Volume
LIPIcs, Volume 244
30th Annual European Symposium on Algorithms (ESA 2022)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA)
Event
ESA 2022, September 5-9, 2022, Berlin/Potsdam, GermanyEditors
Publication Details
- published at: 2022-09-01
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-247-1
- DBLP: db/conf/esa/esa2022
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Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volume
Abstract
LIPIcs, Volume 244, ESA 2022, Complete Volume
Cite as
30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1-1406, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{chechik_et_al:LIPIcs.ESA.2022,
title = {{LIPIcs, Volume 244, ESA 2022, Complete Volume}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {1--1406},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022},
URN = {urn:nbn:de:0030-drops-169374},
doi = {10.4230/LIPIcs.ESA.2022},
annote = {Keywords: LIPIcs, Volume 244, ESA 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 0:i-0:xxii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chechik_et_al:LIPIcs.ESA.2022.0,
author = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {0:i--0:xxii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.0},
URN = {urn:nbn:de:0030-drops-169382},
doi = {10.4230/LIPIcs.ESA.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Enumerating Minimal Connected Dominating Sets
Abstract
The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively.
We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.
Cite as
Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann. Enumerating Minimal Connected Dominating Sets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{abukhzam_et_al:LIPIcs.ESA.2022.1,
author = {Abu-Khzam, Faisal N. and Fernau, Henning and Gras, Benjamin and Liedloff, Mathieu and Mann, Kevin},
title = {{Enumerating Minimal Connected Dominating Sets}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {1:1--1:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.1},
URN = {urn:nbn:de:0030-drops-169390},
doi = {10.4230/LIPIcs.ESA.2022.1},
annote = {Keywords: enumeration problems, connected domination, degenerate graphs}
}
Document
Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries
Abstract
We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (TALG '20). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1± ε) relative error approximation to the number of edges, with query complexity Õ(ε^{-5}log⁵ n), where Õ(⋅) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ε,log n) query complexity. Prior work requires Ω(log² n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ε, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ε^{-2} log^{11} n) queries. Building on our edge estimation result, we give the first {non-adaptive} algorithm for outputting a nearly uniformly sampled edge with query complexity Õ(ε^{-6} log⁶ n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log³ n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a Õ(n log^8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n²) queries.
Cite as
Raghavendra Addanki, Andrew McGregor, and Cameron Musco. Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{addanki_et_al:LIPIcs.ESA.2022.2,
author = {Addanki, Raghavendra and McGregor, Andrew and Musco, Cameron},
title = {{Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {2:1--2:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.2},
URN = {urn:nbn:de:0030-drops-169400},
doi = {10.4230/LIPIcs.ESA.2022.2},
annote = {Keywords: sublinear graph algorithms, bipartite independent set queries, edge sampling and counting, graph connectivity, query adaptivity}
}
Document
Hardness of Token Swapping on Trees
Abstract
Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree.
These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown.
We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2.
Cite as
Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of Token Swapping on Trees. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aichholzer_et_al:LIPIcs.ESA.2022.3,
author = {Aichholzer, Oswin and Demaine, Erik D. and Korman, Matias and Lubiw, Anna and Lynch, Jayson and Mas\'{a}rov\'{a}, Zuzana and Rudoy, Mikhail and Vassilevska Williams, Virginia and Wein, Nicole},
title = {{Hardness of Token Swapping on Trees}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.3},
URN = {urn:nbn:de:0030-drops-169413},
doi = {10.4230/LIPIcs.ESA.2022.3},
annote = {Keywords: Sorting, Token swapping, Trees, NP-hard, Approximation}
}
Document
Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities
Abstract
We study the b-matching problem in bipartite graphs G = (S,R,E). Each vertex s ∈ S is a server with individual capacity b_s. The vertices r ∈ R are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that G is a (k,d)-graph [J. Naor and D. Wajc, 2018], where k specifies a lower bound on the degree of each server and d is an upper bound on the degree of each request. This setting models matching problems in timely applications.
We present tight upper and lower bounds on the performance of deterministic online algorithms. In particular, we develop a new online algorithm via a primal-dual analysis. The optimal competitive ratio tends to 1, for arbitrary k ≥ d, as the server capacities increase. Hence, nearly optimal solutions can be computed online. Our results also hold for the vertex-weighted problem extension, and thus for AdWords and auction problems in which each bidder issues individual, equally valued bids.
Our bounds improve the previous best competitive ratios. The asymptotic competitiveness of 1 is a significant improvement over the previous factor of 1-1/e^{k/d}, for the interesting range where k/d ≥ 1 is small. Recall that 1-1/e ≈ 0.63. Matching problems that admit a competitive ratio arbitrarily close to 1 are rare. Prior results rely on randomization or probabilistic input models.
Cite as
Susanne Albers and Sebastian Schubert. Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{albers_et_al:LIPIcs.ESA.2022.4,
author = {Albers, Susanne and Schubert, Sebastian},
title = {{Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {4:1--4:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.4},
URN = {urn:nbn:de:0030-drops-169420},
doi = {10.4230/LIPIcs.ESA.2022.4},
annote = {Keywords: online algorithms, deterministic algorithms, primal-dual analysis, b-matching, bounded-degree graph, variable vertex capacities, unweighted matching, vertex-weighted matching}
}
Document
TSP in a Simple Polygon
Abstract
We study the Traveling Salesman Problem inside a simple polygon. In this problem, which we call tsp in a simple polygon, we wish to compute a shortest tour that visits a given set S of n sites inside a simple polygon P with m edges while staying inside the polygon. This natural problem has, to the best of our knowledge, not been studied so far from a theoretical perspective. It can be solved exactly in poly(n,m) + 2^O(√nlog n) time, using an algorithm by Marx, Pilipczuk, and Pilipczuk (FOCS 2018) for subset tsp as a subroutine. We present a much simpler algorithm that solves tsp in a simple polygon directly and that has the same running time.
Cite as
Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous. TSP in a Simple Polygon. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{alkema_et_al:LIPIcs.ESA.2022.5,
author = {Alkema, Henk and de Berg, Mark and Monemizadeh, Morteza and Theocharous, Leonidas},
title = {{TSP in a Simple Polygon}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {5:1--5:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.5},
URN = {urn:nbn:de:0030-drops-169434},
doi = {10.4230/LIPIcs.ESA.2022.5},
annote = {Keywords: Traveling Salesman Problem, Subexponential algorithms, TSP with obstacles}
}
Document
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming
Abstract
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value.
Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.
Cite as
Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6,
author = {Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos},
title = {{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.6},
URN = {urn:nbn:de:0030-drops-169444},
doi = {10.4230/LIPIcs.ESA.2022.6},
annote = {Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum}
}
Document
Techniques for Generalized Colorful k-Center Problems
Abstract
Fair clustering enjoyed a surge of interest recently. One appealing way of integrating fairness aspects into classical clustering problems is by introducing multiple covering constraints. This is a natural generalization of the robust (or outlier) setting, which has been studied extensively and is amenable to a variety of classic algorithmic techniques. In contrast, for the case of multiple covering constraints (the so-called colorful setting), specialized techniques have only been developed recently for k-Center clustering variants, which is also the focus of this paper.
While prior techniques assume covering constraints on the clients, they do not address additional constraints on the facilities, which has been extensively studied in non-colorful settings. In this paper, we present a quite versatile framework to deal with various constraints on the facilities in the colorful setting, by combining ideas from the iterative greedy procedure for Colorful k-Center by Inamdar and Varadarajan with new ingredients. To exemplify our framework, we show how it leads, for a constant number γ of colors, to the first constant-factor approximations for both Colorful Matroid Supplier with respect to a linear matroid and Colorful Knapsack Supplier. In both cases, we readily get an O(2^γ)-approximation.
Moreover, for Colorful Knapsack Supplier, we show that it is possible to obtain constant approximation guarantees that are independent of the number of colors γ, as long as γ = O(1), which is needed to obtain a polynomial running time. More precisely, we obtain a 7-approximation by extending a technique recently introduced by Jia, Sheth, and Svensson for Colorful k-Center.
Cite as
Georg Anegg, Laura Vargas Koch, and Rico Zenklusen. Techniques for Generalized Colorful k-Center Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{anegg_et_al:LIPIcs.ESA.2022.7,
author = {Anegg, Georg and Vargas Koch, Laura and Zenklusen, Rico},
title = {{Techniques for Generalized Colorful k-Center Problems}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {7:1--7:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.7},
URN = {urn:nbn:de:0030-drops-169458},
doi = {10.4230/LIPIcs.ESA.2022.7},
annote = {Keywords: Approximation Algorithms, Fair Clustering, Colorful k-Center}
}
Document
Simple Streaming Algorithms for Edge Coloring
Abstract
We revisit the classical edge coloring problem for general graphs in the streaming model. In this model, the input graph is presented as a stream of edges, and the algorithm must report colors assigned to the edges in a streaming fashion, using a memory of size O(n polylog n) on graphs of up to O(n²) edges. In ESA 2019 and SOSA 2021, two elegant randomized algorithms were presented for this problem in the adversarial edge arrival model, where the latest one colors any input graph using O(Δ²/s) colors with high probability in Õ(ns) space. In this short note, we propose two extremely simple streaming algorithms that achieve the same color and space bounds deterministically. Besides being surprisingly simple, our algorithms do not use randomness at all, and are very simple to analyze.
Cite as
Mohammad Ansari, Mohammad Saneian, and Hamid Zarrabi-Zadeh. Simple Streaming Algorithms for Edge Coloring. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 8:1-8:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ansari_et_al:LIPIcs.ESA.2022.8,
author = {Ansari, Mohammad and Saneian, Mohammad and Zarrabi-Zadeh, Hamid},
title = {{Simple Streaming Algorithms for Edge Coloring}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {8:1--8:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.8},
URN = {urn:nbn:de:0030-drops-169468},
doi = {10.4230/LIPIcs.ESA.2022.8},
annote = {Keywords: Edge coloring, streaming model, adversarial order}
}
Document
Computing Smallest Convex Intersecting Polygons
Abstract
A polygon C is an intersecting polygon for a set O of objects in ℝ² if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n.
Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.
Cite as
Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos. Computing Smallest Convex Intersecting Polygons. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{antoniadis_et_al:LIPIcs.ESA.2022.9,
author = {Antoniadis, Antonios and de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Skarlatos, Antonis},
title = {{Computing Smallest Convex Intersecting Polygons}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {9:1--9:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.9},
URN = {urn:nbn:de:0030-drops-169470},
doi = {10.4230/LIPIcs.ESA.2022.9},
annote = {Keywords: convex hull, imprecise points, computational geometry}
}
Document
The Price of Hierarchical Clustering
Abstract
Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of α if the costs of each k-clustering in the hierarchy are at most α times the costs of an optimal k-clustering. We study as cost functions the maximum (discrete) radius of any cluster (k-center problem) and the maximum diameter of any cluster (k-diameter problem).
In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of 1 cannot be achieved. We call the smallest approximation factor that can be achieved for any instance the price of hierarchy. For the k-diameter problem we improve the upper bound on the price of hierarchy to 3+2√2≈ 5.83. Moreover we significantly improve the lower bounds for k-center and k-diameter, proving a price of hierarchy of exactly 4 and 3+2√2, respectively.
Cite as
Anna Arutyunova and Heiko Röglin. The Price of Hierarchical Clustering. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{arutyunova_et_al:LIPIcs.ESA.2022.10,
author = {Arutyunova, Anna and R\"{o}glin, Heiko},
title = {{The Price of Hierarchical Clustering}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {10:1--10:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.10},
URN = {urn:nbn:de:0030-drops-169487},
doi = {10.4230/LIPIcs.ESA.2022.10},
annote = {Keywords: Hierarchical Clustering, approximation Algorithms, k-center Problem}
}
Document
Bounding and Computing Obstacle Numbers of Graphs
Abstract
An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons.
It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)²) [Dujmović and Morin, 2015]. We improve this lower bound to Ω(n/log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n²) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances.
We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.
Cite as
Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, and Alexander Wolff. Bounding and Computing Obstacle Numbers of Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{balko_et_al:LIPIcs.ESA.2022.11,
author = {Balko, Martin and Chaplick, Steven and Ganian, Robert and Gupta, Siddharth and Hoffmann, Michael and Valtr, Pavel and Wolff, Alexander},
title = {{Bounding and Computing Obstacle Numbers of Graphs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {11:1--11:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.11},
URN = {urn:nbn:de:0030-drops-169495},
doi = {10.4230/LIPIcs.ESA.2022.11},
annote = {Keywords: Obstacle representation, Obstacle number, Visibility, NP-hardness, FPT}
}
Document
Computing NP-Hard Repetitiveness Measures via MAX-SAT
Abstract
Repetitiveness measures reveal profound characteristics of datasets, and give rise to compressed data structures and algorithms working in compressed space. Alas, the computation of some of these measures is NP-hard, and straight-forward computation is infeasible for datasets of even small sizes. Three such measures are the smallest size of a string attractor, the smallest size of a bidirectional macro scheme, and the smallest size of a straight-line program. While a vast variety of implementations for heuristically computing approximations exist, exact computation of these measures has received little to no attention. In this paper, we present MAX-SAT formulations that provide the first non-trivial implementations for exact computation of smallest string attractors, smallest bidirectional macro schemes, and smallest straight-line programs. Computational experiments show that our implementations work for texts of length up to a few hundred for straight-line programs and bidirectional macro schemes, and texts even over a million for string attractors.
Cite as
Hideo Bannai, Keisuke Goto, Masakazu Ishihata, Shunsuke Kanda, Dominik Köppl, and Takaaki Nishimoto. Computing NP-Hard Repetitiveness Measures via MAX-SAT. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bannai_et_al:LIPIcs.ESA.2022.12,
author = {Bannai, Hideo and Goto, Keisuke and Ishihata, Masakazu and Kanda, Shunsuke and K\"{o}ppl, Dominik and Nishimoto, Takaaki},
title = {{Computing NP-Hard Repetitiveness Measures via MAX-SAT}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {12:1--12:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.12},
URN = {urn:nbn:de:0030-drops-169505},
doi = {10.4230/LIPIcs.ESA.2022.12},
annote = {Keywords: repetitiveness measures, string attractor, bidirectional macro scheme}
}
Document
Online Metric Allocation and Time-Varying Regularization
Abstract
We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server.
Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm.
We also consider the case of non-convex cost functions. Using a simple 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.
Cite as
Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13,
author = {Bansal, Nikhil and Coester, Christian},
title = {{Online Metric Allocation and Time-Varying Regularization}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.13},
URN = {urn:nbn:de:0030-drops-169515},
doi = {10.4230/LIPIcs.ESA.2022.13},
annote = {Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization}
}
Document
An Upper Bound on the Number of Extreme Shortest Paths in Arbitrary Dimensions
Abstract
Graphs with multiple edge costs arise naturally in the route planning domain when apart from travel time other criteria like fuel consumption or positive height difference are also objectives to be minimized. In such a scenario, this paper investigates the number of extreme shortest paths between a given source-target pair s, t. We show that for a fixed but arbitrary number of cost types d ≥ 1 the number of extreme shortest paths is in n^O(log^{d-1}n) in graphs G with n nodes. This is a generalization of known upper bounds for d = 2 and d = 3.
