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Document
Complete Volume
Authors:
Nikhil Bansal, Emanuela Merelli, and James Worrell
Abstract
LIPIcs, Volume 198, ICALP 2021, Complete Volume
Cite as
48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 1-2622, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@Proceedings{bansal_et_al:LIPIcs.ICALP.2021,
title = {{LIPIcs, Volume 198, ICALP 2021, Complete Volume}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {1--2622},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021},
URN = {urn:nbn:de:0030-drops-140687},
doi = {10.4230/LIPIcs.ICALP.2021},
annote = {Keywords: LIPIcs, Volume 198, ICALP 2021, Complete Volume}
}
Document
Front Matter
Authors:
Nikhil Bansal, Emanuela Merelli, and James Worrell
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 0:i-0:xxxviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bansal_et_al:LIPIcs.ICALP.2021.0,
author = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {0:i--0:xxxviii},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.0},
URN = {urn:nbn:de:0030-drops-140696},
doi = {10.4230/LIPIcs.ICALP.2021.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Authors:
Christel Baier, Clemens Dubslaff, Florian Funke, Simon Jantsch, Rupak Majumdar, Jakob Piribauer, and Robin Ziemek
Abstract
In view of the growing complexity of modern software architectures, formal models are increasingly used to understand why a system works the way it does, opposed to simply verifying that it behaves as intended. This paper surveys approaches to formally explicate the observable behavior of reactive systems. We describe how Halpern and Pearl’s notion of actual causation inspired verification-oriented studies of cause-effect relationships in the evolution of a system. A second focus lies on applications of the Shapley value to responsibility ascriptions, aimed to measure the influence of an event on an observable effect. Finally, formal approaches to probabilistic causation are collected and connected, and their relevance to the understanding of probabilistic systems is discussed.
Cite as
Christel Baier, Clemens Dubslaff, Florian Funke, Simon Jantsch, Rupak Majumdar, Jakob Piribauer, and Robin Ziemek. From Verification to Causality-Based Explications (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{baier_et_al:LIPIcs.ICALP.2021.1,
author = {Baier, Christel and Dubslaff, Clemens and Funke, Florian and Jantsch, Simon and Majumdar, Rupak and Piribauer, Jakob and Ziemek, Robin},
title = {{From Verification to Causality-Based Explications}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {1:1--1:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.1},
URN = {urn:nbn:de:0030-drops-140709},
doi = {10.4230/LIPIcs.ICALP.2021.1},
annote = {Keywords: Model Checking, Causality, Responsibility, Counterfactuals, Shapley value}
}
Document
Invited Talk
Authors:
Andrei A. Bulatov
Abstract
The Constraint Satisfaction Problem (CSP) and a number of problems related to it have seen major advances during the past three decades. In many cases the leading driving force that made these advances possible has been the so-called algebraic approach that uses symmetries of constraint problems and tools from algebra to determine the complexity of problems and design solution algorithms. In this presentation we give a high level overview of the main ideas behind the algebraic approach illustrated by examples ranging from the regular CSP, to counting problems, to optimization and promise problems, to graph isomorphism.
Cite as
Andrei A. Bulatov. Symmetries and Complexity (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bulatov:LIPIcs.ICALP.2021.2,
author = {Bulatov, Andrei A.},
title = {{Symmetries and Complexity}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {2:1--2:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.2},
URN = {urn:nbn:de:0030-drops-140717},
doi = {10.4230/LIPIcs.ICALP.2021.2},
annote = {Keywords: constraint problems, algebraic approach, dichotomy theorems}
}
Document
Invited Talk
Authors:
Keren Censor-Hillel
Abstract
This is a survey of the exciting recent progress made in understanding the complexity of distributed subgraph finding problems. It overviews the results and techniques for assorted variants of subgraph finding problems in various models of distributed computing, and states intriguing open questions.
