel when parameterized by k+d unless NP ⊆ coNP/poly. This lower bound holds even when ℳ is a partition matroid.
- Another set of results concerns the scenario when the graph G is chordal. In this case, our computational lower bound excludes an FPT algorithm when the input matroid is given by its independence oracle. However, we demonstrate that INDEPENDENT STABLE SET can be solved in 2^𝒪(k)⋅‖ℳ‖^𝒪(1) time when ℳ is a linear matroid given by its representation. In the same setting, INDEPENDENT STABLE SET does not have a polynomial kernel when parameterized by k unless NP ⊆ coNP/poly.
Cite as
Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Saket Saurabh. Stability in Graphs with Matroid Constraints. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fomin_et_al:LIPIcs.SWAT.2024.22,
author = {Fomin, Fedor V. and Golovach, Petr A. and Korhonen, Tuukka and Saurabh, Saket},
title = {{Stability in Graphs with Matroid Constraints}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.22},
URN = {urn:nbn:de:0030-drops-200629},
doi = {10.4230/LIPIcs.SWAT.2024.22},
annote = {Keywords: frameworks, independent stable sets, parameterized complexity, kernelization}
}
Document
A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree
Abstract
In the classic Directed Steiner Tree problem (DST), we are given an edge-weighted directed graph G = (V,E) with n nodes, a specified root node r ∈ V, and k terminals X ⊆ V-{r}. The goal is to find the cheapest F ⊆ E such that r can reach any terminal using only edges in F.
Designing approximation algorithms for DST is quite challenging, to date the best approximation guarantee of a polynomial-time algorithm for DST is O(k^ε) for any constant ε > 0 [Charikar et al., 1999]. For network design problems like DST, one often relies on natural cut-based linear programming (LP) relaxations to design approximation algorithms. In general, the integrality gap of such an LP for DST is known to have a polynomial integrality gap lower bound [Zosin and Khuller, 2002; Li and Laekhanukit, 2021]. So particular interest has been invested in special cases or in strengthenings of this LP.
In this work, we show the integrality gap is only O(log k) for instances of DST where no Steiner node has both an edge from another Steiner node and an edge to another Steiner node, i.e. the longest path using only Steiner nodes has length at most 1. This generalizes the well-studied case of quasi-bipartite DST where no edge has both endpoints being Steiner nodes. Our result is also optimal in the sense that the integrality gap can be as bad as poly(n) even if the longest path with only Steiner nodes has length 2.
Cite as
Zachary Friggstad and Hao Sun. A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{friggstad_et_al:LIPIcs.SWAT.2024.23,
author = {Friggstad, Zachary and Sun, Hao},
title = {{A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.23},
URN = {urn:nbn:de:0030-drops-200638},
doi = {10.4230/LIPIcs.SWAT.2024.23},
annote = {Keywords: Steiner Tree, Approximation Algorithms, Linear Programming}
}
Document
Optimizing Symbol Visibility Through Displacement
Abstract
In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols.
Specifically, we consider unit square symbols that need to be placed at specified y-coordinates. We optimize the drawing order of the symbols as well as their x-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most 2, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in O(nlog n) time. If the width is at most 2, there is a vertical line that stabs all squares, and in this case, we give a 2-approximation algorithm (assuming fixed container height) that runs in O(nlog n) time. As a minimum visible perimeter of 2 is always trivially achievable, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.
Cite as
Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy, Wouter Meulemans, Bettina Speckmann, and Miloš Stojaković. Optimizing Symbol Visibility Through Displacement. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{gartner_et_al:LIPIcs.SWAT.2024.24,
author = {G\"{a}rtner, Bernd and Kalani, Vishwas and M. Reddy, Meghana and Meulemans, Wouter and Speckmann, Bettina and Stojakovi\'{c}, Milo\v{s}},
title = {{Optimizing Symbol Visibility Through Displacement}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {24:1--24:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.24},
URN = {urn:nbn:de:0030-drops-200643},
doi = {10.4230/LIPIcs.SWAT.2024.24},
annote = {Keywords: symbol placement, visibility, jittering, stacking order}
}
Document
Delaunay Triangulations in the Hilbert Metric
Abstract
The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of n points in the Hilbert geometry defined by a convex m-gon. The algorithm runs in O(n (log n + log³ m)) expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time O(n h log² m), where h is the number of points on the hull’s boundary.