Cite as
Florian Barth, Stefan Funke, and Claudius Proissl. An Upper Bound on the Number of Extreme Shortest Paths in Arbitrary Dimensions. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{barth_et_al:LIPIcs.ESA.2022.14,
author = {Barth, Florian and Funke, Stefan and Proissl, Claudius},
title = {{An Upper Bound on the Number of Extreme Shortest Paths in Arbitrary Dimensions}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {14:1--14:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.14},
URN = {urn:nbn:de:0030-drops-169525},
doi = {10.4230/LIPIcs.ESA.2022.14},
annote = {Keywords: Parametric Shortest Paths, Extreme Shortest Paths}
}
Document
Galactic Token Sliding
Abstract
Given a graph G and two independent sets I_s and I_t of size k, the Independent Set Reconfiguration problem asks whether there exists a sequence of independent sets (each of size k) I_s = I₀, I₁, I₂, …, I_𝓁 = I_t such that each independent set is obtained from the previous one using a so-called reconfiguration step. Viewing each independent set as a collection of k tokens placed on the vertices of a graph G, the two most studied reconfiguration steps are token jumping and token sliding. In the Token Jumping variant of the problem, a single step allows a token to jump from one vertex to any other vertex in the graph. In the Token Sliding variant, a token is only allowed to slide from a vertex to one of its neighbors. Like the Independent Set problem, both of the aforementioned problems are known to be W[1]-hard on general graphs (for parameter k). A very fruitful line of research [Bodlaender, 1988; Grohe et al., 2017; Telle and Villanger, 2019; Pilipczuk and Siebertz, 2021] has showed that the Independent Set problem becomes fixed-parameter tractable when restricted to sparse graph classes, such as planar, bounded treewidth, nowhere-dense, and all the way to biclique-free graphs. Over a series of papers, the same was shown to hold for the Token Jumping problem [Ito et al., 2014; Lokshtanov et al., 2018; Siebertz, 2018; Bousquet et al., 2017]. As for the Token Sliding problem, which is mentioned in most of these papers, almost nothing is known beyond the fact that the problem is polynomial-time solvable on trees [Demaine et al., 2015] and interval graphs [Marthe Bonamy and Nicolas Bousquet, 2017]. We remedy this situation by introducing a new model for the reconfiguration of independent sets, which we call galactic reconfiguration. Using this new model, we show that (standard) Token Sliding is fixed-parameter tractable on graphs of bounded degree, planar graphs, and chordal graphs of bounded clique number. We believe that the galactic reconfiguration model is of independent interest and could potentially help in resolving the remaining open questions concerning the (parameterized) complexity of Token Sliding.
Cite as
Valentin Bartier, Nicolas Bousquet, and Amer E. Mouawad. Galactic Token Sliding. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bartier_et_al:LIPIcs.ESA.2022.15,
author = {Bartier, Valentin and Bousquet, Nicolas and Mouawad, Amer E.},
title = {{Galactic Token Sliding}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {15:1--15:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.15},
URN = {urn:nbn:de:0030-drops-169535},
doi = {10.4230/LIPIcs.ESA.2022.15},
annote = {Keywords: reconfiguration, independent set, galactic reconfiguration, sparse graphs, token sliding, parameterized complexity}
}
Document
When Are Cache-Oblivious Algorithms Cache Adaptive? A Case Study of Matrix Multiplication and Sorting
Abstract
Cache-adaptive algorithms are a class of algorithms that achieve optimal utilization of dynamically changing memory. These memory fluctuations are the norm in today’s multi-threaded shared-memory machines and time-sharing caches.
Bender et al. [Bender et al., 2014] proved that many cache-oblivious algorithms are optimally cache-adaptive, but that some cache-oblivious algorithms can be relatively far from optimally cache-adaptive on worst-case memory fluctuations. This worst-case gap between cache obliviousness and cache adaptivity depends on a highly-structured, adversarial memory profile. Existing cache-adaptive analysis does not predict the relative performance of cache-oblivious and cache-adaptive algorithms on non-adversarial profiles. Does the worst-case gap appear in practice, or is it an artifact of an unrealistically powerful adversary?
This paper sheds light on the question of whether cache-oblivious algorithms can effectively adapt to realistically fluctuating memory sizes; the paper focuses on matrix multiplication and sorting. The two matrix-multiplication algorithms in this paper are canonical examples of "(a, b, c)-regular" cache-oblivious algorithms, which underlie much of the existing theory on cache-adaptivity. Both algorithms have the same asymptotic I/O performance when the memory size remains fixed, but one is optimally cache-adaptive, and the other is not. In our experiments, we generate both adversarial and non-adversarial memory workloads. The performance gap between the algorithms for matrix multiplication grows with problem size (up to 3.8×) on the adversarial profiles, but the gap does not grow with problem size (stays at 2×) on non-adversarial profiles. The sorting algorithms in this paper are not "(a, b, c)-regular," but they have been well-studied in the classical external-memory model when the memory size does not fluctuate. The relative performance of a non-oblivious (cache-aware) sorting algorithm degrades with the problem size: it incurs up to 6 × the number of disk I/Os compared to an oblivious adaptive algorithm on both adversarial and non-adversarial profiles.
To summarize, in all our experiments, the cache-oblivious matrix-multiplication and sorting algorithms that we tested empirically adapt well to memory fluctuations. We conjecture that cache-obliviousness will empirically help achieve adaptivity for other problems with similar structures.
Cite as
Arghya Bhattacharya, Abiyaz Chowdhury, Helen Xu, Rathish Das, Rezaul A. Chowdhury, Rob Johnson, Rishab Nithyanand, and Michael A. Bender. When Are Cache-Oblivious Algorithms Cache Adaptive? A Case Study of Matrix Multiplication and Sorting. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2022.16,
author = {Bhattacharya, Arghya and Chowdhury, Abiyaz and Xu, Helen and Das, Rathish and Chowdhury, Rezaul A. and Johnson, Rob and Nithyanand, Rishab and Bender, Michael A.},
title = {{When Are Cache-Oblivious Algorithms Cache Adaptive? A Case Study of Matrix Multiplication and Sorting}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {16:1--16:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.16},
URN = {urn:nbn:de:0030-drops-169543},
doi = {10.4230/LIPIcs.ESA.2022.16},
annote = {Keywords: Cache-adaptive algorithms, cache-oblivious algorithms}
}
Document
Simple Dynamic Spanners with Near-Optimal Recourse Against an Adaptive Adversary
Abstract
Designing dynamic algorithms against an adaptive adversary whose performance match the ones assuming an oblivious adversary is a major research program in the field of dynamic graph algorithms. One of the prominent examples whose oblivious-vs-adaptive gap remains maximally large is the fully dynamic spanner problem; there exist algorithms assuming an oblivious adversary with near-optimal size-stretch trade-off using only polylog(n) update time [Baswana, Khurana, and Sarkar TALG'12; Forster and Goranci STOC'19; Bernstein, Forster, and Henzinger SODA'20], while against an adaptive adversary, even when we allow infinite time and only count recourse (i.e. the number of edge changes per update in the maintained spanner), all previous algorithms with stretch at most log⁵(n) require at least Ω(n) amortized recourse [Ausiello, Franciosa, and Italiano ESA'05].
In this paper, we completely close this gap with respect to recourse by showing algorithms against an adaptive adversary with near-optimal size-stretch trade-off and recourse. More precisely, for any k ≥ 1, our algorithm maintains a (2k-1)-spanner of size O(n^{1+1/k}log n) with O(log n) amortized recourse, which is optimal in all parameters up to a O(log n) factor. As a step toward algorithms with small update time (not just recourse), we show another algorithm that maintains a 3-spanner of size Õ(n^{1.5}) with polylog(n) amortized recourse and simultaneously Õ(√n) worst-case update time.
Cite as
Sayan Bhattacharya, Thatchaphol Saranurak, and Pattara Sukprasert. Simple Dynamic Spanners with Near-Optimal Recourse Against an Adaptive Adversary. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2022.17,
author = {Bhattacharya, Sayan and Saranurak, Thatchaphol and Sukprasert, Pattara},
title = {{Simple Dynamic Spanners with Near-Optimal Recourse Against an Adaptive Adversary}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {17:1--17:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.17},
URN = {urn:nbn:de:0030-drops-169555},
doi = {10.4230/LIPIcs.ESA.2022.17},
annote = {Keywords: Algorithms, Dynamic Algorithms, Spanners, Recourse}
}
Document
Online Spanners in Metric Spaces
Abstract
Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight.
Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm.
Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.
Cite as
Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth. Online Spanners in Metric Spaces. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhore_et_al:LIPIcs.ESA.2022.18,
author = {Bhore, Sujoy and Filtser, Arnold and Khodabandeh, Hadi and T\'{o}th, Csaba D.},
title = {{Online Spanners in Metric Spaces}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {18:1--18:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.18},
URN = {urn:nbn:de:0030-drops-169564},
doi = {10.4230/LIPIcs.ESA.2022.18},
annote = {Keywords: spanner, online algorithm, lightness, sparsity, minimum weight}
}
Document
Sparse Temporal Spanners with Low Stretch
Abstract
A temporal graph is an undirected graph G = (V,E) along with a function λ : E → ℕ^+ that assigns a time-label to each edge in E. A path in G such that the traversed time-labels are non-decreasing is called a temporal path. Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is measured as the number of its edges.
In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^{1+1/k}) edges, where k > 1 is an integer parameter of choice. Choosing k = ⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity.
We then turn our attention to general temporal graphs. Since Ω(n²) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω(n²) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of Õ(n² / β) on the size of any temporal β-additive spanner, which we show to be tight up to polylogarithmic factors.
Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.
Cite as
Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse Temporal Spanners with Low Stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bilo_et_al:LIPIcs.ESA.2022.19,
author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Rossi, Mirko},
title = {{Sparse Temporal Spanners with Low Stretch}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {19:1--19:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.19},
URN = {urn:nbn:de:0030-drops-169575},
doi = {10.4230/LIPIcs.ESA.2022.19},
annote = {Keywords: temporal spanners, temporal graphs, graph sparsification, approximate distances}
}
Document
Resource Sharing Revisited: Local Weak Duality and Optimal Convergence
Abstract
We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an 𝓁_∞-minimal element in a Minkowski sum 𝒳 = ∑_{C ∈ 𝒞} X_C of non-empty closed convex sets X_C ⊆ ℝ^ℛ_{≥ 0}, where 𝒞 and ℛ are finite sets. We assume that an oracle for approximate linear minimization over X_C is given.
We improve on the currently fastest known FPTAS in various ways. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately 𝓁_{∞}-minimal, but among such solutions, also its second-highest entry is approximately minimal.
Based on a result by Klein and Young [Klein and Young, 2015], we provide a lower bound of Ω((𝒞|+|ℛ|)/δ² log |ℛ|) required oracle calls for a natural class of algorithms. Our FPTAS is optimal within this class - its running time matches the lower bound precisely, and thus improves on the previously best-known running time for the primal as well as the dual problem.
Cite as
Daniel Blankenburg. Resource Sharing Revisited: Local Weak Duality and Optimal Convergence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blankenburg:LIPIcs.ESA.2022.20,
author = {Blankenburg, Daniel},
title = {{Resource Sharing Revisited: Local Weak Duality and Optimal Convergence}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.20},
URN = {urn:nbn:de:0030-drops-169582},
doi = {10.4230/LIPIcs.ESA.2022.20},
annote = {Keywords: Resource sharing, Dantzig-Wolfe-type algorithms, Decreasing minimization}
}
Document
On the External Validity of Average-Case Analyses of Graph Algorithms
Abstract
The number one criticism of average-case analysis is that we do not actually know the probability distribution of real-world inputs. Thus, analyzing an algorithm on some random model has no implications for practical performance. At its core, this criticism doubts the existence of external validity, i.e., it assumes that algorithmic behavior on the somewhat simple and clean models does not translate beyond the models to practical performance real-world input.
With this paper, we provide a first step towards studying the question of external validity systematically. To this end, we evaluate the performance of six graph algorithms on a collection of 2751 sparse real-world networks depending on two properties; the heterogeneity (variance in the degree distribution) and locality (tendency of edges to connect vertices that are already close). We compare this with the performance on generated networks with varying locality and heterogeneity. We find that the performance in the idealized setting of network models translates surprisingly well to real-world networks. Moreover, heterogeneity and locality appear to be the core properties impacting the performance of many graph algorithms.
Cite as
Thomas Bläsius and Philipp Fischbeck. On the External Validity of Average-Case Analyses of Graph Algorithms. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2022.21,
author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp},
title = {{On the External Validity of Average-Case Analyses of Graph Algorithms}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.21},
URN = {urn:nbn:de:0030-drops-169590},
doi = {10.4230/LIPIcs.ESA.2022.21},
annote = {Keywords: Average Case, Network Models, Empirical Evaluation}
}
Document
On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations
Abstract
For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today’s computation, we employ one of the most successful models of such precomputation - the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations.
We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of {Subset TSP}, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.
Cite as
Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla. On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blazej_et_al:LIPIcs.ESA.2022.22,
author = {Bla\v{z}ej, V\'{a}clav and Choudhary, Pratibha and Knop, Du\v{s}an and Schierreich, \v{S}imon and Such\'{y}, Ond\v{r}ej and Valla, Tom\'{a}\v{s}},
title = {{On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.22},
URN = {urn:nbn:de:0030-drops-169600},
doi = {10.4230/LIPIcs.ESA.2022.22},
annote = {Keywords: Traveling Salesperson, Subset TSP, Waypoint Routing, Kernelization}
}
Document
Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case
Abstract
Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function 𝓁. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM).
Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function 𝓁 is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function 𝓁 is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).
Cite as
Kobi Bodek and Moran Feldman. Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bodek_et_al:LIPIcs.ESA.2022.23,
author = {Bodek, Kobi and Feldman, Moran},
title = {{Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {23:1--23:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.23},
URN = {urn:nbn:de:0030-drops-169618},
doi = {10.4230/LIPIcs.ESA.2022.23},
annote = {Keywords: Unconstrained submodular maximization, regularization, double greedy, non-oblivious local search, inapproximability}
}
Document
List Colouring Trees in Logarithmic Space
Abstract
We show that List Colouring can be solved on n-vertex trees by a deterministic Turing machine using O(log n) bits on the worktape. Given an n-vertex graph G = (V,E) and a list L(v) ⊆ {1,… ,n} of available colours for each v ∈ V, a list colouring for G is a proper colouring c such that c(v) ∈ L(v) for all v.
