19 Search Results for "Schmidt, G�nter"

Document
DOI: 10.4230/LIPIcs.ESA.2023.55
Noisy k-Means++ Revisited

Authors: Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
The k-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the k-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is O(log k)-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in k-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of O(log² k). In this paper, we close the gap by providing a tight, O(log k)-approximate guarantee for the k-means++ algorithm with noise.
Cite as

Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň. Noisy k-Means++ Revisited. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 55:1-55:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grunau_et_al:LIPIcs.ESA.2023.55,
  author =	{Grunau, Christoph and \"{O}z\"{u}do\u{g}ru, Ahmet Alper and Rozho\v{n}, V\'{a}clav},
  title =	{{Noisy k-Means++ Revisited}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{55:1--55:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.55},
  URN =		{urn:nbn:de:0030-drops-187080},
  doi =		{10.4230/LIPIcs.ESA.2023.55},
  annote =	{Keywords: clustering, k-means, k-means++, adversarial noise}
}
Document
DOI: 10.4230/LIPIcs.SEA.2023.10
Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover

Authors: Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
In the Directed Feedback Vertex Set (DFVS) problem, one is given a directed graph G = (V,E) and wants to find a minimum cardinality set S ⊆ V such that G-S is acyclic. DFVS is a fundamental problem in computer science and finds applications in areas such as deadlock detection. The problem was the subject of the 2022 PACE coding challenge. We develop a novel exact algorithm for the problem that is tailored to perform well on instances that are mostly bi-directed. For such instances, we adapt techniques from the well-researched vertex cover problem. Our core idea is an iterative reduction to vertex cover. To this end, we also develop a new reduction rule that reduces the number of not bi-directed edges. With the resulting algorithm, we were able to win third place in the exact track of the PACE challenge. We perform computational experiments and compare the running time to other exact algorithms, in particular to the winning algorithm in PACE. Our experiments show that we outpace the other algorithms on instances that have a low density of uni-directed edges.
Cite as

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angrick_et_al:LIPIcs.SEA.2023.10,
  author =	{Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo},
  title =	{{Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.10},
  URN =		{urn:nbn:de:0030-drops-183602},
  doi =		{10.4230/LIPIcs.SEA.2023.10},
  annote =	{Keywords: directed feedback vertex set, vertex cover, reduction rules}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.50
Connected k-Center and k-Diameter Clustering

Authors: Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.
Cite as

Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drexler_et_al:LIPIcs.ICALP.2023.50,
  author =	{Drexler, Lukas and Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian},
  title =	{{Connected k-Center and k-Diameter Clustering}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.50},
  URN =		{urn:nbn:de:0030-drops-181024},
  doi =		{10.4230/LIPIcs.ICALP.2023.50},
  annote =	{Keywords: Approximation algorithms, Clustering, Connectivity constraints}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.17
Cut Paths and Their Remainder Structure, with Applications

Authors: Massimo Cairo, Shahbaz Khan, Romeo Rizzi, Sebastian Schmidt, Alexandru I. Tomescu, and Elia C. Zirondelli

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
In a strongly connected graph G = (V,E), a cut arc (also called strong bridge) is an arc e ∈ E whose removal makes the graph no longer strongly connected. Equivalently, there exist u,v ∈ V, such that all u-v walks contain e. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems. In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths P for which there exist u,v ∈ V, such that all u-v walks contain P as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple O(m)-time verification algorithm for a cut path (|V| = n, |E| = m). Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple O(m)-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time, since they are related in a structured way. These properties yield an O(mn)-time algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time O(m²+n³). The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc.
Cite as