Cite as
Auguste H. Gezalyan, Soo H. Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, and David M. Mount. Delaunay Triangulations in the Hilbert Metric. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{gezalyan_et_al:LIPIcs.SWAT.2024.25,
author = {Gezalyan, Auguste H. and Kim, Soo H. and Lopez, Carlos and Skora, Daniel and Stefankovic, Zofia and Mount, David M.},
title = {{Delaunay Triangulations in the Hilbert Metric}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.25},
URN = {urn:nbn:de:0030-drops-200657},
doi = {10.4230/LIPIcs.SWAT.2024.25},
annote = {Keywords: Delaunay Triangulations, Hilbert metric, convexity, randomized algorithms}
}
Document
No-Dimensional Tverberg Partitions Revisited
Abstract
Given a set P ⊂ ℝ^d of n points, with diameter Δ, and a parameter δ ∈ (0,1), it is known that there is a partition of P into sets P_1, …, P_t, each of size O(1/δ²), such that their convex hulls all intersect a common ball of radius δΔ. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm (i.e., O(dn)). We also provide a deterministic algorithm with running time O(dn log n). Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better.
We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a "fuzzy" centerpoint, and prove a no-dimensional weak ε-net theorem with an improved constant.
Cite as
Sariel Har-Peled and Eliot W. Robson. No-Dimensional Tverberg Partitions Revisited. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{harpeled_et_al:LIPIcs.SWAT.2024.26,
author = {Har-Peled, Sariel and Robson, Eliot W.},
title = {{No-Dimensional Tverberg Partitions Revisited}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {26:1--26:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.26},
URN = {urn:nbn:de:0030-drops-200664},
doi = {10.4230/LIPIcs.SWAT.2024.26},
annote = {Keywords: Points, partitions, convex hull, high dimension}
}
Document
Optimizing Visibility-Based Search in Polygonal Domains
Abstract
Given a geometric domain P, visibility-based search problems seek routes for one or more mobile agents ("watchmen") to move within P in order to be able to see a portion (or all) of P, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within P according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of P. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain P in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of P (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon P we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains P (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in P to guarantee some specified probability of detection of a static target within P, randomly distributed in P according to a given prior distribution.
Cite as
Kien C. Huynh, Joseph S. B. Mitchell, Linh Nguyen, and Valentin Polishchuk. Optimizing Visibility-Based Search in Polygonal Domains. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{huynh_et_al:LIPIcs.SWAT.2024.27,
author = {Huynh, Kien C. and Mitchell, Joseph S. B. and Nguyen, Linh and Polishchuk, Valentin},
title = {{Optimizing Visibility-Based Search in Polygonal Domains}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.27},
URN = {urn:nbn:de:0030-drops-200671},
doi = {10.4230/LIPIcs.SWAT.2024.27},
annote = {Keywords: Quota watchman route problem, budgeted watchman route problem, visibility-based search, approximation}
}
Document
Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion
Abstract
For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most c ⋅ opt contain v. Recent work showed that polynomial-time algorithms to detect c-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected n-vertex graph G in time 2^𝒪(𝓁³)⋅n^𝒪(1), where 𝓁 is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that (2-ε)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.