Cite as
Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. List Colouring Trees in Logarithmic Space. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bodlaender_et_al:LIPIcs.ESA.2022.24,
author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo},
title = {{List Colouring Trees in Logarithmic Space}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {24:1--24:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.24},
URN = {urn:nbn:de:0030-drops-169620},
doi = {10.4230/LIPIcs.ESA.2022.24},
annote = {Keywords: List colouring, trees, space complexity, logspace, graph algorithms, tree-partition-width}
}
Document
Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget
Abstract
In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of 𝒪({k⁷ log n}) while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bartłomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of Ω(n) on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented.
As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs 𝒜. We show that if there is a set 𝒜' of non-intersecting unit arcs of size L²-1 such that 𝒜 ∪ 𝒜' does not contain L+1 arcs intersecting in one point, then it is possible to color 𝒜 with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color 𝒜 with L+1 colors or more.
Cite as
Bartłomiej Bosek and Anna Zych-Pawlewicz. Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bosek_et_al:LIPIcs.ESA.2022.25,
author = {Bosek, Bart{\l}omiej and Zych-Pawlewicz, Anna},
title = {{Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.25},
URN = {urn:nbn:de:0030-drops-169637},
doi = {10.4230/LIPIcs.ESA.2022.25},
annote = {Keywords: dynamic algorithms, unit interval graphs, coloring, recourse budget, parametrized dynamic algorithms}
}
Document
Polynomial Kernel for Immersion Hitting in Tournaments
Abstract
For a fixed simple digraph H without isolated vertices, we consider the problem of deleting arcs from a given tournament to get a digraph which does not contain H as an immersion. We prove that for every H, this problem admits a polynomial kernel when parameterized by the number of deleted arcs. The degree of the bound on the kernel size depends on H.
Cite as
Łukasz Bożyk and Michał Pilipczuk. Polynomial Kernel for Immersion Hitting in Tournaments. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bozyk_et_al:LIPIcs.ESA.2022.26,
author = {Bo\.{z}yk, {\L}ukasz and Pilipczuk, Micha{\l}},
title = {{Polynomial Kernel for Immersion Hitting in Tournaments}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {26:1--26:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.26},
URN = {urn:nbn:de:0030-drops-169642},
doi = {10.4230/LIPIcs.ESA.2022.26},
annote = {Keywords: kernelization, graph immersion, tournament, protrusion}
}
Document
A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension
Abstract
We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism.
It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the Fitting group and π-radicals. A low dimensional WL-algorithm additionally determines the isomorphism type of the socle as well as the factors in the derives series and the upper and lower central series.
We also analyze the behavior of the WL-algorithm for group extensions and prove that a low dimensional WL-algorithm determines the isomorphism types of the composition factors of a group.
Finally we develop a new tool to define a canonical maximal central decomposition for groups. This allows us to show that the Weisfeiler-Leman dimension of a group is at most one larger than the dimensions of its direct indecomposable factors. In other words the Weisfeiler-Leman dimension increases by at most 1 when taking direct products.
Cite as
Jendrik Brachter and Pascal Schweitzer. A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{brachter_et_al:LIPIcs.ESA.2022.27,
author = {Brachter, Jendrik and Schweitzer, Pascal},
title = {{A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {27:1--27:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.27},
URN = {urn:nbn:de:0030-drops-169653},
doi = {10.4230/LIPIcs.ESA.2022.27},
annote = {Keywords: group isomorphism problem, Weisfeiler-Leman algorithms, group invariants, direct product decompositions}
}
Document
Faster Approximate Covering of Subcurves Under the Fréchet Distance
Abstract
Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given Δ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of O(klog(k)) line segments that cover a given polygonal curve of n vertices under Fréchet distance at most O(Δ). We show that the algorithm runs in Õ(k² n + k n³) time in expectation and uses Õ(k n + n³) space. For input curves that are c-packed and lie in the plane, we bound the expected running time by Õ(k² c² n) and the space by Õ(kn + c² n). In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in n without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.
Cite as
Frederik Brüning, Jacobus Conradi, and Anne Driemel. Faster Approximate Covering of Subcurves Under the Fréchet Distance. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bruning_et_al:LIPIcs.ESA.2022.28,
author = {Br\"{u}ning, Frederik and Conradi, Jacobus and Driemel, Anne},
title = {{Faster Approximate Covering of Subcurves Under the Fr\'{e}chet Distance}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {28:1--28:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.28},
URN = {urn:nbn:de:0030-drops-169660},
doi = {10.4230/LIPIcs.ESA.2022.28},
annote = {Keywords: Clustering, Set cover, Fr\'{e}chet distance, Approximation algorithms}
}
Document
Efficient Fréchet Distance Queries for Segments
Abstract
We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(nlog²n) storage, such queries take O(log³ n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk^{3+ε}+n²) size data structure, where k ∈ [1,n] is a parameter the user can choose, and ε > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k)log²n+log⁴ n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments.
We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n^{5/2+ε}) time, improving a previous O(n³) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P.
Cite as
Maike Buchin, Ivor van der Hoog, Tim Ophelders, Lena Schlipf, Rodrigo I. Silveira, and Frank Staals. Efficient Fréchet Distance Queries for Segments. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{buchin_et_al:LIPIcs.ESA.2022.29,
author = {Buchin, Maike and van der Hoog, Ivor and Ophelders, Tim and Schlipf, Lena and Silveira, Rodrigo I. and Staals, Frank},
title = {{Efficient Fr\'{e}chet Distance Queries for Segments}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {29:1--29:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.29},
URN = {urn:nbn:de:0030-drops-169671},
doi = {10.4230/LIPIcs.ESA.2022.29},
annote = {Keywords: Computational Geometry, Data Structures, Fr\'{e}chet distance}
}
Document
Search-Space Reduction via Essential Vertices
Abstract
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.
Cite as
Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-Space Reduction via Essential Vertices. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bumpus_et_al:LIPIcs.ESA.2022.30,
author = {Bumpus, Benjamin Merlin and Jansen, Bart M. P. and de Kroon, Jari J. H.},
title = {{Search-Space Reduction via Essential Vertices}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {30:1--30:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.30},
URN = {urn:nbn:de:0030-drops-169687},
doi = {10.4230/LIPIcs.ESA.2022.30},
annote = {Keywords: fixed-parameter tractability, essential vertices, covering versus packing}
}
Document
Width Helps and Hinders Splitting Flows
Abstract
Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.
Cite as
Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{caceres_et_al:LIPIcs.ESA.2022.31,
author = {C\'{a}ceres, Manuel and Cairo, Massimo and Grigorjew, Andreas and Khan, Shahbaz and Mumey, Brendan and Rizzi, Romeo and Tomescu, Alexandru I. and Williams, Lucia},
title = {{Width Helps and Hinders Splitting Flows}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.31},
URN = {urn:nbn:de:0030-drops-169695},
doi = {10.4230/LIPIcs.ESA.2022.31},
annote = {Keywords: Flow decomposition, approximation algorithms, graph width}
}
Document
Counting Simplices in Hypergraph Streams
Abstract
We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H.
We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs.
Cite as
Amit Chakrabarti and Themistoklis Haris. Counting Simplices in Hypergraph Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chakrabarti_et_al:LIPIcs.ESA.2022.32,
author = {Chakrabarti, Amit and Haris, Themistoklis},
title = {{Counting Simplices in Hypergraph Streams}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {32:1--32:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.32},
URN = {urn:nbn:de:0030-drops-169705},
doi = {10.4230/LIPIcs.ESA.2022.32},
annote = {Keywords: data streaming, graph algorithms, hypergraphs, sub-linear algorithms, triangle counting}
}
Document
Approximation Algorithms for Continuous Clustering and Facility Location Problems
Abstract
In this paper, we consider center-based clustering problems where C, the set of points to be clustered, lies in a metric space (X,d), and the set X of candidate centers is potentially infinite-sized. We call such problems continuous clustering problems to differentiate them from the discrete clustering problems where the set of candidate centers is explicitly given. It is known that for many objectives, when one restricts the set of centers to C itself and applies an α_dis-approximation algorithm for the discrete version, one obtains a β ⋅ α_{dis}-approximation algorithm for the continuous version via the triangle inequality property of the distance function. Here β depends on the objective, and for many objectives such as k-median, β = 2, while for some others such as k-means, β = 4. The motivating question in this paper is whether this gap of factor β between continuous and discrete problems is inherent, or can one design better algorithms for continuous clustering than simply reducing to the discrete case as mentioned above? In a recent SODA 2021 paper, Cohen-Addad, Karthik, and Lee prove a factor-2 and a factor-4 hardness, respectively, for the continuous versions of the k-median and k-means problems, even when the number of cluster centers is a constant. The discrete problem for a constant number of centers is easily solvable exactly using enumeration, and therefore, in certain regimes, the "β-factor loss" seems unavoidable.
In this paper, we describe a technique based on the round-or-cut framework to approach continuous clustering problems. We show that, for the continuous versions of some clustering problems, we can design approximation algorithms attaining a better factor than the β-factor blow-up mentioned above. In particular, we do so for: the uncapacitated facility location problem with uniform facility opening costs (λ-UFL); the k-means problem; the individually fair k-median problem; and the k-center with outliers problem. Notably, for λ-UFL, where β = 2 and the discrete version is NP-hard to approximate within a factor of 1.27, we describe a 2.32-approximation for the continuous version, and indeed 2.32 < 2 × 1.27. Also, for k-means, where β = 4 and the best known approximation factor for the discrete version is 9, we obtain a 32-approximation for the continuous version, which is better than 4 × 9 = 36.
The main challenge one faces is that most algorithms for the discrete clustering problems, including the state of the art solutions, depend on Linear Program (LP) relaxations that become infinite-sized in the continuous version. To overcome this, we design new linear program relaxations for the continuous clustering problems which, although having exponentially many constraints, are amenable to the round-or-cut framework.
Cite as
Deeparnab Chakrabarty, Maryam Negahbani, and Ankita Sarkar. Approximation Algorithms for Continuous Clustering and Facility Location Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chakrabarty_et_al:LIPIcs.ESA.2022.33,
author = {Chakrabarty, Deeparnab and Negahbani, Maryam and Sarkar, Ankita},
title = {{Approximation Algorithms for Continuous Clustering and Facility Location Problems}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.33},
URN = {urn:nbn:de:0030-drops-169710},
doi = {10.4230/LIPIcs.ESA.2022.33},
annote = {Keywords: Approximation Algorithms, Clustering, Facility Location, Fairness, Outliers}
}
Document
Distinct Elements in Streams: An Algorithm for the (Text) Book
Abstract
Given a data stream 𝒟 = ⟨ a₁, a₂, …, a_m ⟩ of m elements where each a_i ∈ [n], the Distinct Elements problem is to estimate the number of distinct elements in 𝒟. Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it. All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.
Cite as
Sourav Chakraborty, N. V. Vinodchandran¹, and Kuldeep S. Meel. Distinct Elements in Streams: An Algorithm for the (Text) Book. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 34:1-34:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2022.34,
author = {Chakraborty, Sourav and Vinodchandran¹, N. V. and Meel, Kuldeep S.},
title = {{Distinct Elements in Streams: An Algorithm for the (Text) Book}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {34:1--34:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.34},
URN = {urn:nbn:de:0030-drops-169725},
doi = {10.4230/LIPIcs.ESA.2022.34},
annote = {Keywords: F₀ Estimation, Streaming, Sampling}
}
Document
Approximate Circular Pattern Matching
Abstract
We investigate the complexity of approximate circular pattern matching (CPM, in short) under the Hamming and edit distance. Under each of these two basic metrics, we are given a length-n text T, a length-m pattern P, and a positive integer threshold k, and we are to report all starting positions (called occurrences) of fragments of T that are at distance at most k from some cyclic rotation of P. In the decision version of the problem, we are to check if there is any such occurrence. All previous results for approximate CPM were either average-case upper bounds or heuristics, with the exception of the work of Charalampopoulos et al. [CKP^+, JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP^+, JCSS'21] from 𝒪(n+(n/m) ⋅ k⁴) to 𝒪(n+(n/m) ⋅ k³ log log k) time; for the edit distance, we give an 𝒪(nk²)-time algorithm. Notably, for the decision versions and wide parameter-ranges, we give algorithms whose complexities are almost identical to the state-of-the-art for standard (i.e., non-circular) approximate pattern matching:
- For the decision version of the approximate CPM problem under the Hamming distance, we obtain an 𝒪(n+(n/m) ⋅ k² log k / log log k)-time algorithm, which works in 𝒪(n) time whenever k = 𝒪(√{m log log m / log m}). In comparison, the fastest algorithm for the standard counterpart of the problem, by Chan et al. [CGKKP, STOC’20], runs in 𝒪(n) time only for k = 𝒪(√m). We achieve this result via a reduction to a geometric problem by building on ideas from [CKP^+, JCSS'21] and Charalampopoulos et al. [CKW, FOCS'20].
- For the decision version of the approximate CPM problem under the edit distance, the 𝒪(nklog³ k) runtime of our algorithm near matches the 𝒪(nk) runtime of the Landau-Vishkin algorithm [LV, J. Algorithms'89] for approximate pattern matching under edit distance; the latter algorithm remains the fastest known for k = Ω(m^{2/5}). As a stepping stone, we propose an 𝒪(nklog³ k)-time algorithm for solving the Longest Prefix k'-Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all k' ∈ {1,…,k}. Our algorithm is based on Tiskin’s theory of seaweeds [Tiskin, Math. Comput. Sci.'08], with recent advancements (see Charalampopoulos et al. [CKW, FOCS'22]), and on exploiting the seaweeds' relation to Monge matrices.
In contrast, we obtain a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute the Strong Exponential Time Hypothesis.
Cite as
Panagiotis Charalampopoulos, Tomasz Kociumaka, Jakub Radoszewski, Solon P. Pissis, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba. Approximate Circular Pattern Matching. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{charalampopoulos_et_al:LIPIcs.ESA.2022.35,
author = {Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Radoszewski, Jakub and Pissis, Solon P. and Rytter, Wojciech and Wale\'{n}, Tomasz and Zuba, Wiktor},
title = {{Approximate Circular Pattern Matching}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {35:1--35:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.35},
URN = {urn:nbn:de:0030-drops-169738},
doi = {10.4230/LIPIcs.ESA.2022.35},
annote = {Keywords: approximate circular pattern matching, Hamming distance, edit distance}
}
Document
Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space
Abstract
We investigate the Euclidean 𝖽-Dimensional Stable Roommates problem, which asks whether a given set V of 𝖽⋅ n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Π = {V₁,…,V_{n}} with |V_i| = 𝖽 for each V_i ∈ Π such that Π is {stable}. Here, {stability} means that no point subset W ⊆ V is blocking Π, and W is said to be {blocking} Π if |W| = 𝖽 such that ∑_{w' ∈ W}δ(w,w') < ∑_{v ∈ Π(w)}δ(w,v) holds for each point w ∈ W, where Π(w) denotes the subset V_i ∈ Π which contains w and δ(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for 𝖽 = 2, we show that such polynomial-time algorithms cannot exist for any fixed number 𝖽 ≥ 3 unless P=NP. Our result for 𝖽 = 3 answers a decade-long open question in the theory of Stable Matching and Hedonic Games [Iwama et al., 2007; Arkin et al., 2009; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; David F. Manlove, 2013].