Massimo Cairo, Shahbaz Khan, Romeo Rizzi, Sebastian Schmidt, Alexandru I. Tomescu, and Elia C. Zirondelli. Cut Paths and Their Remainder Structure, with Applications. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cairo_et_al:LIPIcs.STACS.2023.17,
  author =	{Cairo, Massimo and Khan, Shahbaz and Rizzi, Romeo and Schmidt, Sebastian and Tomescu, Alexandru I. and Zirondelli, Elia C.},
  title =	{{Cut Paths and Their Remainder Structure, with Applications}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.17},
  URN =		{urn:nbn:de:0030-drops-176690},
  doi =		{10.4230/LIPIcs.STACS.2023.17},
  annote =	{Keywords: reachability, cut arc, strong bridge, covering walk, safety, persistence, essentiality, genome assembly}
}
Document
PACE Solver Description
DOI: 10.4230/LIPIcs.IPEC.2022.28
PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set

Authors: Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Abstract
In this document we describe the techniques we used and implemented for our submission to the Parameterized Algorithms and Computational Experiments Challenge (PACE) 2022. The given problem is Directed Feedback Vertex Set (DFVS), where one is given a directed graph G = (V,E) and wants to find a minimum S ⊆ V such that G-S is acyclic. We approach this problem by first exhaustively applying a set of reduction rules. In order to find a minimum DFVS on the remaining instance, we create and solve a series of Vertex Cover instances.
Cite as

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 28:1-28:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{angrick_et_al:LIPIcs.IPEC.2022.28,
  author =	{Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo},
  title =	{{PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{28:1--28:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.28},
  URN =		{urn:nbn:de:0030-drops-173847},
  doi =		{10.4230/LIPIcs.IPEC.2022.28},
  annote =	{Keywords: directed feedback vertex set, vertex cover, reduction rules}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2021.22
Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Authors: Nicolas Bousquet, Laurent Feuilloley, and Théo Pierron

Published in: LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Abstract
Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds using a new proof technique.
Cite as

Nicolas Bousquet, Laurent Feuilloley, and Théo Pierron. Local Certification of Graph Decompositions and Applications to Minor-Free Classes. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bousquet_et_al:LIPIcs.OPODIS.2021.22,
  author =	{Bousquet, Nicolas and Feuilloley, Laurent and Pierron, Th\'{e}o},
  title =	{{Local Certification of Graph Decompositions and Applications to Minor-Free Classes}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.22},
  URN =		{urn:nbn:de:0030-drops-157970},
  doi =		{10.4230/LIPIcs.OPODIS.2021.22},
  annote =	{Keywords: Local certification, proof-labeling schemes, locally checkable proofs, graph decompositions, minor-free graphs}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.51
Sampling Multiple Edges Efficiently

Authors: Talya Eden, Saleet Mossel, and Ronitt Rubinfeld

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.
Cite as

Talya Eden, Saleet Mossel, and Ronitt Rubinfeld. Sampling Multiple Edges Efficiently. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{eden_et_al:LIPIcs.APPROX/RANDOM.2021.51,
  author =	{Eden, Talya and Mossel, Saleet and Rubinfeld, Ronitt},
  title =	{{Sampling Multiple Edges Efficiently}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{51:1--51:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.51},
  URN =		{urn:nbn:de:0030-drops-147441},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.51},
  annote =	{Keywords: Sampling edges, graph algorithm, sublinear algorithms}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.55
Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time

Authors: Amartya Shankha Biswas, Talya Eden, and Ronitt Rubinfeld

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H) = {O_{k₁},...,O_{k_q}, S_{p₁},...,S_{p_𝓁}}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.
Cite as

Amartya Shankha Biswas, Talya Eden, and Ronitt Rubinfeld. Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{biswas_et_al:LIPIcs.APPROX/RANDOM.2021.55,
  author =	{Biswas, Amartya Shankha and Eden, Talya and Rubinfeld, Ronitt},
  title =	{{Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{55:1--55:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.55},
  URN =		{urn:nbn:de:0030-drops-147480},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.55},
  annote =	{Keywords: Sublinear time algorithms, Graph algorithms, Sampling subgraphs, Approximate counting}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.52
Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

Authors: Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be Θ̃(n ⋅ L). Prior to our work, a better space complexity was only known with Õ(√L) passes. We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is Θ̃(n ⋅ √L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses Ω̃(n ⋅ L^{1/p}) space. In addition, we show a similar Θ̃(n ⋅ √L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph.
Cite as

Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2021.52,
  author =	{Chen, Lijie and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R. and Song, Zhao and Yu, Huacheng},
  title =	{{Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{52:1--52:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.52},
  URN =		{urn:nbn:de:0030-drops-141218},
  doi =		{10.4230/LIPIcs.ICALP.2021.52},
  annote =	{Keywords: streaming algorithms, random walk sampling}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.23
Online Carpooling Using Expander Decompositions

Authors: Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Sahil Singla

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We consider the online carpooling problem: given n vertices, a sequence of edges arrive over time. When an edge e_t = (u_t, v_t) arrives at time step t, the algorithm must orient the edge either as v_t → u_t or u_t → v_t, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in. In this paper, we design efficient algorithms which can maintain polylog(n,T) maximum discrepancy (w.h.p) over any sequence of T arrivals, when the arriving edges are sampled independently and uniformly from any given graph G. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were O(√{n log n})-discrepancy for any adversarial sequence of arrivals, or O(log log n)-discrepancy bounds for the stochastic arrivals when G is the complete graph. The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when G is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm.
Cite as

Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Sahil Singla. Online Carpooling Using Expander Decompositions. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2020.23,
  author =	{Gupta, Anupam and Krishnaswamy, Ravishankar and Kumar, Amit and Singla, Sahil},
  title =	{{Online Carpooling Using Expander Decompositions}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.23},
  URN =		{urn:nbn:de:0030-drops-132647},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.23},
  annote =	{Keywords: Online Algorithms, Discrepancy Minimization, Carpooling}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.43
Minimum Scan Cover with Angular Transition Costs

Authors: Sándor P. Fekete, Linda Kleist, and Dominik Krupke

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (msc), we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex takes some time proportional to the corresponding turn angle. Our goal is to minimize the time until all scans are completed, i.e., to compute a schedule of minimum makespan. We show that msc is closely related to both graph coloring and the minimum (directed and undirected) cut cover problem; in particular, we show that the minimum scan time for instances in 1D and 2D lies in Θ(log χ(G)), while for 3D the minimum scan time is not upper bounded by χ(G). We use this relationship to prove that the existence of a constant-factor approximation implies P=NP, even for one-dimensional instances. In 2D, we show that it is NP-hard to approximate a minimum scan cover within less than a factor of 3/2, even for bipartite graphs; conversely, we present a 9/2-approximation algorithm for this scenario. Generally, we give an O(c)-approximation for k-colored graphs with k ≤ χ(G)^c. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph.
Cite as

Sándor P. Fekete, Linda Kleist, and Dominik Krupke. Minimum Scan Cover with Angular Transition Costs. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2020.43,
  author =	{Fekete, S\'{a}ndor P. and Kleist, Linda and Krupke, Dominik},
  title =	{{Minimum Scan Cover with Angular Transition Costs}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.43},
  URN =		{urn:nbn:de:0030-drops-122014},
  doi =		{10.4230/LIPIcs.SoCG.2020.43},
  annote =	{Keywords: Graph scanning, graph coloring, angular metric, complexity, approximation, scheduling}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.45
Algorithms and Adaptivity Gaps for Stochastic k-TSP

Authors: Haotian Jiang, Jian Li, Daogao Liu, and Sahil Singla

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Given a metric (V,d) and a root ∈ V, the classic k-TSP problem is to find a tour originating at the root of minimum length that visits at least k nodes in V. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of k-TSP. In Stoch-Reward k-TSP, originally defined by Ene-Nagarajan-Saket [Ene et al., 2018], each vertex v in the given metric (V,d) contains a stochastic reward R_v. The goal is to adaptively find a tour of minimum expected length that collects at least reward k; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an O(log k)-approximation adaptive algorithm for this problem, and left open if there is an O(1)-approximation algorithm. We totally resolve their open question, and even give an O(1)-approximation non-adaptive algorithm for Stoch-Reward k-TSP. We also introduce and obtain similar results for the Stoch-Cost k-TSP problem. In this problem each vertex v has a stochastic cost C_v, and the goal is to visit and select at least k vertices to minimize the expected sum of tour length and cost of selected vertices. Besides being a natural stochastic generalization of k-TSP, this problem is also interesting because it generalizes the Price of Information framework [Singla, 2018] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". In general, replacing a random variable with its expectation leads to very poor results. We show that for our problems, if we truncate the random variables at an ideal threshold, then their expected values form a good surrogate. Here, we rely on running several repetitions of our algorithm with the same threshold, and then argue concentration using Freedman’s and Jogdeo-Samuels' inequalities. Unfortunately, this ideal threshold depends on how far we are from achieving our target k, which a non-adaptive algorithm does not know. To overcome this barrier, we truncate the random variables at various different scales and identify a "critical" scale.
Cite as