Cite as
Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{jansen_et_al:LIPIcs.SWAT.2024.28,
author = {Jansen, Bart M. P. and Verhaegh, Ruben F. A.},
title = {{Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.28},
URN = {urn:nbn:de:0030-drops-200683},
doi = {10.4230/LIPIcs.SWAT.2024.28},
annote = {Keywords: fixed-parameter tractability, essential vertices, integrality gap}
}
Document
Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs
Abstract
It is known that the weighted version of Edge Multiway Cut (also known as Multiterminal Cut) is NP-complete on planar graphs of maximum degree 3. In contrast, for the unweighted version, NP-completeness is only known for planar graphs of maximum degree 11. In fact, the complexity of unweighted Edge Multiway Cut was open for graphs of maximum degree 3 for over twenty years. We prove that the unweighted version is NP-complete even for planar graphs of maximum degree 3. As weighted Edge Multiway Cut is polynomial-time solvable for graphs of maximum degree at most 2, we have now closed the complexity gap. We also prove that (unweighted) Node Multiway Cut (both with and without deletable terminals) is NP-complete for planar graphs of maximum degree 3. By combining our results with known results, we can apply two meta-classifications on graph containment from the literature. This yields full dichotomies for all three problems on H-topological-minor-free graphs and, should H be finite, on H-subgraph-free graphs as well. Previously, such dichotomies were only implied for H-minor-free graphs.
Cite as
Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{johnson_et_al:LIPIcs.SWAT.2024.29,
author = {Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan},
title = {{Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {29:1--29:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.29},
URN = {urn:nbn:de:0030-drops-200699},
doi = {10.4230/LIPIcs.SWAT.2024.29},
annote = {Keywords: multiway cut, planar subcubic graph, complexity dichotomy, graph containment}
}
Document
Parameterized Complexity of Submodular Minimization Under Uncertainty
Abstract
This paper studies the computational complexity of a robust variant of a two-stage submodular minimization problem that we call Robust Submodular Minimizer. In this problem, we are given k submodular functions f_1,… ,f_k over a set family 2^V, which represent k possible scenarios in the future when we will need to find an optimal solution for one of these scenarios, i.e., a minimizer for one of the functions. The present task is to find a set X ⊆ V that is close to some optimal solution for each f_i in the sense that some minimizer of f_i can be obtained from X by adding/removing at most d elements for a given integer d ∈ ℕ. The main contribution of this paper is to provide a complete computational map of this problem with respect to parameters k and d, which reveals a tight complexity threshold for both parameters:
- Robust Submodular Minimizer can be solved in polynomial time when k ≤ 2, but is NP-hard if k is a constant with k ≥ 3.
- Robust Submodular Minimizer can be solved in polynomial time when d = 0, but is NP-hard if d is a constant with d ≥ 1.
- Robust Submodular Minimizer is fixed-parameter tractable when parameterized by (k,d). We also show that if some submodular function f_i has a polynomial number of minimizers, then the problem becomes fixed-parameter tractable when parameterized by d. We remark that all our hardness results hold even if each submodular function is given by a cut function of a directed graph.
Cite as
Naonori Kakimura and Ildikó Schlotter. Parameterized Complexity of Submodular Minimization Under Uncertainty. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kakimura_et_al:LIPIcs.SWAT.2024.30,
author = {Kakimura, Naonori and Schlotter, Ildik\'{o}},
title = {{Parameterized Complexity of Submodular Minimization Under Uncertainty}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {30:1--30:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.30},
URN = {urn:nbn:de:0030-drops-200702},
doi = {10.4230/LIPIcs.SWAT.2024.30},
annote = {Keywords: Submodular minimization, optimization under uncertainty, parameterized complexity, cut function}
}
Document
Optimal In-Place Compaction of Sliding Cubes
Abstract
The sliding cubes model is a well-established theoretical framework that supports the analysis of reconfiguration algorithms for modular robots consisting of face-connected cubes. As is common in the literature, we focus on reconfiguration via an intermediate canonical shape. Specifically, we present an in-place algorithm that reconfigures any n-cube configuration into a compact canonical shape using a number of moves proportional to the sum of coordinates of the input cubes. This result is asymptotically optimal and strictly improves on all prior work. Furthermore, our algorithm directly extends to dimensions higher than three.