Cite as
Jiehua Chen and Sanjukta Roy. Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chen_et_al:LIPIcs.ESA.2022.36,
author = {Chen, Jiehua and Roy, Sanjukta},
title = {{Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {36:1--36:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.36},
URN = {urn:nbn:de:0030-drops-169741},
doi = {10.4230/LIPIcs.ESA.2022.36},
annote = {Keywords: stable matchings, multidimensional stable roommates, Euclidean preferences, coalition formation games, stable cores, NP-hardness}
}
Document
Spanner Approximations in Practice
Abstract
A multiplicative α-spanner H is a subgraph of G = (V,E) with the same vertices and fewer edges that preserves distances up to the factor α, i.e., d_H(u,v) ≤ α⋅ d_G(u,v) for all vertices u, v. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch.
Cite as
Markus Chimani and Finn Stutzenstein. Spanner Approximations in Practice. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chimani_et_al:LIPIcs.ESA.2022.37,
author = {Chimani, Markus and Stutzenstein, Finn},
title = {{Spanner Approximations in Practice}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {37:1--37:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.37},
URN = {urn:nbn:de:0030-drops-169750},
doi = {10.4230/LIPIcs.ESA.2022.37},
annote = {Keywords: Graph spanners, experimental study, algorithm engineering}
}
Document
Determinants from Homomorphisms
Abstract
We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.
Cite as
Radu Curticapean. Determinants from Homomorphisms. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 38:1-38:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{curticapean:LIPIcs.ESA.2022.38,
author = {Curticapean, Radu},
title = {{Determinants from Homomorphisms}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {38:1--38:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.38},
URN = {urn:nbn:de:0030-drops-169767},
doi = {10.4230/LIPIcs.ESA.2022.38},
annote = {Keywords: determinant, homomorphisms, matrix trace, Newton identities}
}
Document
Conditional Lower Bounds for Dynamic Geometric Measure Problems
Abstract
We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector Multiplication problem [Henzinger et al., STOC 2015]. In particular we get lower bounds in the incremental and fully-dynamic settings for counting maximal or extremal points in ℝ³, different variants of Klee’s Measure Problem, problems related to finding the largest empty disk in a set of points, and querying the size of the i'th convex layer in a planar set of points. We also answer a question of Chan et al. [SODA 2022] by giving a conditional lower bound for dynamic approximate square set cover. While many conditional lower bounds for dynamic data structures have been proven since the seminal work of Pătraşcu [STOC 2010], few of them relate to computational geometry problems. This is the first paper focusing on this topic. Most problems we consider can be solved in O(nlog n) time in the static case and their dynamic versions have only been approached from the perspective of improving known upper bounds. One exception to this is Klee’s measure problem in ℝ², for which Chan [CGTA 2010] gave an unconditional Ω(√n) lower bound on the worst-case update time. By a similar approach, we show that such a lower bound also holds for an important special case of Klee’s measure problem in ℝ³ known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.
Cite as
Justin Dallant and John Iacono. Conditional Lower Bounds for Dynamic Geometric Measure Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dallant_et_al:LIPIcs.ESA.2022.39,
author = {Dallant, Justin and Iacono, John},
title = {{Conditional Lower Bounds for Dynamic Geometric Measure Problems}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {39:1--39:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.39},
URN = {urn:nbn:de:0030-drops-169777},
doi = {10.4230/LIPIcs.ESA.2022.39},
annote = {Keywords: Computational geometry, Fine-grained complexity, Dynamic data structures}
}
Document
A Simpler QPTAS for Scheduling Jobs with Precedence Constraints
Abstract
We study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a (2-1/m)-approximation. Interestingly, it is open whether the problem is NP-hard if m = 3 which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that m is a constant. In a break-through paper, Levey and Rothvoss presented a (1+ε)-approximation with a running time of n^{(log n)^{O((m²/ε²)log log n)}} [STOC 2016, SICOMP 2019] and this running time was improved to quasi-polynomial by Garg [ICALP 2018] and to even n^O_{m,ε}(log³log n) by Li [SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules.
In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly.
Cite as
Syamantak Das and Andreas Wiese. A Simpler QPTAS for Scheduling Jobs with Precedence Constraints. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 40:1-40:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{das_et_al:LIPIcs.ESA.2022.40,
author = {Das, Syamantak and Wiese, Andreas},
title = {{A Simpler QPTAS for Scheduling Jobs with Precedence Constraints}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {40:1--40:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.40},
URN = {urn:nbn:de:0030-drops-169782},
doi = {10.4230/LIPIcs.ESA.2022.40},
annote = {Keywords: makespan minimization, precedence constraints, QPTAS}
}
Document
A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games
Abstract
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [Tsaknakis and Spirakis, 2008], with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+δ)-Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [Tsaknakis and Spirakis, 2008], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.
Cite as
Argyrios Deligkas, Michail Fasoulakis, and Evangelos Markakis. A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{deligkas_et_al:LIPIcs.ESA.2022.41,
author = {Deligkas, Argyrios and Fasoulakis, Michail and Markakis, Evangelos},
title = {{A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {41:1--41:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.41},
URN = {urn:nbn:de:0030-drops-169790},
doi = {10.4230/LIPIcs.ESA.2022.41},
annote = {Keywords: bimatrix games, approximate Nash equilibria}
}
Document
Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem
Abstract
In a graph G with a source s, we design a distance oracle that can answer the following query: Query(s,t,e) - find the length of shortest path from a fixed source s to any destination vertex t while avoiding any edge e. We design a deterministic algorithm that builds such an oracle in Õ(m √n) time. Our oracle uses Õ(n √n) space and can answer queries in Õ(1) time. Our oracle is an improvement of the work of Bilò et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in Õ(m √n+n²) time.
Using our distance oracle, we also solve the single source replacement path problem (Ssrp problem). Chechik and Cohen (SODA 2019) designed a randomized combinatorial algorithm to solve the Ssrp problem. The running time of their algorithm is Õ(m √n + n²). In this paper, we show that the Ssrp problem can be solved in Õ(m √n + |ℛ|) time, where ℛ is the output set of the Ssrp problem in G. Our Ssrp algorithm is optimal (upto polylogarithmic factor) as there is a conditional lower bound of Ω(m √n) for any combinatorial algorithm that solves this problem.
Cite as
Dipan Dey and Manoj Gupta. Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dey_et_al:LIPIcs.ESA.2022.42,
author = {Dey, Dipan and Gupta, Manoj},
title = {{Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {42:1--42:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.42},
URN = {urn:nbn:de:0030-drops-169800},
doi = {10.4230/LIPIcs.ESA.2022.42},
annote = {Keywords: distance sensitivity oracle, single-source replacement paths}
}
Document
Fast Computation of Zigzag Persistence
Abstract
Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a Δ-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.
Cite as
Tamal K. Dey and Tao Hou. Fast Computation of Zigzag Persistence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dey_et_al:LIPIcs.ESA.2022.43,
author = {Dey, Tamal K. and Hou, Tao},
title = {{Fast Computation of Zigzag Persistence}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {43:1--43:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.43},
URN = {urn:nbn:de:0030-drops-169813},
doi = {10.4230/LIPIcs.ESA.2022.43},
annote = {Keywords: zigzag persistence, persistent homology, fast computation}
}
Document
Turbocharging Heuristics for Weak Coloring Numbers
Abstract
Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number is breached. We provide hardness and tractability results on the corresponding computational subproblems. We implemented several of the resulting algorithms and show them to be competitive with previous approaches on a previously studied set of benchmark instances containing 86 graphs with up to 183831 edges. We obtain improved weak coloring numbers for over half of the instances.
Cite as
Alexander Dobler, Manuel Sorge, and Anaïs Villedieu. Turbocharging Heuristics for Weak Coloring Numbers. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dobler_et_al:LIPIcs.ESA.2022.44,
author = {Dobler, Alexander and Sorge, Manuel and Villedieu, Ana\"{i}s},
title = {{Turbocharging Heuristics for Weak Coloring Numbers}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {44:1--44:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.44},
URN = {urn:nbn:de:0030-drops-169820},
doi = {10.4230/LIPIcs.ESA.2022.44},
annote = {Keywords: Structural sparsity, parameterized algorithms, parameterized complexity, fixed-parameter tractability}
}
Document
A Local Search Algorithm for Large Maximum Weight Independent Set Problems
Abstract
Motivated by a real-world vehicle routing application, we consider the maximum-weight independent set problem: Given a node-weighted graph, find a set of independent (mutually nonadjacent) nodes whose node-weight sum is maximum. Some of the graphs arising in the vehicle routing application are large, having hundreds of thousands of nodes and hundreds of millions of edges.
To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic based on the greedy randomized adaptive search (GRASP) framework. This algorithm, named METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path-relinking is introduced to escape local optima and so is a new alternating augmenting-path local search move that improves algorithm performance.
We compare an implementation of our algorithm with a state-of-the-art publicly available code on public benchmark sets, including some large instances. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances of the maximum weight independent set problem. We hope that our results will lead to even better maximum-weight independent set algorithms.
Cite as
Yuanyuan Dong, Andrew V. Goldberg, Alexander Noe, Nikos Parotsidis, Mauricio G.C. Resende, and Quico Spaen. A Local Search Algorithm for Large Maximum Weight Independent Set Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dong_et_al:LIPIcs.ESA.2022.45,
author = {Dong, Yuanyuan and Goldberg, Andrew V. and Noe, Alexander and Parotsidis, Nikos and Resende, Mauricio G.C. and Spaen, Quico},
title = {{A Local Search Algorithm for Large Maximum Weight Independent Set Problems}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {45:1--45:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.45},
URN = {urn:nbn:de:0030-drops-169839},
doi = {10.4230/LIPIcs.ESA.2022.45},
annote = {Keywords: GRASP, local search, maximum-weight independent set, path-relinking, heuristic, metaheuristic}
}
Document
SAT Backdoors: Depth Beats Size
Abstract
For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. Bounded backdoor size implies bounded backdoor depth, but there are formulas of constant backdoor depth and arbitrarily large backdoor size.
We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. We base our FPT approximation algorithm on a sophisticated notion of obstructions, extending Mählmann et al.’s obstruction trees in various ways, including the addition of separator obstructions. We develop the algorithm through a new game-theoretic framework that simplifies the reasoning about backdoors.
Finally, we show that bounded backdoor depth captures tractable classes of CNF formulas not captured by any known method.
Cite as
Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. SAT Backdoors: Depth Beats Size. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 46:1-46:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dreier_et_al:LIPIcs.ESA.2022.46,
author = {Dreier, Jan and Ordyniak, Sebastian and Szeider, Stefan},
title = {{SAT Backdoors: Depth Beats Size}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {46:1--46:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.46},
URN = {urn:nbn:de:0030-drops-169840},
doi = {10.4230/LIPIcs.ESA.2022.46},
annote = {Keywords: satisfiability, backdoor (depth)}
}
Document
Finding a Cluster in Incomplete Data
Abstract
We study two variants of the fundamental problem of finding a cluster in incomplete data. In the problems under consideration, we are given a multiset of incomplete d-dimensional vectors over the binary domain and integers k and r, and the goal is to complete the missing vector entries so that the multiset of complete vectors either contains (i) a cluster of k vectors of radius at most r, or (ii) a cluster of k vectors of diameter at most r. We give tight characterizations of the parameterized complexity of the problems under consideration with respect to the parameters k, r, and a third parameter that captures the missing vector entries.
Cite as
Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. Finding a Cluster in Incomplete Data. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{eiben_et_al:LIPIcs.ESA.2022.47,
author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan},
title = {{Finding a Cluster in Incomplete Data}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {47:1--47:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.47},
URN = {urn:nbn:de:0030-drops-169858},
doi = {10.4230/LIPIcs.ESA.2022.47},
annote = {Keywords: Parameterized complexity, incomplete data, clustering}
}
Document
Lyndon Arrays Simplified
Abstract
A Lyndon word is a string that is lexicographically smaller than all of its proper suffixes (e.g., "airbus" is a Lyndon word; "amtrak" is not a Lyndon word because its suffix "ak" is lexicographically smaller than "amtrak"). The Lyndon array (sometimes called Lyndon table) identifies the longest Lyndon prefix of each suffix of a string. It is well known that the Lyndon array of a length-n string can be computed in O(n) time. However, most of the existing algorithms require the suffix array, which has theoretical and practical disadvantages. The only known algorithms that compute the Lyndon array in O(n) time without the suffix array (or similar data structures) do so in a particularly space efficient way (Bille et al., ICALP 2020), or in an online manner (Badkobeh et al., CPM 2022). Due to the additional goals of space efficiency and online computation, these algorithms are complicated in technical detail. Using the main ideas of the aforementioned algorithms, we provide a simpler and easier to understand algorithm that computes the Lyndon array in O(n) time.
Cite as
Jonas Ellert. Lyndon Arrays Simplified. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ellert:LIPIcs.ESA.2022.48,
author = {Ellert, Jonas},
title = {{Lyndon Arrays Simplified}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {48:1--48:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.48},
URN = {urn:nbn:de:0030-drops-169863},
doi = {10.4230/LIPIcs.ESA.2022.48},
annote = {Keywords: Lyndon table, Lyndon array, Lyndon word, nearest smaller suffixes, lexicographical ordering, general ordered alphabets, combinatorial algorithms}
}
Document
Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty
Abstract
We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral γ ≥ 2, we present algorithms that are γ-robust and (1+1/γ)-consistent, meaning that they use at most γOPT queries if the predictions are arbitrarily wrong and at most (1+1/γ)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those.
Cite as
Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{erlebach_et_al:LIPIcs.ESA.2022.49,
author = {Erlebach, Thomas and de Lima, Murilo Santos and Megow, Nicole and Schl\"{o}ter, Jens},
title = {{Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {49:1--49:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.49},
URN = {urn:nbn:de:0030-drops-169872},
doi = {10.4230/LIPIcs.ESA.2022.49},
annote = {Keywords: explorable uncertainty, queries, untrusted predictions}
}
Document
Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search
Abstract
We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time.
We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1.
We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].