Haotian Jiang, Jian Li, Daogao Liu, and Sahil Singla. Algorithms and Adaptivity Gaps for Stochastic k-TSP. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 45:1-45:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jiang_et_al:LIPIcs.ITCS.2020.45,
  author =	{Jiang, Haotian and Li, Jian and Liu, Daogao and Singla, Sahil},
  title =	{{Algorithms and Adaptivity Gaps for Stochastic k-TSP}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{45:1--45:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.45},
  URN =		{urn:nbn:de:0030-drops-117308},
  doi =		{10.4230/LIPIcs.ITCS.2020.45},
  annote =	{Keywords: approximation algorithms, stochastic optimization, travelling salesman problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.12
Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors: Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.
Cite as

Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bannink_et_al:LIPIcs.STACS.2019.12,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Buhrman, Harry and Labib, Farrokh and Lee, Troy},
  title =	{{Bounding Quantum-Classical Separations for Classes of Nonlocal Games}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{12:1--12:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.12},
  URN =		{urn:nbn:de:0030-drops-102512},
  doi =		{10.4230/LIPIcs.STACS.2019.12},
  annote =	{Keywords: Nonlocal games, communication complexity, bounded separations, semidefinite programming, pseudorandomness, Gowers norms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.38
A Cut Tree Representation for Pendant Pairs

Authors: On-Hei S. Lo and Jens M. Schmidt

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today. Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs.
Cite as

On-Hei S. Lo and Jens M. Schmidt. A Cut Tree Representation for Pendant Pairs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 38:1-38:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lo_et_al:LIPIcs.ISAAC.2018.38,
  author =	{Lo, On-Hei S. and Schmidt, Jens M.},
  title =	{{A Cut Tree Representation for Pendant Pairs}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{38:1--38:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.38},
  URN =		{urn:nbn:de:0030-drops-99869},
  doi =		{10.4230/LIPIcs.ISAAC.2018.38},
  annote =	{Keywords: Pendant Pairs, Pendant Tree, Maximal Adjacency Ordering, Maximum Cardinality Search, Testing Edge-Connectivity, Gomory-Hu Tree}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2016.5
Linear-Time Recognition of Map Graphs with Outerplanar Witness

Authors: Matthias Mnich, Ignaz Rutter, and Jens M. Schmidt

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract
Map graphs generalize planar graphs and were introduced by Chen, Grigni and Papadimitriou [STOC 1998, J.ACM 2002]. They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup [FOCS 1998] published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Omega(n^{120}) for n-vertex graphs, and a full description of its details remains unpublished. We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup's approach, it computes the witness graph W in the affirmative case.
Cite as

Matthias Mnich, Ignaz Rutter, and Jens M. Schmidt. Linear-Time Recognition of Map Graphs with Outerplanar Witness. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mnich_et_al:LIPIcs.SWAT.2016.5,
  author =	{Mnich, Matthias and Rutter, Ignaz and Schmidt, Jens M.},
  title =	{{Linear-Time Recognition of Map Graphs with Outerplanar Witness}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.5},
  URN =		{urn:nbn:de:0030-drops-60349},
  doi =		{10.4230/LIPIcs.SWAT.2016.5},
  annote =	{Keywords: Algorithms and data structures, map graphs, recognition, planar graphs}
}
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