Cite as
Irina Kostitsyna, Tim Ophelders, Irene Parada, Tom Peters, Willem Sonke, and Bettina Speckmann. Optimal In-Place Compaction of Sliding Cubes. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kostitsyna_et_al:LIPIcs.SWAT.2024.31,
author = {Kostitsyna, Irina and Ophelders, Tim and Parada, Irene and Peters, Tom and Sonke, Willem and Speckmann, Bettina},
title = {{Optimal In-Place Compaction of Sliding Cubes}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.31},
URN = {urn:nbn:de:0030-drops-200713},
doi = {10.4230/LIPIcs.SWAT.2024.31},
annote = {Keywords: Sliding cubes, Reconfiguration algorithm, Modular robots}
}
Document
Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman
Abstract
In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in TC². Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width k admits a rank decomposition of width ≤ 2k and height O(log n) (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation.
Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in TC¹. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by FO + C formulas with 6k+4 variables and quantifier depth O(log n). Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in NC.
Cite as
Michael Levet, Puck Rombach, and Nicholas Sieger. Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{levet_et_al:LIPIcs.SWAT.2024.32,
author = {Levet, Michael and Rombach, Puck and Sieger, Nicholas},
title = {{Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {32:1--32:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.32},
URN = {urn:nbn:de:0030-drops-200724},
doi = {10.4230/LIPIcs.SWAT.2024.32},
annote = {Keywords: Graph Isomorphism, Weisfeiler-Leman, Rank-Width, Canonization, Descriptive Complexity, Circuit Complexity}
}
Document
Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes
Abstract
There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to k-uniform hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [Louis, Makarychev - TOC'16; Chan, Louis, Tang, Zhang - JACM'18]. In the second approach, one can view a hypergraph as a simplicial complex and study its various topological properties [Linial, Meshulam - Combinatorica'06; Meshulam, Wallach - RSA'09; Dotterrer, Kaufman, Wagner - SoCG'16; Parzanchevski, Rosenthal - RSA'17] and spectral properties [Kaufman, Mass - ITCS'17; Dinur, Kaufman - FOCS'17; Kaufman, Openheim - STOC'18; Oppenheim - DCG'18; Kaufman, Openheim - Combinatorica'20].
In this work, we attempt to bridge these two directions of study by relating the spectrum of up-down walks and swap walks on the simplicial complex, a downward closed set system, to hypergraph expansion. More precisely, we study the simplicial complex obtained by downward closing the given hypergraph and random walks between its levels X(l), i.e., the sets of cardinality l. In surprising contrast to random walks on graphs, we show that the spectral gap of swap walks and up-down walks between level m and l with 1 < m ⩽ l cannot be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap walks between X(1) and X(k-1) cannot be used to infer any bounds on hypergraph conductance. In contrast, we give a Cheeger-like inequality relating the spectra of walks between level 1 and l for any l ⩽ k to hypergraph expansion. This is a surprising difference between swaps walks and up-down walks!
Finally, we also give a construction to show that the well-studied notion of link expansion in simplicial complexes cannot be used to bound hypergraph expansion in a Cheeger-like manner.
Cite as
Anand Louis, Rameesh Paul, and Arka Ray. Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{louis_et_al:LIPIcs.SWAT.2024.33,
author = {Louis, Anand and Paul, Rameesh and Ray, Arka},
title = {{Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.33},
URN = {urn:nbn:de:0030-drops-200739},
doi = {10.4230/LIPIcs.SWAT.2024.33},
annote = {Keywords: Sparse Cuts, Random Walks, Link Expansion, Hypergraph Expansion, Simplicial Complexes, High Dimensional Expander, Threshold Rank}
}
Document
Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints
Abstract
We introduce and analyze a model for self-reconfigurable robots made up of unit-cube modules. Compared to past models, our model aims to newly capture two important practical aspects of real-world robots. First, modules often do not occupy an exact unit cube, but rather have features like bumps extending outside the allotted space so that modules can interlock. Thus, for example, our model forbids modules from squeezing in between two other modules that are one unit distance apart. Second, our model captures the practical scenario of many passive modules assembled by a single robot, instead of requiring all modules to be able to move on their own.
We prove two universality results. First, with a supply of auxiliary modules, we show that any connected polycube structure can be constructed by a carefully aligned plane sweep. Second, without additional modules, we show how to construct any structure for which a natural notion of external feature size is at least a constant; this property largely consolidates forbidden-pattern properties used in previous works on reconfigurable modular robots.