Cite as
Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{esmer_et_al:LIPIcs.ESA.2022.50,
author = {Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani},
title = {{Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {50:1--50:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.50},
URN = {urn:nbn:de:0030-drops-169887},
doi = {10.4230/LIPIcs.ESA.2022.50},
annote = {Keywords: parameterized approximations, exponential approximations, monotone local search}
}
Document
Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection
Abstract
We develop data structures for intersection detection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study two main problems: (i) Preprocess a set of n tetrahedra in {ℝ}⁴ into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as segment-tetrahedron intersection queries), and (ii) Preprocess a set of n triangles in {ℝ}⁴ into a data structure that supports triangle-intersection queries amid the input triangles (referred to as triangle-triangle intersection queries). As far as we can tell, these problems have not been previously studied.
For problem (i), we first present a "standard" solution which, for any prespecified value n ≤ s ≤ n⁶ of a so-called storage parameter s, yields a data structure with O^*(s) storage and expected preprocessing, which answers an intersection query in O^*(n/s^{1/6}) time (here and in what follows, the O^*(⋅) notation hides subpolynomial factors). For problem (ii), using similar arguments, we present a solution that has the same asymptotic performance bounds.
We then improve the solution for problem (i), and present a more intricate data structure that uses O^*(n²) storage and expected preprocessing, and answers a segment-tetrahedron intersection query in O^*(n^{1/2}) time. Using the parametric search technique of Agarwal and Matoušek [P. K. Agarwal and J. Matoušek, 1993], we can obtain data structures with similar performance bounds for the ray-shooting problem amid tetrahedra in {ℝ}⁴. Unfortunately, so far we do not know how to obtain a similar improvement for problem (ii).
Our algorithms are based on a primal-dual technique for range searching with semi-algebraic sets, based on recent advances in this area [P. K. Agarwal et al., 2021; J. Matoušek and Z. Patáková, 2015]. As this is a result of independent interest, we spell out the details of this technique.
As an application, we present a solution to the problem of "continuous collision detection" amid moving tetrahedra in 3-space. That is, the workspace consists of n tetrahedra, each moving at its own fixed velocity, and the goal is to detect a collision between some pair of moving tetrahedra. Using our solutions to problems (i) and (ii), we obtain an algorithm that detects a collision in O^*(n^{12/7}) expected time. We also present further applications, including an output-sensitive algorithm for constructing the arrangement of n tetrahedra in ℝ⁴ and an output-sensitive algorithm for constructing the intersection or union of two or several nonconvex polyhedra in ℝ⁴.
Cite as
Esther Ezra and Micha Sharir. Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ezra_et_al:LIPIcs.ESA.2022.51,
author = {Ezra, Esther and Sharir, Micha},
title = {{Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {51:1--51:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.51},
URN = {urn:nbn:de:0030-drops-169895},
doi = {10.4230/LIPIcs.ESA.2022.51},
annote = {Keywords: Computational geometry, Ray shooting, Tetrahedra in \{\mathbb{R}\}⁴, Intersection queries in \{\mathbb{R}\}⁴, Polynomial partitioning, Range searching, Semi-algebraic sets, Tradeoff}
}
Document
Submodular Maximization Subject to Matroid Intersection on the Fly
Abstract
Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to k matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in k is needed to obtain an o(k/(log k))-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obtain a constant-factor approximation by maintaining a set of elements whose size is independent of the stream size. Finally, for bipartite matching constraints, a well-known special case of matroid intersection, we present a new technique to obtain hardness bounds that are significantly stronger than those obtained with prior approaches. Prior results left it open whether a 2-approximation may exist in this setting, and only a complexity-theoretic hardness of 1.91 was known. We prove an unconditional hardness of 2.69.
Cite as
Moran Feldman, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Submodular Maximization Subject to Matroid Intersection on the Fly. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{feldman_et_al:LIPIcs.ESA.2022.52,
author = {Feldman, Moran and Norouzi-Fard, Ashkan and Svensson, Ola and Zenklusen, Rico},
title = {{Submodular Maximization Subject to Matroid Intersection on the Fly}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.52},
URN = {urn:nbn:de:0030-drops-169902},
doi = {10.4230/LIPIcs.ESA.2022.52},
annote = {Keywords: Submodular Maximization, Matroid Intersection, Streaming Algorithms}
}
Document
There and Back Again: On Applying Data Reduction Rules by Undoing Others
Abstract
Data reduction rules are an established method in the algorithmic toolbox for tackling computationally challenging problems. A data reduction rule is a polynomial-time algorithm that, given a problem instance as input, outputs an equivalent, typically smaller instance of the same problem. The application of data reduction rules during the preprocessing of problem instances allows in many cases to considerably shrink their size, or even solve them directly. Commonly, these data reduction rules are applied exhaustively and in some fixed order to obtain irreducible instances. It was often observed that by changing the order of the rules, different irreducible instances can be obtained. We propose to "undo" data reduction rules on irreducible instances, by which they become larger, and then subsequently apply data reduction rules again to shrink them. We show that this somewhat counter-intuitive approach can lead to significantly smaller irreducible instances. The process of undoing data reduction rules is not limited to "rolling back" data reduction rules applied to the instance during preprocessing. Instead, we formulate so-called backward rules, which essentially undo a data reduction rule, but without using any information about which data reduction rules were applied to it previously. In particular, based on the example of Vertex Cover we propose two methods applying backward rules to shrink the instances further. In our experiments we show that this way smaller irreducible instances consisting of real-world graphs from the SNAP and DIMACS datasets can be computed.
Cite as
Aleksander Figiel, Vincent Froese, André Nichterlein, and Rolf Niedermeier. There and Back Again: On Applying Data Reduction Rules by Undoing Others. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{figiel_et_al:LIPIcs.ESA.2022.53,
author = {Figiel, Aleksander and Froese, Vincent and Nichterlein, Andr\'{e} and Niedermeier, Rolf},
title = {{There and Back Again: On Applying Data Reduction Rules by Undoing Others}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {53:1--53:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.53},
URN = {urn:nbn:de:0030-drops-169914},
doi = {10.4230/LIPIcs.ESA.2022.53},
annote = {Keywords: Kernelization, Preprocessing, Vertex Cover}
}
Document
Improved Search of Relevant Points for Nearest-Neighbor Classification
Abstract
Given a training set P ⊂ ℝ^d, the nearest-neighbor classifier assigns any query point q ∈ ℝ^d to the class of its closest point in P. To answer these classification queries, some training points are more relevant than others. We say a training point is relevant if its omission from the training set could induce the misclassification of some query point in ℝ^d. These relevant points are commonly known as border points, as they define the boundaries of the Voronoi diagram of P that separate points of different classes. Being able to compute this set of points efficiently is crucial to reduce the size of the training set without affecting the accuracy of the nearest-neighbor classifier.
Improving over a decades-long result by Clarkson (FOCS'94), Eppstein (SOSA’22) recently proposed an output-sensitive algorithm to find the set of border points of P in 𝒪(n² + nk²) time, where k is the size of such set. In this paper, we improve this algorithm to have time complexity equal to 𝒪(nk²) by proving that the first phase of their algorithm, which requires 𝒪(n²) time, are unnecessary.
Cite as
Alejandro Flores-Velazco. Improved Search of Relevant Points for Nearest-Neighbor Classification. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 54:1-54:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{floresvelazco:LIPIcs.ESA.2022.54,
author = {Flores-Velazco, Alejandro},
title = {{Improved Search of Relevant Points for Nearest-Neighbor Classification}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {54:1--54:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.54},
URN = {urn:nbn:de:0030-drops-169922},
doi = {10.4230/LIPIcs.ESA.2022.54},
annote = {Keywords: nearest-neighbor classification, nearest-neighbor rule, decision boundaries, border points, relevant points}
}
Document
Longest Cycle Above Erdős-Gallai Bound
Abstract
In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2^𝒪(k)⋅n^𝒪(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.
Cite as
Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Longest Cycle Above Erdős-Gallai Bound. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fomin_et_al:LIPIcs.ESA.2022.55,
author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill},
title = {{Longest Cycle Above Erd\H{o}s-Gallai Bound}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {55:1--55:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.55},
URN = {urn:nbn:de:0030-drops-169935},
doi = {10.4230/LIPIcs.ESA.2022.55},
annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization, average degree, Erd\H{o}s and Gallai theorem}
}
Document
Improved Polynomial-Time Approximations for Clustering with Minimum Sum of Radii or Diameters
Abstract
We give an improved approximation algorithm for two related clustering problems. In the Minimum Sum of Radii clustering problem (MSR), we are to select k balls in a metric space to cover all points while minimizing the sum of the radii of these balls. In the Minimum Sum of Diameters clustering problem (MSD), we are to simply partition the points of a metric space into k parts while minimizing the sum of the diameters of these parts. We present a 3.389-approximation for MSR and a 6.546-approximation for MSD, improving over their respective 3.504 and 7.008 approximations developed by Charikar and Panigrahy (2001). In particular, our guarantee for MSD is better than twice our guarantee for MSR.
Our approach refines a so-called bipoint rounding procedure of Charikar and Panigrahy’s algorithm by considering centering balls at some points that were not necessarily centers in the bipoint solution. This added versatility enables the analysis of our improved approximation guarantees. We also provide an alternative approach to finding the bipoint solution using a straightforward LP rounding procedure rather than a primal-dual algorithm.
Cite as
Zachary Friggstad and Mahya Jamshidian. Improved Polynomial-Time Approximations for Clustering with Minimum Sum of Radii or Diameters. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{friggstad_et_al:LIPIcs.ESA.2022.56,
author = {Friggstad, Zachary and Jamshidian, Mahya},
title = {{Improved Polynomial-Time Approximations for Clustering with Minimum Sum of Radii or Diameters}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {56:1--56:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.56},
URN = {urn:nbn:de:0030-drops-169946},
doi = {10.4230/LIPIcs.ESA.2022.56},
annote = {Keywords: Approximation Algorithms, Clustering, Linear Programming}
}
Document
Simple Worst-Case Optimal Adaptive Prefix-Free Coding
Abstract
We give a new and simple worst-case optimal algorithm for adaptive prefix-free coding that matches Gagie and Nekrich’s (2009) bounds except for lower-order terms, and uses no data structures more complicated than a lookup table.
Cite as
Travis Gagie. Simple Worst-Case Optimal Adaptive Prefix-Free Coding. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 57:1-57:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gagie:LIPIcs.ESA.2022.57,
author = {Gagie, Travis},
title = {{Simple Worst-Case Optimal Adaptive Prefix-Free Coding}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {57:1--57:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.57},
URN = {urn:nbn:de:0030-drops-169959},
doi = {10.4230/LIPIcs.ESA.2022.57},
annote = {Keywords: Adaptive prefix-free coding, Shannon coding, Lookup tables}
}
Document
Taming Graphs with No Large Creatures and Skinny Ladders
Abstract
We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class 𝒢 there exists a constant k such that no member of 𝒢 contains a k-creature as an induced subgraph or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every G ∈ 𝒢 contains at most p(|V(G)|) minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from 𝒢. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).
Cite as
Jakub Gajarský, Lars Jaffke, Paloma T. Lima, Jana Novotná, Marcin Pilipczuk, Paweł Rzążewski, and Uéverton S. Souza. Taming Graphs with No Large Creatures and Skinny Ladders. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 58:1-58:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gajarsky_et_al:LIPIcs.ESA.2022.58,
author = {Gajarsk\'{y}, Jakub and Jaffke, Lars and Lima, Paloma T. and Novotn\'{a}, Jana and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l} and Souza, U\'{e}verton S.},
title = {{Taming Graphs with No Large Creatures and Skinny Ladders}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {58:1--58:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.58},
URN = {urn:nbn:de:0030-drops-169969},
doi = {10.4230/LIPIcs.ESA.2022.58},
annote = {Keywords: Minimal separator, hereditary graph class}
}
Document
Faster Path Queries in Colored Trees via Sparse Matrix Multiplication and Min-Plus Product
Abstract
Let T be an ordinal tree on n nodes in which each node is assigned a color. We consider the batched colored path counting problem and the batched path mode/least frequent element query problem, in which given n query paths, each identified by a pair of nodes in T, one is asked to answer queries of the following forms: How many distinct colors are there on each query path (i.e. the colored path counting problem); what is the color on each query path that occurs at least/most as frequently as any other colors (i.e. the path mode/least frequent element query problem). By reducing the batched colored path counting problem to sparse matrix multiplication, we design a solution that answers n colored path counting queries in Õ(n^{2ω/(ω+1)}) = O(n^1.40704) time in total, while we reduce batched path mode/least frequent element query to the min-plus-query-witness problem so that we can answer a batch of n queries in Õ(n^{{24+2ω}/{17+ω}}) = O(n^1.483814) time. Previously, both problems could only be solved in Õ(n^1.5) time.
Based on similar techniques, we design a dynamic colored path counting structure supporting both queries and updates in Õ(n^{{ω+1}/{ω+3}}) = O(n^0.627759) time, while our dynamic path mode/least frequent element query structures support each operation in Õ(n^{{16+ω(1,2,1)}/{26+ω(1,2,1)}}) = O(n^0.658139) time, where ω(1, 2, 1) denotes the minimum value such that the product of an n × n² matrix and an n² × n matrix can be computed in O(n^{ω(1, 2, 1)+ε}) time for any constant ε > 0. We also solve batched range mode/least frequent element query problems over arrays in Õ(n^{{18+2ω}/{13+ω}}) = O(n^1.479603) time. Both problems can be viewed as special cases of these batched path queries, and previously, the fastest algorithm for batched range mode queries and batched range least frequent element queries use O(n^1.4805) and Õ(n^1.5) time, respectively.
Cite as
Younan Gao and Meng He. Faster Path Queries in Colored Trees via Sparse Matrix Multiplication and Min-Plus Product. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gao_et_al:LIPIcs.ESA.2022.59,
author = {Gao, Younan and He, Meng},
title = {{Faster Path Queries in Colored Trees via Sparse Matrix Multiplication and Min-Plus Product}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {59:1--59:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.59},
URN = {urn:nbn:de:0030-drops-169971},
doi = {10.4230/LIPIcs.ESA.2022.59},
annote = {Keywords: min-plus product, range mode queries, range least frequent queries, path queries, colored path counting, path mode queries, path least frequent queries}
}
Document
Computing the 4-Edge-Connected Components of a Graph: An Experimental Study
Abstract
The notions of edge-cuts and k-edge-connected components are fundamental in graph theory with numerous practical applications. Very recently, the first linear-time algorithms for computing all the 3-edge cuts and the 4-edge-connected components of a graph have been introduced. In this paper we present carefully engineered implementations of these algorithms and evaluate their efficiency in practice, by performing a thorough empirical study using both real-world graphs taken from a variety of application areas, as well as artificial graphs. To the best of our knowledge, this is the first experimental study for these problems, which highlights the merits and weaknesses of each technique. Furthermore, we present an improved algorithm for computing the 4-edge-connected components of an undirected graph in linear time. The new algorithm uses only elementary data structures, and is implementable in the pointer machine model of computation.