Cite as
Keren Censor-Hillel. Distributed Subgraph Finding: Progress and Challenges (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{censorhillel:LIPIcs.ICALP.2021.3,
author = {Censor-Hillel, Keren},
title = {{Distributed Subgraph Finding: Progress and Challenges}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {3:1--3:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.3},
URN = {urn:nbn:de:0030-drops-140726},
doi = {10.4230/LIPIcs.ICALP.2021.3},
annote = {Keywords: distributed algorithms, subgraph finding, limited bandwidth}
}
Document
Invited Talk
Authors:
Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, and Ran J. Tessler
Abstract
In this paper we consider a new type of space partitioning which bridges the gap between continuous and discrete spaces in an error resilient way. It is motivated by the problem of rounding noisy measurements from some continuous space such as ℝ^d to a discrete subset of representative values, in which each tile in the partition is defined as the preimage of one of the output points. Standard rounding schemes seem to be inherently discontinuous across tile boundaries, but in this paper we show how to make it perfectly consistent (with error resilience ε) by guaranteeing that any pair of consecutive measurements X₁ and X₂ whose L₂ distance is bounded by ε will be rounded to the same nearby representative point in the discrete output space. We achieve this resilience by allowing a few bits of information about the first measurement X₁ to be unidirectionally communicated to and used by the rounding process of the second measurement X₂. Minimizing this revealed information can be particularly important in privacy-sensitive applications such as COVID-19 contact tracing, in which we want to find out all the cases in which two persons were at roughly the same place at roughly the same time, by comparing cryptographically hashed versions of their itineraries in an error resilient way.
The main problem we study in this paper is characterizing the achievable tradeoffs between the amount of information provided and the error resilience for various dimensions. We analyze the problem by considering the possible colored tilings of the space with k available colors, and use the color of the tile in which X₁ resides as the side information. We obtain our upper and lower bounds with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Sperner’s lemma, and Čech cohomology. In particular, we show that when X_i ∈ ℝ^d, communicating log₂(d+1) bits of information is both sufficient and necessary (in the worst case) to achieve positive resilience, and when d=3 we obtain a tight upper and lower asymptotic bound of (0.561 …)k^{1/3} on the achievable error resilience when we provide log₂(k) bits of information about X₁’s color.
Cite as
Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, and Ran J. Tessler. Error Resilient Space Partitioning (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dunkelman_et_al:LIPIcs.ICALP.2021.4,
author = {Dunkelman, Orr and Geyzel, Zeev and Keller, Chaya and Keller, Nathan and Ronen, Eyal and Shamir, Adi and Tessler, Ran J.},
title = {{Error Resilient Space Partitioning}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {4:1--4:22},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.4},
URN = {urn:nbn:de:0030-drops-140731},
doi = {10.4230/LIPIcs.ICALP.2021.4},
annote = {Keywords: space partition, high-dimensional rounding, error resilience, sphere packing, Sperner’s lemma, Brunn-Minkowski theorem, \v{C}ech cohomology}
}
Document
Invited Talk
Authors:
Toniann Pitassi
Abstract
Given a set of polynomial equations over a field F, how hard is it to prove that they are simultaneously unsolvable? In the last twenty years, algebraic proof systems for refuting such systems of equations have been extensively studied, revealing close connections to both upper bounds (connections between short refutations and efficient approximation algorithms) and lower bounds (connections to fundamental questions in circuit complexity.)
The Ideal Proof System (IPS) is a simple yet powerful algebraic proof system, with very close connections to circuit lower bounds: [Joshua A. Grochow and Toniann Pitassi, 2018] proved that lower bounds for IPS imply VNP ≠ VP, and very recently connections in the other direction have been made, showing that circuit lower bounds imply IPS lower bounds [Rahul Santhanam and Iddo Tzameret, 2021; Yaroslav Alekseev et al., 2020].
In this talk I will survey the landscape of algebraic proof systems, focusing on their connections to complexity theory, derandomization, and standard proposional proof complexity. I will discuss the state-of-the-art lower bounds, as well as the relationship between algebraic systems and textbook style propositional proof systems. Finally we end with open problems, and some recent progress towards proving superpolynomial lower bounds for bounded-depth Frege systems with modular gates (a major open problem in propositional proof complexity).