Cite as
MIT-NASA Space Robots Team, Josh Brunner, Kenneth C. Cheung, Erik D. Demaine, Jenny Diomidova, Christine Gregg, Della H. Hendrickson, and Irina Kostitsyna. Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mitnasaspacerobotsteam_et_al:LIPIcs.SWAT.2024.34,
author = {MIT-NASA Space Robots Team and Brunner, Josh and Cheung, Kenneth C. and Demaine, Erik D. and Diomidova, Jenny and Gregg, Christine and Hendrickson, Della H. and Kostitsyna, Irina},
title = {{Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {34:1--34:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.34},
URN = {urn:nbn:de:0030-drops-200742},
doi = {10.4230/LIPIcs.SWAT.2024.34},
annote = {Keywords: Modular robotics, programmable matter, digital materials, motion planning}
}
Document
Solving a Family Of Multivariate Optimization and Decision Problems on Classes of Bounded Expansion
Abstract
For some time, it has been known that the model checking problem for first-order formulas is fixed-parameter tractable on nowhere dense graph classes, so we shall ask in which direction there is space for improvements. One of the possible directions is to go beyond first-order formulas: Augmenting first-order logic with general counting quantifiers increases the expressiveness by far, but makes the model checking problem hard even on graphs of bounded tree-depth. The picture is different if we allow only "simple" - but arbitrarily nested - counting terms of the form #y φ(x^- ,y) > N. Even then, only approximate model checking is possible on graph classes of bounded expansion. Here, the largest known logic fragment, on which exact model checking is still fpt, consists of formulas of the form ∃x_1 … ∃x_k #y φ(x^- ,y) > N, where φ(x^- ,y) is a first-order formula without counting terms. An example of a problem that can be expressed in this way is partial dominating set: Are there k vertices that dominate at least a given number of vertices in the graph? The complexity of the same problem is open if you replace at least with exactly. Likewise, the complexity of "are there k vertices that dominate at least half of the blue and half of the red vertices?" is also open. We answer both questions by providing an fpt algorithm that solves the model checking problem for formulas of the more general form ψ ≡ ∃x_1 … ∃x_k P(#y φ_1(x^- ,y), …, #y φ_ℓ(x^- ,y)), where P is an arbitrary polynomially computable predicate on numbers. The running time is f(|ψ|)n^{𝓁+1} polylog(n) on graph classes of bounded expansion. Under SETH, this running time is tight up to almost linear factor.
Cite as
Daniel Mock and Peter Rossmanith. Solving a Family Of Multivariate Optimization and Decision Problems on Classes of Bounded Expansion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mock_et_al:LIPIcs.SWAT.2024.35,
author = {Mock, Daniel and Rossmanith, Peter},
title = {{Solving a Family Of Multivariate Optimization and Decision Problems on Classes of Bounded Expansion}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {35:1--35:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.35},
URN = {urn:nbn:de:0030-drops-200754},
doi = {10.4230/LIPIcs.SWAT.2024.35},
annote = {Keywords: bounded expansion, parameterized algorithms, sparsity, counting logic, dominating set, model checking, multivariate optimization}
}
Document
Path-Reporting Distance Oracles with Linear Size
Abstract
Given an undirected weighted graph, an (approximate) distance oracle is a data structure that can (approximately) answer distance queries. A Path-Reporting Distance Oracle, or PRDO, is a distance oracle that must also return a path between the queried vertices. Given a graph on n vertices and an integer parameter k ≥ 1, Thorup and Zwick [M. Thorup and U. Zwick, 2001] showed a PRDO with stretch 2k-1, size O(k⋅n^{1+1/k}) and query time O(k) (for the query time of PRDOs, we omit the time needed to report the path itself). Subsequent works [Mendel and Naor, 2007; Shiri Chechik, 2014; Shiri Chechik, 2015] improved the size to O(n^{1+1/k}) and the query time to O(1). However, these improvements produce distance oracles which are not path-reporting. Several other works [Michael Elkin et al., 2016; Michael Elkin and Seth Pettie, 2016] focused on small size PRDO for general graphs, but all known results on distance oracles with linear size suffer from polynomial stretch, polynomial query time, or not being path-reporting.