Cite as
Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-Edge-Connected Components of a Graph: An Experimental Study. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2022.60,
author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Kosinas, Evangelos},
title = {{Computing the 4-Edge-Connected Components of a Graph: An Experimental Study}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {60:1--60:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.60},
URN = {urn:nbn:de:0030-drops-169988},
doi = {10.4230/LIPIcs.ESA.2022.60},
annote = {Keywords: Connectivity Cuts, Edge Connectivity, Graph Algorithms}
}
Document
Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited
Abstract
Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + 𝓁, where 𝓁 is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if 𝓁 is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case where 𝓁 is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3.
Cite as
Tatsuya Gima, Takehiro Ito, Yasuaki Kobayashi, and Yota Otachi. Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gima_et_al:LIPIcs.ESA.2022.61,
author = {Gima, Tatsuya and Ito, Takehiro and Kobayashi, Yasuaki and Otachi, Yota},
title = {{Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {61:1--61:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.61},
URN = {urn:nbn:de:0030-drops-169991},
doi = {10.4230/LIPIcs.ESA.2022.61},
annote = {Keywords: Combinatorial reconfiguration, monadic second-order logic, fixed-parameter tractability, treedepth, neighborhood diversity}
}
Document
Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs
Abstract
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an n-vertex planar graph, augments this graph in 𝒪(n²) steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized (Anti)factor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case.
Cite as
Miriam Goetze, Paul Jungeblut, and Torsten Ueckerdt. Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goetze_et_al:LIPIcs.ESA.2022.62,
author = {Goetze, Miriam and Jungeblut, Paul and Ueckerdt, Torsten},
title = {{Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {62:1--62:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.62},
URN = {urn:nbn:de:0030-drops-170007},
doi = {10.4230/LIPIcs.ESA.2022.62},
annote = {Keywords: edge colorings, planar graphs, cubic graphs, generalized factors, SPQR-tree}
}
Document
Adaptive-Adversary-Robust Algorithms via Small Copy Tree Embeddings
Abstract
Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and ill-suited against adaptive adversaries.
In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing O(log n) copies of each vertex while preserving the connectivity structure of every subgraph and O(log² n)-approximating the cost of every subgraph. Using this embedding we obtain the first deterministic bicriteria approximation algorithm for the online covering Steiner problem as well as the first poly-log approximations for demand-robust Steiner forest, group Steiner tree and group Steiner forest.
Cite as
Bernhard Haepler, D. Ellis Hershkowitz, and Goran Zuzic. Adaptive-Adversary-Robust Algorithms via Small Copy Tree Embeddings. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{haepler_et_al:LIPIcs.ESA.2022.63,
author = {Haepler, Bernhard and Hershkowitz, D. Ellis and Zuzic, Goran},
title = {{Adaptive-Adversary-Robust Algorithms via Small Copy Tree Embeddings}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {63:1--63:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.63},
URN = {urn:nbn:de:0030-drops-170016},
doi = {10.4230/LIPIcs.ESA.2022.63},
annote = {Keywords: Tree metrics, metric embeddings, approximation algorithms, group Steiner forest, group Steiner tree, demand-robust algorithms, online algorithms, deterministic algorithms}
}
Document
Hedonic Games and Treewidth Revisited
Abstract
We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently?
We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree Δ. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2^{O(Δ⁵t)}. We present an algorithm with parameter dependence (Δ t)^{O(Δ t)}, significantly improving upon the parameter dependence on Δ given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence t^{o(t)} for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on Δ and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH.
We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided.
Cite as
Tesshu Hanaka and Michael Lampis. Hedonic Games and Treewidth Revisited. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hanaka_et_al:LIPIcs.ESA.2022.64,
author = {Hanaka, Tesshu and Lampis, Michael},
title = {{Hedonic Games and Treewidth Revisited}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {64:1--64:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.64},
URN = {urn:nbn:de:0030-drops-170025},
doi = {10.4230/LIPIcs.ESA.2022.64},
annote = {Keywords: Hedonic Games, Nash Equilibrium, Treewidth}
}
Document
Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs
Abstract
A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ε}) update time and O(m^{1 -ε}) query time, for any small ε > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.
Cite as
Monika Henzinger, Ami Paz, and A. R. Sricharan. Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{henzinger_et_al:LIPIcs.ESA.2022.65,
author = {Henzinger, Monika and Paz, Ami and Sricharan, A. R.},
title = {{Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {65:1--65:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.65},
URN = {urn:nbn:de:0030-drops-170035},
doi = {10.4230/LIPIcs.ESA.2022.65},
annote = {Keywords: Dynamic graph algorithms, Expander graphs, Power-law graphs}
}
Document
O(1) Steiner Point Removal in Series-Parallel Graphs
Abstract
We study how to vertex-sparsify a graph while preserving both the graph’s metric and structure. Specifically, we study the Steiner point removal (SPR) problem where we are given a weighted graph G = (V,E,w) and terminal set V' ⊆ V and must compute a weighted minor G' = (V',E', w') of G which approximates G’s metric on V'. A major open question in the area of metric embeddings is the existence of O(1) multiplicative distortion SPR solutions for every (non-trivial) minor-closed family of graphs. To this end prior work has studied SPR on trees, cactus and outerplanar graphs and showed that in these graphs such a minor exists with O(1) distortion.
We give O(1) distortion SPR solutions for series-parallel graphs, extending the frontier of this line of work. The main engine of our approach is a new metric decomposition for series-parallel graphs which we call a hammock decomposition. Roughly, a hammock decomposition is a forest-like structure that preserves certain critical parts of the metric induced by a series-parallel graph.
Cite as
D. Ellis Hershkowitz and Jason Li. O(1) Steiner Point Removal in Series-Parallel Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 66:1-66:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hershkowitz_et_al:LIPIcs.ESA.2022.66,
author = {Hershkowitz, D. Ellis and Li, Jason},
title = {{O(1) Steiner Point Removal in Series-Parallel Graphs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {66:1--66:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.66},
URN = {urn:nbn:de:0030-drops-170041},
doi = {10.4230/LIPIcs.ESA.2022.66},
annote = {Keywords: Metric embeddings, graph algorithms, vertex sparsification}
}
Document
Chromatic k-Nearest Neighbor Queries
Abstract
Let P be a set of n colored points. We develop efficient data structures that store P and can answer chromatic k-nearest neighbor (k-NN) queries. Such a query consists of a query point q and a number k, and asks for the color that appears most frequently among the k points in P closest to q. Answering such queries efficiently is the key to obtain fast k-NN classifiers. Our main aim is to obtain query times that are independent of k while using near-linear space.
We show that this is possible using a combination of two data structures. The first data structure allow us to compute a region containing exactly the k-nearest neighbors of a query point q, and the second data structure can then report the most frequent color in such a region. This leads to linear space data structures with query times of O(n^{1/2} log n) for points in ℝ¹, and with query times varying between O(n^{2/3}log^{2/3} n) and O(n^{5/6} polylog n), depending on the distance measure used, for points in ℝ². These results can be extended to work in higher dimensions as well. Since the query times are still fairly large we also consider approximations. If we are allowed to report a color that appears at least (1-ε)f^* times, where f^* is the frequency of the most frequent color, we obtain a query time of O(log n + log log_{1/(1-ε)} n) in ℝ¹ and expected query times ranging between Õ(n^{1/2}ε^{-3/2}) and Õ(n^{1/2}ε^{-5/2}) in ℝ² using near-linear space (ignoring polylogarithmic factors).
Cite as
Thijs van der Horst, Maarten Löffler, and Frank Staals. Chromatic k-Nearest Neighbor Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{vanderhorst_et_al:LIPIcs.ESA.2022.67,
author = {van der Horst, Thijs and L\"{o}ffler, Maarten and Staals, Frank},
title = {{Chromatic k-Nearest Neighbor Queries}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {67:1--67:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.67},
URN = {urn:nbn:de:0030-drops-170055},
doi = {10.4230/LIPIcs.ESA.2022.67},
annote = {Keywords: data structure, nearest neighbor, classification}
}
Document
Maximum Weight b-Matchings in Random-Order Streams
Abstract
We consider the maximum weight b-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from [1,W], we present a 2 - 1/(2W) + ε approximation algorithm, using a memory of O(max(|M_G|, n) ⋅ poly(log(m),W,1/ε)), where |M_G| denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Aaron Bernstein, 2020], which achieves a 3/2 + ε approximation for the maximum cardinality simple matching. When W is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a 2 - δ approximation (for some small constant δ ∼ 10^{-17}) for the maximum weight simple matching. In particular, for the weighted b-matching problem, ours is the first result beating the approximation ratio of 2. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Aaron Bernstein and Cliff Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a 2 - 1/(2W) + ε approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry’s duality theorem.
Cite as
Chien-Chung Huang and François Sellier. Maximum Weight b-Matchings in Random-Order Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{huang_et_al:LIPIcs.ESA.2022.68,
author = {Huang, Chien-Chung and Sellier, Fran\c{c}ois},
title = {{Maximum Weight b-Matchings in Random-Order Streams}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {68:1--68:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.68},
URN = {urn:nbn:de:0030-drops-170062},
doi = {10.4230/LIPIcs.ESA.2022.68},
annote = {Keywords: Maximum weight matching, b-matching, streaming, random order}
}
Document
Embedding Phylogenetic Trees in Networks of Low Treewidth
Abstract
Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input network N as parameter, showing that the problem can be solved in 2^O(t²)⋅|N| time and space.
Cite as
Leo van Iersel, Mark Jones, and Mathias Weller. Embedding Phylogenetic Trees in Networks of Low Treewidth. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{vaniersel_et_al:LIPIcs.ESA.2022.69,
author = {van Iersel, Leo and Jones, Mark and Weller, Mathias},
title = {{Embedding Phylogenetic Trees in Networks of Low Treewidth}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {69:1--69:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.69},
URN = {urn:nbn:de:0030-drops-170070},
doi = {10.4230/LIPIcs.ESA.2022.69},
annote = {Keywords: fixed-parameter tractability, treewidth, phylogenetic tree, phylogenetic network, display graph, tree containment, embedding}
}
Document
Vertex Sparsifiers for Hyperedge Connectivity
Abstract
Recently, Chalermsook et al. {[}SODA'21{]} introduces a notion of vertex sparsifiers for c-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for c-edge st-connectivity {[}Jin and Sun FOCS'22{]}.
We study a natural extension called vertex sparsifiers for c-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph G = (V,E) with n vertices and m hyperedges with k terminal vertices and a parameter c, there exists a hypergraph H containing only O(kc³) hyperedges that preserves all minimum cuts (up to value c) between all subset of terminals. This matches the best bound of O(kc³) edges for normal graphs by [Liu'20]. Moreover, H can be constructed in almost-linear O(p^{1+o(1)} + n(rclog n)^{O(rc)}log m) time where r = max_{e ∈ E}|e| is the rank of G and p = ∑_{e ∈ E}|e| is the total size of G, or in poly(m, n) time if we slightly relax the size to O(kc³log^{1.5}(kc)) hyperedges.
Cite as
Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang. Vertex Sparsifiers for Hyperedge Connectivity. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{jiang_et_al:LIPIcs.ESA.2022.70,
author = {Jiang, Han and Huang, Shang-En and Saranurak, Thatchaphol and Zhang, Tian},
title = {{Vertex Sparsifiers for Hyperedge Connectivity}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {70:1--70:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.70},
URN = {urn:nbn:de:0030-drops-170081},
doi = {10.4230/LIPIcs.ESA.2022.70},
annote = {Keywords: Vertex sparsifier, hypergraph, connectivity}
}
Document
Approximation Algorithms for Round-UFP and Round-SAP
Abstract
We study Round-UFP and Round-SAP, two generalizations of the classical Bin Packing problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of jobs where for each job we are given a demand and a subpath. In Round-UFP, the goal is to find a packing of all jobs into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of jobs on any edge does not exceed the capacity of the respective edge. In Round-SAP, the jobs are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges.
We show that in contrast to Bin Packing, both problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic (2+ε)-approximations for both problems. For the general case, we obtain an O(log log n)-approximation algorithm and an O(log log 1/δ)-approximation under (1+δ)-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum job demand is at most the minimum edge capacity), we obtain an absolute 12- and an asymptotic (16+ε)-approximation algorithm for Round-UFP and Round-SAP, respectively.
Cite as
Debajyoti Kar, Arindam Khan, and Andreas Wiese. Approximation Algorithms for Round-UFP and Round-SAP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kar_et_al:LIPIcs.ESA.2022.71,
author = {Kar, Debajyoti and Khan, Arindam and Wiese, Andreas},
title = {{Approximation Algorithms for Round-UFP and Round-SAP}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {71:1--71:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.71},
URN = {urn:nbn:de:0030-drops-170098},
doi = {10.4230/LIPIcs.ESA.2022.71},
annote = {Keywords: Approximation Algorithms, Scheduling, Rectangle Packing}
}
Document
Optimizing Safe Flow Decompositions in DAGs
Abstract
Network flow is one of the most studied combinatorial optimization problems having innumerable applications. Any flow on a directed acyclic graph G having n vertices and m edges can be decomposed into a set of O(m) paths. The applications of such a flow decomposition range from network routing to the assembly of biological sequences. However, in some applications, each solution (decomposition) corresponds to some particular data that generated the original flow. Given the possibility of multiple optimal solutions, no optimization criterion ensures the identification of the correct decomposition. Hence, recently flow decomposition was studied [RECOMB22] in the Safe and Complete framework, particularly for RNA Assembly. The proposed solution reported all the safe paths, i.e., the paths which are subpath of every possible solution of flow decomposition.
They presented a characterization of the safe paths, resulting in an O(mn+out_R) time algorithm to compute all safe paths, where out_R is the size of the raw output reporting each safe path explicitly. They also showed that out_R can be Ω(mn²) in the worst case but O(m) in the best case. Hence, they further presented an algorithm to report a concise representation of the output out_C in O(mn+out_C) time, where out_C can be Ω(mn) in the worst case but O(m) in the best case.
In this work, we study how different safe paths interact, resulting in optimal output-sensitive algorithms requiring O(m+out_R) and O(m+out_C) time for computing the existing representations of the safe paths. Our algorithm uses a novel data structure called Path Tries, which may be of independent interest. Further, we propose a new characterization of the safe paths resulting in the optimal representation of safe paths out_O, which can be Ω(mn) in the worst case but requires optimal O(1) space for every safe path reported. We also present a near-optimal algorithm to compute all the safe paths in O(m+out_Olog n) time. The new representation also establishes tighter worst case bounds Θ(mn²) and Θ(mn) bounds for out_R and out_C (along with out_O), respectively.
Overall we further develop the theory of safe and complete solutions for the flow decomposition problem, giving an optimal algorithm for the explicit representation, and a near-optimal algorithm for the optimal representation of the safe paths.
Cite as
Shahbaz Khan and Alexandru I. Tomescu. Optimizing Safe Flow Decompositions in DAGs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{khan_et_al:LIPIcs.ESA.2022.72,
author = {Khan, Shahbaz and Tomescu, Alexandru I.},
title = {{Optimizing Safe Flow Decompositions in DAGs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {72:1--72:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.72},
URN = {urn:nbn:de:0030-drops-170101},
doi = {10.4230/LIPIcs.ESA.2022.72},
annote = {Keywords: safety, flows, networks, directed acyclic graphs}
}
Document
Scheduling Kernels via Configuration LP
Abstract
Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max?
We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter.
Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel.
Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight.
Cite as
Dušan Knop and Martin Koutecký. Scheduling Kernels via Configuration LP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{knop_et_al:LIPIcs.ESA.2022.73,
author = {Knop, Du\v{s}an and Kouteck\'{y}, Martin},
title = {{Scheduling Kernels via Configuration LP}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {73:1--73:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.73},
URN = {urn:nbn:de:0030-drops-170118},
doi = {10.4230/LIPIcs.ESA.2022.73},
annote = {Keywords: Scheduling, Kernelization}
}
Document
Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams
Abstract
This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive.
Cite as
Lex de Kogel, Marc van Kreveld, and Jordi L. Vermeulen. Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 74:1-74:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dekogel_et_al:LIPIcs.ESA.2022.74,
author = {de Kogel, Lex and van Kreveld, Marc and Vermeulen, Jordi L.},
title = {{Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {74:1--74:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.74},
URN = {urn:nbn:de:0030-drops-170120},
doi = {10.4230/LIPIcs.ESA.2022.74},
annote = {Keywords: Morphing, Hausdorff distance, Voronoi diagrams}
}
Document
Average Sensitivity of the Knapsack Problem
Abstract
In resource allocation, we often require that the output allocation of an algorithm is stable against input perturbation because frequent reallocation is costly and untrustworthy. Varma and Yoshida (SODA'21) formalized this requirement for algorithms as the notion of average sensitivity. Here, the average sensitivity of an algorithm on an input instance is, roughly speaking, the average size of the symmetric difference of the output for the instance and that for the instance with one item deleted, where the average is taken over the deleted item.
In this work, we consider the average sensitivity of the knapsack problem, a representative example of a resource allocation problem. We first show a (1-ε)-approximation algorithm for the knapsack problem with average sensitivity O(ε^{-1}log ε^{-1}). Then, we complement this result by showing that any (1-ε)-approximation algorithm has average sensitivity Ω(ε^{-1}). As an application of our algorithm, we consider the incremental knapsack problem in the random-order setting, where the goal is to maintain a good solution while items arrive one by one in a random order. Specifically, we show that for any ε > 0, there exists a (1-ε)-approximation algorithm with amortized recourse O(ε^{-1}log ε^{-1}) and amortized update time O(log n+f_ε), where n is the total number of items and f_ε > 0 is a value depending on ε.
Cite as
Soh Kumabe and Yuichi Yoshida. Average Sensitivity of the Knapsack Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 75:1-75:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kumabe_et_al:LIPIcs.ESA.2022.75,
author = {Kumabe, Soh and Yoshida, Yuichi},
title = {{Average Sensitivity of the Knapsack Problem}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {75:1--75:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.75},
URN = {urn:nbn:de:0030-drops-170136},
doi = {10.4230/LIPIcs.ESA.2022.75},
annote = {Keywords: Average Sensitivity, Knapsack Problem, FPRAS}
}
Document
Cardinality Estimation Using Gumbel Distribution
Abstract
Cardinality estimation is the task of approximating the number of distinct elements in a large dataset with possibly repeating elements. LogLog and HyperLogLog (c.f. Durand and Flajolet [ESA 2003], Flajolet et al. [Discrete Math Theor. 2007]) are small space sketching schemes for cardinality estimation, which have both strong theoretical guarantees of performance and are highly effective in practice. This makes them a highly popular solution with many implementations in big-data systems (e.g. Algebird, Apache DataSketches, BigQuery, Presto and Redis). However, despite having simple and elegant formulation, both the analysis of LogLog and HyperLogLog are extremely involved - spanning over tens of pages of analytic combinatorics and complex function analysis.
We propose a modification to both LogLog and HyperLogLog that replaces discrete geometric distribution with the continuous Gumbel distribution. This leads to a very short, simple and elementary analysis of estimation guarantees, and smoother behavior of the estimator.
Cite as
Aleksander Łukasiewicz and Przemysław Uznański. Cardinality Estimation Using Gumbel Distribution. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 76:1-76:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lukasiewicz_et_al:LIPIcs.ESA.2022.76,
author = {{\L}ukasiewicz, Aleksander and Uzna\'{n}ski, Przemys{\l}aw},
title = {{Cardinality Estimation Using Gumbel Distribution}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {76:1--76:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.76},
URN = {urn:nbn:de:0030-drops-170140},
doi = {10.4230/LIPIcs.ESA.2022.76},
annote = {Keywords: Streaming algorithms, Cardinality estimation, Sketching, Gumbel distribution}
}
Document
(In-)Approximability Results for Interval, Resource Restricted, and Low Rank Scheduling
Abstract
We consider variants of the restricted assignment problem where a set of jobs has to be assigned to a set of machines, for each job a size and a set of eligible machines is given, and the jobs may only be assigned to eligible machines with the goal of makespan minimization. For the variant with interval restrictions, where the machines can be arranged on a path such that each job is eligible on a subpath, we present the first better than 2-approximation and an improved inapproximability result. In particular, we give a (2-1/24)-approximation and show that no better than 9/8-approximation is possible, unless P=NP. Furthermore, we consider restricted assignment with R resource restrictions and rank D unrelated scheduling. In the former problem, a machine may process a job if it can meet its resource requirements regarding R (renewable) resources. In the latter, the size of a job is dependent on the machine it is assigned to and the corresponding processing time matrix has rank at most D. The problem with interval restrictions includes the 1 resource variant, is encompassed by the 2 resource variant, and regarding approximation the R resource variant is essentially a special case of the rank R+1 problem. We show that no better than 3/2, 8/7, and 3/2-approximation is possible (unless P=NP) for the 3 resource, 2 resource, and rank 3 variant, respectively. Both the approximation result for the interval case and the inapproximability result for the rank 3 variant are solutions to open challenges stated in previous works. Lastly, we also consider the reverse objective, that is, maximizing the minimal load any machine receives, and achieve similar results.
Cite as
Marten Maack, Simon Pukrop, and Anna Rodriguez Rasmussen. (In-)Approximability Results for Interval, Resource Restricted, and Low Rank Scheduling. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{maack_et_al:LIPIcs.ESA.2022.77,
author = {Maack, Marten and Pukrop, Simon and Rasmussen, Anna Rodriguez},
title = {{(In-)Approximability Results for Interval, Resource Restricted, and Low Rank Scheduling}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {77:1--77:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.77},
URN = {urn:nbn:de:0030-drops-170152},
doi = {10.4230/LIPIcs.ESA.2022.77},
annote = {Keywords: Scheduling, Restricted Assignment, Approximation, Inapproximability}
}
Document
Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds
Abstract
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements.
This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations.
In this paper, we propose a new method to find a triangulation that admits a solution to the gluing equations, using convex optimization and localized combinatorial modifications. It is based on Casson and Rivin’s reformulation of the equations. We provide a novel approach to modify a triangulation and update its geometry, along with experimental results to support the new method.
Cite as
Clément Maria and Owen Rouillé. Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{maria_et_al:LIPIcs.ESA.2022.78,
author = {Maria, Cl\'{e}ment and Rouill\'{e}, Owen},
title = {{Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {78:1--78:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.78},
URN = {urn:nbn:de:0030-drops-170168},
doi = {10.4230/LIPIcs.ESA.2022.78},
annote = {Keywords: knots and 3-manifolds, triangulation, hyperbolic structure, Thurston equations}
}
Document
Computing Treedepth in Polynomial Space and Linear FPT Time
Abstract
The treedepth of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent in G is bound by the ancestor/descendant relation. We propose an algorithm that given a graph G and an integer d, either finds an elimination forest of G of depth at most d or concludes that no such forest exists; thus the algorithm decides whether the treedepth of G is at most d. The running time is 2^𝒪(d²)⋅n^𝒪(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2^𝒪(d²)⋅n and d^𝒪(1)⋅n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2^𝒪(d²)⋅n, but uses exponential space.
Cite as
Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz. Computing Treedepth in Polynomial Space and Linear FPT Time. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{nadara_et_al:LIPIcs.ESA.2022.79,
author = {Nadara, Wojciech and Pilipczuk, Micha{\l} and Smulewicz, Marcin},
title = {{Computing Treedepth in Polynomial Space and Linear FPT Time}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {79:1--79:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.79},
URN = {urn:nbn:de:0030-drops-170175},
doi = {10.4230/LIPIcs.ESA.2022.79},
annote = {Keywords: treedepth, FPT, polynomial space}
}
Document
The Pareto Cover Problem
Abstract
We introduce the problem of finding a set B of k points in [0,1]ⁿ such that the expected cost of the cheapest point in B that dominates a random point from [0,1]ⁿ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for k = 2 when each coordinate is drawn from {0,1}, and obtain an FPTAS for general fixed k under mild assumptions on the distributions.
Cite as
Bento Natura, Meike Neuwohner, and Stefan Weltge. The Pareto Cover Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 80:1-80:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{natura_et_al:LIPIcs.ESA.2022.80,
author = {Natura, Bento and Neuwohner, Meike and Weltge, Stefan},
title = {{The Pareto Cover Problem}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {80:1--80:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.80},
URN = {urn:nbn:de:0030-drops-170186},
doi = {10.4230/LIPIcs.ESA.2022.80},
annote = {Keywords: Pareto, Covering, Optimization, Approximation Algorithm}
}
Document
A Unified Framework for Hopsets
Abstract
Given an undirected graph G = (V,E), an (α,β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, d_G(u,v) ≤ d_{G∪H}^(β)(u,v) ≤ α⋅ d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α.
In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm.
In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^ε),O(k^{1-ε}))-hopset of size Õ(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ε < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α,β)-hopset of size O(n^{1+1/k}) must have α⋅β ≥ Ω(k).
Cite as
Ofer Neiman and Idan Shabat. A Unified Framework for Hopsets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{neiman_et_al:LIPIcs.ESA.2022.81,
author = {Neiman, Ofer and Shabat, Idan},
title = {{A Unified Framework for Hopsets}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {81:1--81:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.81},
URN = {urn:nbn:de:0030-drops-170192},
doi = {10.4230/LIPIcs.ESA.2022.81},
annote = {Keywords: Graph Algorithms, Shortest Paths, Hopsets}
}
Document
Data Structures for Node Connectivity Queries
Abstract
Let κ(s,t) denote the maximum number of internally disjoint st-paths in an undirected graph G. We consider designing a data structure that includes a list of cuts, and answers the following query: given s,t ∈ V, determine whether κ(s,t) ≤ k, and if so, return a pointer to an st-cut of size ≤ k (or to a minimum st-cut) in the list. A trivial data structure that includes a list of n(n-1)/2 cuts and requires Θ(kn²) space can answer each query in O(1) time. We obtain the following results.
- In the case when G is k-connected, we show that 2n cuts suffice, and that these cuts can be partitioned into 2k+1 laminar families. Thus using space O(kn) we can answers each min-cut query in O(1) time, slightly improving and substantially simplifying the proof of a recent result of Pettie and Yin [S. Pettie and L. Yin, 2021]. We then extend this data structure to subset k-connectivity.
- In the general case we show that (2k+1)n cuts suffice to return an st-cut of size ≤ k, and a list of size k(k+2)n contains a minimum st-cut for every s,t ∈ V. Combining our subset k-connectivity data structure with the data structure of Hsu and Lu [T-H. Hsu and H-I. Lu, 2009] for checking k-connectivity, we give an O(k² n) space data structure that returns an st-cut of size ≤ k in O(log k) time, while O(k³ n) space enables to return a minimum st-cut.
Cite as
Zeev Nutov. Data Structures for Node Connectivity Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 82:1-82:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{nutov:LIPIcs.ESA.2022.82,
author = {Nutov, Zeev},
title = {{Data Structures for Node Connectivity Queries}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {82:1--82:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.82},
URN = {urn:nbn:de:0030-drops-170205},
doi = {10.4230/LIPIcs.ESA.2022.82},
annote = {Keywords: node connectivity, minimum cuts, data structure, connectivity queries}
}
Document
Improved Bounds for Online Balanced Graph Re-Partitioning
Abstract
We study the online balanced graph re-partitioning problem (OBGR) which was introduced by Avin, Bienkowski, Loukas, Pacut, and Schmid [Avin et al., 2020] and has recently received significant attention [Pacut et al., 2021; Henzinger et al., 2021; Henzinger et al., 2019; Tobias Forner et al., 2021; Bienkowski et al., 2021] owing to potential applications in large-scale, data-intensive distributed computing. In OBGR, we have a set of 𝓁 clusters, each with k vertices (representing processes or virtual machines), and an online sequence of communication requests, each represented by a pair of vertices. Any request (u,v) incurs unit communication cost if u and v are located in different clusters (and zero otherwise). Any vertex can be migrated from one cluster to another at a migration cost of α ≥ 1. We consider the objective of minimizing the total communication and migration cost in the competitive analysis framework. The only known algorithms (which run in exponential time) include an O(k²𝓁²) competitive [Avin et al., 2020] and an O(k𝓁 2^O(k)) competitive algorithm [Bienkowski et al., 2021]. A lower bound of Ω(k𝓁) is known [Pacut et al., 2021]. In an effort to bridge the gap, recent results have considered beyond worst case analyses including resource augmentation (with augmented cluster capacity [Avin et al., 2020; Henzinger et al., 2019; Henzinger et al., 2021]) and restricted request sequences (the learning model [Henzinger et al., 2019; Henzinger et al., 2021; Pacut et al., 2021]).
In this paper, we give deterministic, polynomial-time algorithms for OBGR, which mildly exploit resource augmentation (i.e. augmented cluster capacity of (1+ε) k for arbitrary ε > 0). We improve beyond O(k²𝓁²)-competitiveness (for general 𝓁, k) by first giving a simple algorithm with competitive ratio O(k𝓁²log k). Our main result is an algorithm with a significantly improved competitive ratio of O(k𝓁 log k). At a high level, we achieve this by employing i) an ILP framework to guide the allocation of large components, ii) a simple "any fit" style assignment of small components and iii) a charging argument which allows us to bound the cost of migrations. Like previous work on OBGR, our algorithm and analysis are phase-based, where each phase solves an independent instance of the learning model. Finally, we give an Ω(α k𝓁 log k) lower bound on the total cost incurred by any algorithm for OBGR under the learning model, which quantifies the limitation of a phase-based approach.