In this paper we devise the first linear size PRDO with poly-logarithmic stretch and low query time O(log log n). More generally, for any integer k ≥ 1, we obtain a PRDO with stretch at most O(k^4.82), size O(n^{1+1/k}), and query time O(log k). In addition, we can make the size of our PRDO as small as n+o(n), at the cost of increasing the query time to poly-logarithmic. For unweighted graphs, we improve the stretch to O(k²).
We also consider pairwise PRDO, which is a PRDO that is only required to answer queries from a given set of pairs P. An exact PRDO of size O(n+|P|²) and constant query time was provided in [Michael Elkin and Seth Pettie, 2016]. In this work we dramatically improve the size, at the cost of slightly increasing the stretch. Specifically, given any ε > 0, we devise a pairwise PRDO with stretch 1+ε, constant query time, and near optimal size n^o(1)⋅ (n+|P|).
Cite as
Ofer Neiman and Idan Shabat. Path-Reporting Distance Oracles with Linear Size. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{neiman_et_al:LIPIcs.SWAT.2024.36,
author = {Neiman, Ofer and Shabat, Idan},
title = {{Path-Reporting Distance Oracles with Linear Size}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {36:1--36:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.36},
URN = {urn:nbn:de:0030-drops-200760},
doi = {10.4230/LIPIcs.SWAT.2024.36},
annote = {Keywords: Graph Algorithms, Shortest Paths, Distance Oracles}
}
Document
Toward Grünbaum’s Conjecture
Abstract
Given a spanning tree T of a planar graph G, the co-tree of T is the spanning tree of the dual graph G^* with edge set (E(G)-E(T))^*. Grünbaum conjectured in 1970 that every planar 3-connected graph G contains a spanning tree T such that both T and its co-tree have maximum degree at most 3.
While Grünbaum’s conjecture remains open, Biedl proved that there is a spanning tree T such that T and its co-tree have maximum degree at most 5. By using new structural insights into Schnyder woods, we prove that there is a spanning tree T such that T and its co-tree have maximum degree at most 4. This tree can be computed in linear time.
Cite as
Christian Ortlieb and Jens M. Schmidt. Toward Grünbaum’s Conjecture. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ortlieb_et_al:LIPIcs.SWAT.2024.37,
author = {Ortlieb, Christian and Schmidt, Jens M.},
title = {{Toward Gr\"{u}nbaum’s Conjecture}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {37:1--37:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.37},
URN = {urn:nbn:de:0030-drops-200777},
doi = {10.4230/LIPIcs.SWAT.2024.37},
annote = {Keywords: Planar graph, spanning tree, maximum degree, Schnyder wood}
}
Document
Finding Induced Subgraphs from Graphs with Small Mim-Width
Abstract
In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and the complexity status of many problems of finding dense induced subgraphs remains open when parameterized by mim-width. In this paper, we investigate the complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width. We first give a meta-theorem implying that various induced subgraph problems are NP-hard for bounded mim-width graphs. Moreover, we show that some problems, including Clique and Induced Cluster Subgraph, remain NP-hard even for graphs with (linear) mim-width at most 2. In contrast to the intractability, we provide an algorithm that, given a graph and its branch decomposition with mim-width at most 1, solves Induced Cluster Subgraph in polynomial time. We emphasize that our algorithmic technique is applicable to other problems such as Induced Polar Subgraph and Induced Split Subgraph. Since a branch decomposition with mim-width at most 1 can be constructed in polynomial time for block graphs, interval graphs, permutation graphs, cographs, distance-hereditary graphs, convex graphs, and their complement graphs, our positive results reveal the polynomial-time solvability of various problems for these graph classes.