Cite as
Rajmohan Rajaraman and Omer Wasim. Improved Bounds for Online Balanced Graph Re-Partitioning. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{rajaraman_et_al:LIPIcs.ESA.2022.83,
author = {Rajaraman, Rajmohan and Wasim, Omer},
title = {{Improved Bounds for Online Balanced Graph Re-Partitioning}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {83:1--83:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.83},
URN = {urn:nbn:de:0030-drops-170210},
doi = {10.4230/LIPIcs.ESA.2022.83},
annote = {Keywords: online algorithms, graph partitioning, competitive analysis}
}
Document
An Empirical Evaluation of k-Means Coresets
Abstract
Coresets are among the most popular paradigms for summarizing data. In particular, there exist many high performance coresets for clustering problems such as k-means in both theory and practice. Curiously, there exists no work on comparing the quality of available k-means coresets.
In this paper we perform such an evaluation. There currently is no algorithm known to measure the distortion of a candidate coreset. We provide some evidence as to why this might be computationally difficult. To complement this, we propose a benchmark for which we argue that computing coresets is challenging and which also allows us an easy (heuristic) evaluation of coresets. Using this benchmark and real-world data sets, we conduct an exhaustive evaluation of the most commonly used coreset algorithms from theory and practice.
Cite as
Chris Schwiegelshohn and Omar Ali Sheikh-Omar. An Empirical Evaluation of k-Means Coresets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{schwiegelshohn_et_al:LIPIcs.ESA.2022.84,
author = {Schwiegelshohn, Chris and Sheikh-Omar, Omar Ali},
title = {{An Empirical Evaluation of k-Means Coresets}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {84:1--84:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.84},
URN = {urn:nbn:de:0030-drops-170225},
doi = {10.4230/LIPIcs.ESA.2022.84},
annote = {Keywords: coresets, k-means coresets, evaluation, benchmark}
}
Document
An Improved Algorithm for Finding the Shortest Synchronizing Words
Abstract
A synchronizing word of a deterministic finite complete automaton is a word whose action maps every state to a single one. Finding a shortest or a short synchronizing word is a central computational problem in the theory of synchronizing automata and is applied in other areas such as model-based testing and the theory of codes. Because the problem of finding a shortest synchronizing word is computationally hard, among exact algorithms only exponential ones are known. We redesign the previously fastest known exact algorithm based on the bidirectional breadth-first search and improve it with respect to time and space in a practical sense. We develop new algorithmic enhancements and adapt the algorithm to multithreaded and GPU computing. Our experiments show that the new algorithm is multiple times faster than the previously fastest one and its advantage quickly grows with the hardness of the problem instance. Given a modest time limit, we compute the lengths of the shortest synchronizing words for random binary automata up to 570 states, significantly beating the previous record. We refine the experimental estimation of the average reset threshold of these automata. Finally, we develop a general computational package devoted to the problem, where an efficient and practical implementation of our algorithm is included, together with several well-known heuristics.
Cite as
Marek Szykuła and Adam Zyzik. An Improved Algorithm for Finding the Shortest Synchronizing Words. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{szykula_et_al:LIPIcs.ESA.2022.85,
author = {Szyku{\l}a, Marek and Zyzik, Adam},
title = {{An Improved Algorithm for Finding the Shortest Synchronizing Words}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {85:1--85:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.85},
URN = {urn:nbn:de:0030-drops-170237},
doi = {10.4230/LIPIcs.ESA.2022.85},
annote = {Keywords: \v{C}ern\{\'{y}\} conjecture, reset threshold, reset word, subset checking, synchronizing automaton, synchronizing word}
}
Document
Fast RSK Correspondence by Doubling Search
Abstract
The Robinson-Schensted-Knuth (RSK) correspondence is a fundamental concept in combinatorics and representation theory. It is defined as a certain bijection between permutations and pairs of Young tableaux of a given order. We consider the RSK correspondence as an algorithmic problem, along with the closely related k-chain problem. We give a simple, direct description of the symmetric RSK algorithm, which is implied by the k-chain algorithms of Viennot and of Felsner and Wernisch. We also show how the doubling search of Bentley and Yao can be used as a subroutine by the symmetric RSK algorithm, replacing the default binary search. Surprisingly, such a straightforward replacement improves the asymptotic worst-case running time for the RSK correspondence that has been best known since 1998. A similar improvement also holds for the average running time of RSK on uniformly random permutations.
Cite as
Alexander Tiskin. Fast RSK Correspondence by Doubling Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{tiskin:LIPIcs.ESA.2022.86,
author = {Tiskin, Alexander},
title = {{Fast RSK Correspondence by Doubling Search}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {86:1--86:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.86},
URN = {urn:nbn:de:0030-drops-170249},
doi = {10.4230/LIPIcs.ESA.2022.86},
annote = {Keywords: combinatorics of permutations, Robinson-Schensted-Knuth correspondence, k-chains, RSK algorithm}
}
Document
Insertion Time of Random Walk Cuckoo Hashing below the Peeling Threshold
Abstract
Most hash tables have an insertion time of 𝒪(1), often qualified as "expected" and/or "amortised". While insertions into cuckoo hash tables indeed seem to take 𝒪(1) expected time in practice, only polylogarithmic guarantees are proven in all but the simplest of practically relevant cases. Given the widespread use of cuckoo hashing to implement compact dictionaries and Bloom filter alternatives, closing this gap is an important open problem for theoreticians.
In this paper, we show that random walk insertions into cuckoo hash tables take 𝒪(1) expected amortised time when any number k ≥ 3 of hash functions is used and the load factor is below the corresponding peeling threshold (e.g. ≈0.81 for k = 3). To our knowledge, this is the first meaningful guarantee for constant time insertion for cuckoo hashing that works for k ∈ {3,…,9}.
In addition to being useful in its own right, we hope that our key-centred analysis method can be a stepping stone on the path to the true end goal: 𝒪(1) time insertions for all load factors below the load threshold (e.g. ≈0.91 for k = 3).
Cite as
Stefan Walzer. Insertion Time of Random Walk Cuckoo Hashing below the Peeling Threshold. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 87:1-87:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{walzer:LIPIcs.ESA.2022.87,
author = {Walzer, Stefan},
title = {{Insertion Time of Random Walk Cuckoo Hashing below the Peeling Threshold}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {87:1--87:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.87},
URN = {urn:nbn:de:0030-drops-170250},
doi = {10.4230/LIPIcs.ESA.2022.87},
annote = {Keywords: Cuckoo Hashing, Random Walk, Random Hypergraph, Peeling, Cores}
}
Document
ParGeo: A Library for Parallel Computational Geometry
Abstract
This paper presents ParGeo, a multicore library for computational geometry. ParGeo contains modules for fundamental tasks including kd-tree based spatial search, spatial graph generation, and algorithms in computational geometry.
We focus on three new algorithmic contributions provided in the library. First, we present a new parallel convex hull algorithm based on a reservation technique to enable parallel modifications to the hull. We also provide the first parallel implementations of the randomized incremental convex hull algorithm as well as a divide-and-conquer convex hull algorithm in ℝ³. Second, for the smallest enclosing ball problem, we propose a new sampling-based algorithm to quickly reduce the size of the data set. We also provide the first parallel implementation of Welzl’s classic algorithm for smallest enclosing ball. Third, we present the BDL-tree, a parallel batch-dynamic kd-tree that allows for efficient parallel updates and k-NN queries over dynamically changing point sets. BDL-trees consist of a log-structured set of kd-trees which can be used to efficiently insert, delete, and query batches of points in parallel.
On 36 cores with two-way hyper-threading, our fastest convex hull algorithm achieves up to 44.7x self-relative parallel speedup and up to 559x speedup against the best existing sequential implementation. Our smallest enclosing ball algorithm using our sampling-based algorithm achieves up to 27.1x self-relative parallel speedup and up to 178x speedup against the best existing sequential implementation. Our implementation of the BDL-tree achieves self-relative parallel speedup of up to 46.1x. Across all of the algorithms in ParGeo, we achieve self-relative parallel speedup of 8.1-46.61x.
Cite as
Yiqiu Wang, Rahul Yesantharao, Shangdi Yu, Laxman Dhulipala, Yan Gu, and Julian Shun. ParGeo: A Library for Parallel Computational Geometry. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 88:1-88:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{wang_et_al:LIPIcs.ESA.2022.88,
author = {Wang, Yiqiu and Yesantharao, Rahul and Yu, Shangdi and Dhulipala, Laxman and Gu, Yan and Shun, Julian},
title = {{ParGeo: A Library for Parallel Computational Geometry}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {88:1--88:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.88},
URN = {urn:nbn:de:0030-drops-170265},
doi = {10.4230/LIPIcs.ESA.2022.88},
annote = {Keywords: Computational Geometry, Parallel Algorithms, Libraries}
}
Document
Combining Predicted and Live Traffic with Time-Dependent A* Potentials
Abstract
We study efficient and exact shortest path algorithms for routing on road networks with realistic traffic data. For navigation applications, both current (i.e., live) traffic events and predictions of future traffic flows play an important role in routing. While preprocessing-based speedup techniques have been employed successfully to both settings individually, a combined model poses significant challenges. Supporting predicted traffic typically requires expensive preprocessing while live traffic requires fast updates for regular adjustments. We propose an A*-based solution to this problem. By generalizing A* potentials to time dependency, i.e. the estimate of the distance from a vertex to the target also depends on the time of day when the vertex is visited, we achieve significantly faster query times than previously possible. Our evaluation shows that our approach enables interactive query times on continental-sized road networks while allowing live traffic updates within a fraction of a minute. We achieve a speedup of at least two orders of magnitude over Dijkstra’s algorithm and up to one order of magnitude over state-of-the-art time-independent A* potentials.
Cite as
Nils Werner and Tim Zeitz. Combining Predicted and Live Traffic with Time-Dependent A* Potentials. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 89:1-89:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{werner_et_al:LIPIcs.ESA.2022.89,
author = {Werner, Nils and Zeitz, Tim},
title = {{Combining Predicted and Live Traffic with Time-Dependent A* Potentials}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {89:1--89:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.89},
URN = {urn:nbn:de:0030-drops-170271},
doi = {10.4230/LIPIcs.ESA.2022.89},
annote = {Keywords: realistic road networks, shortest paths, live traffic, time-dependent routing}
}
Document
Approximating Dynamic Time Warping Distance Between Run-Length Encoded Strings
Abstract
Dynamic Time Warping (DTW) is a widely used similarity measure for comparing strings that encode time series data, with applications to areas including bioinformatics, signature verification, and speech recognition. The standard dynamic-programming algorithm for DTW takes O(n²) time, and there are conditional lower bounds showing that no algorithm can do substantially better.
In many applications, however, the strings x and y may contain long runs of repeated letters, meaning that they can be compressed using run-length encoding. A natural question is whether the DTW-distance between these compressed strings can be computed efficiently in terms of the lengths k and 𝓁 of the compressed strings. Recent work has shown how to achieve O(k𝓁² + 𝓁 k²) time, leaving open the question of whether a near-quadratic Õ(k𝓁)-time algorithm might exist.
We show that, if a small approximation loss is permitted, then a near-quadratic time algorithm is indeed possible: our algorithm computes a (1 + ε)-approximation for DTW(x, y) in Õ(k𝓁 / ε³) time, where k and 𝓁 are the number of runs in x and y. Our algorithm allows for DTW to be computed over any metric space (Σ, δ) in which distances are O(log n)-bit integers. Surprisingly, the algorithm also works even if δ does not induce a metric space on Σ (e.g., δ need not satisfy the triangle inequality).
Cite as
Zoe Xi and William Kuszmaul. Approximating Dynamic Time Warping Distance Between Run-Length Encoded Strings. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 90:1-90:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{xi_et_al:LIPIcs.ESA.2022.90,
author = {Xi, Zoe and Kuszmaul, William},
title = {{Approximating Dynamic Time Warping Distance Between Run-Length Encoded Strings}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {90:1--90:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.90},
URN = {urn:nbn:de:0030-drops-170281},
doi = {10.4230/LIPIcs.ESA.2022.90},
annote = {Keywords: Dynamic time warping distance, approximation algorithms, run-length encodings, computational geometry}
}
Document
Correlated Stochastic Knapsack with a Submodular Objective
Abstract
We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time (1 - 1/√e)/2 ≃ 0.1967 approximation algorithm with or without those additional constraints, eliminating the need of a key assumption and improving on the (1 - 1/∜e)/2 ≃ 0.1106 approximation by Fukunaga et al. [AAAI 2019].
Cite as
Sheng Yang, Samir Khuller, Sunav Choudhary, Subrata Mitra, and Kanak Mahadik. Correlated Stochastic Knapsack with a Submodular Objective. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 91:1-91:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{yang_et_al:LIPIcs.ESA.2022.91,
author = {Yang, Sheng and Khuller, Samir and Choudhary, Sunav and Mitra, Subrata and Mahadik, Kanak},
title = {{Correlated Stochastic Knapsack with a Submodular Objective}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {91:1--91:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.91},
URN = {urn:nbn:de:0030-drops-170296},
doi = {10.4230/LIPIcs.ESA.2022.91},
annote = {Keywords: Stochastic Knapsack, Submodular Optimization, Stochastic Optimization}
}
Document
Faster Algorithm for Unique (k,2)-CSP
Abstract
In a (k,2)-Constraint Satisfaction Problem we are given a set of arbitrary constraints on pairs of k-ary variables, and are asked to find an assignment of values to these variables such that all constraints are satisfied. The (k,2)-CSP problem generalizes problems like k-coloring and k-list-coloring. In the Unique (k,2)-CSP problem, we add the assumption that the input set of constraints has at most one satisfying assignment.
Beigel and Eppstein gave an algorithm for (k,2)-CSP running in time O((0.4518k)^n) for k > 3 and O (1.356ⁿ) for k = 3, where n is the number of variables. Feder and Motwani improved upon the Beigel-Eppstein algorithm for k ≥ 11. Hertli, Hurbain, Millius, Moser, Scheder and Szedl{á}k improved these bounds for Unique (k,2)-CSP for every k ≥ 5.
We improve the result of Hertli et al. and obtain better bounds for Unique (k,2)-CSP for k ≥ 5. In particular, we improve the running time of Unique (5,2)-CSP from O (2.254ⁿ) to O (2.232^n) and Unique (6,2)-CSP from O (2.652^n) to O (2.641^n).
Recently, Li and Scheder also published an improvement over the algorithm of Hertli et al. in the same regime as ours. Their improvement does not include quantitative bounds, we compare the works in the paper.
Cite as
Or Zamir. Faster Algorithm for Unique (k,2)-CSP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 92:1-92:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{zamir:LIPIcs.ESA.2022.92,
author = {Zamir, Or},
title = {{Faster Algorithm for Unique (k,2)-CSP}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {92:1--92:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.92},
URN = {urn:nbn:de:0030-drops-170309},
doi = {10.4230/LIPIcs.ESA.2022.92},
annote = {Keywords: Algorithms, Constraint Satisfaction Problem}
}