Cite as
Yota Otachi, Akira Suzuki, and Yuma Tamura. Finding Induced Subgraphs from Graphs with Small Mim-Width. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{otachi_et_al:LIPIcs.SWAT.2024.38,
author = {Otachi, Yota and Suzuki, Akira and Tamura, Yuma},
title = {{Finding Induced Subgraphs from Graphs with Small Mim-Width}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {38:1--38:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.38},
URN = {urn:nbn:de:0030-drops-200788},
doi = {10.4230/LIPIcs.SWAT.2024.38},
annote = {Keywords: mim-width, graph algorithm, NP-hardness, induced subgraph problem, cluster vertex deletion}
}
Document
A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads
Abstract
The capacitated tree cover problem with edge loads is a variant of the tree cover problem, where we are given facility opening costs, edge costs and loads, as well as vertex loads. We try to find a tree cover of minimum cost such that the total edge and vertex load of each tree does not exceed a given bound. We present an 𝒪(mlog n) time 3-approximation algorithm for this problem.
This is achieved by starting with a certain LP formulation. We give a combinatorial algorithm that solves the LP optimally in time 𝒪(mlog n). Then, we show that a linear time rounding and splitting technique leads to an integral solution that costs at most 3 times as much as the LP solution. Finally, we prove that the integrality gap of the LP is 3, which shows that we can not improve the rounding step in general.
Cite as
Benjamin Rockel-Wolff. A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{rockelwolff:LIPIcs.SWAT.2024.39,
author = {Rockel-Wolff, Benjamin},
title = {{A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {39:1--39:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.39},
URN = {urn:nbn:de:0030-drops-200793},
doi = {10.4230/LIPIcs.SWAT.2024.39},
annote = {Keywords: Approximation Algorithms, Tree Cover, LP}
}
Document
Approximation Algorithms for the Airport and Railway Problem
Abstract
In this paper, we present approximation algorithms for the airport and railway problem (AR) on several classes of graphs. The AR problem, introduced by [Anna Adamaszek et al., 2016], is a combination of the Capacitated Facility Location problem (CFL) and the network design problem. An AR instance consists of a set of points (cities) V in a metric d(.,.), each of which is associated with a non-negative cost f_v and a number k, which represent respectively the cost of establishing an airport (facility) in the corresponding point, and the universal airport capacity. A feasible solution is a network of airports and railways providing services to all cities without violating any capacity, where railways are edges connecting pairs of points, with their costs equivalent to the distance between the respective points. The objective is to find such a network with the least cost. In other words, find a forest, each component having at most k points and one open facility, minimizing the total cost of edges and airport opening costs. Adamaszek et al. [Anna Adamaszek et al., 2016] presented a PTAS for AR in the two-dimensional Euclidean metric ℝ² with a uniform opening cost. In subsequent work [Anna Adamaszek et al., 2018] presented a bicriteria 4/3 (2+1/α)-approximation algorithm for AR with non-uniform opening costs but violating the airport capacity by a factor of 1+α, i.e. (1+α)k capacity where 0 < α ≤ 1, a (2+k/(k-1)+ε)-approximation algorithm and a bicriteria Quasi-Polynomial Time Approximation Scheme (QPTAS) for the same problem in the Euclidean plane ℝ². In this work, we give a 2-approximation for AR with a uniform opening cost for general metrics and an O(log n)-approximation for non-uniform opening costs. We also give a QPTAS for AR with a uniform opening cost in graphs of bounded treewidth and a QPTAS for a slightly relaxed version in the non-uniform setting. The latter implies O(1)-approximation on graphs of bounded doubling dimensions, graphs of bounded highway dimensions and planar graphs in quasi-polynomial time.
Cite as
Mohammad R. Salavatipour and Lijiangnan Tian. Approximation Algorithms for the Airport and Railway Problem. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{salavatipour_et_al:LIPIcs.SWAT.2024.40,
author = {Salavatipour, Mohammad R. and Tian, Lijiangnan},
title = {{Approximation Algorithms for the Airport and Railway Problem}},
booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
pages = {40:1--40:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-318-8},
ISSN = {1868-8969},
year = {2024},
volume = {294},
editor = {Bodlaender, Hans L.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.40},
URN = {urn:nbn:de:0030-drops-200806},
doi = {10.4230/LIPIcs.SWAT.2024.40},
annote = {Keywords: Facility Location, Approximation Algorithms, Dynamic Programming}
}