Cite as
Toniann Pitassi. Algebraic Proof Systems (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, p. 5:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{pitassi:LIPIcs.ICALP.2021.5,
author = {Pitassi, Toniann},
title = {{Algebraic Proof Systems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {5:1--5:1},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.5},
URN = {urn:nbn:de:0030-drops-140747},
doi = {10.4230/LIPIcs.ICALP.2021.5},
annote = {Keywords: complexity theory, proof complexity, algebraic circuits}
}
Document
Invited Talk
Authors:
David P. Woodruff
Abstract
We give an overview of dimensionality reduction methods, or sketching, for a number of problems in optimization, first surveying work using these methods for classical problems, which gives near optimal algorithms for regression, low rank approximation, and natural variants. We then survey recent work applying sketching to column subset selection, kernel methods, sublinear algorithms for structured matrices, tensors, trace estimation, and so on. The focus is on fast algorithms. This is a short survey accompanying an invited talk at ICALP, 2021.
Cite as
David P. Woodruff. A Very Sketchy Talk (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 6:1-6:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{woodruff:LIPIcs.ICALP.2021.6,
author = {Woodruff, David P.},
title = {{A Very Sketchy Talk}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {6:1--6:8},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.6},
URN = {urn:nbn:de:0030-drops-140755},
doi = {10.4230/LIPIcs.ICALP.2021.6},
annote = {Keywords: dimensionality reduction, optimization, randomized numerical linear algebra, sketching}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Amir Abboud and Virginia Vassilevska Williams
Abstract
Nearly all quadratic lower bounds conditioned on the Strong Exponential Time Hypothesis (SETH) start by reducing k-SAT to the Orthogonal Vectors (OV) problem: Given two sets A,B of n binary vectors, decide if there is an orthogonal pair a ∈ A, b ∈ B. In this paper, we give an alternative reduction in which the set A does not depend on the input to k-SAT; thus, the quadratic lower bound for OV holds even if one of the sets is fixed in advance.
Using the reductions in the literature from OV to other problems such as computing similarity measures on strings, we get hardness results of a stronger kind: there is a family of sequences {S_n}_{n = 1}^{∞}, |S_n| = n such that computing the Edit Distance between an input sequence X of length n and the (fixed) sequence S_n requires n^{2-o(1)} time under SETH.
Cite as
Amir Abboud and Virginia Vassilevska Williams. Fine-Grained Hardness for Edit Distance to a Fixed Sequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{abboud_et_al:LIPIcs.ICALP.2021.7,
author = {Abboud, Amir and Vassilevska Williams, Virginia},
title = {{Fine-Grained Hardness for Edit Distance to a Fixed Sequence}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {7:1--7:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.7},
URN = {urn:nbn:de:0030-drops-140768},
doi = {10.4230/LIPIcs.ICALP.2021.7},
annote = {Keywords: SAT, edit distance, fine-grained complexity, conditional lower bound, sequence alignment}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Dimitris Achlioptas and Kostas Zampetakis
Abstract
The independent set polynomial of a graph has one variable for each vertex and one monomial for each independent set, comprising the product of the corresponding variables. Given a graph G on n vertices and a vector p ∈ [0,1)ⁿ, a central problem in statistical mechanics is determining whether the independent set polynomial of G is non-vanishing in the polydisk of p, i.e., whether |Z_G(x)| > 0 for every x ∈ ℂⁿ such that |x_i| ≤ p_i. Remarkably, when this holds, Z_G(-p) is a lower bound for the avoidance probability when G is a dependency graph for n events whose probabilities form vector p. A local sufficient condition for |Z_G| > 0 in the polydisk of p is the Lovász Local Lemma (LLL).
In this work we derive several new results on the efficient evaluation and bounding of Z_G. Our starting point is a monotone mapping from subgraphs of G to truncations of the tree of self-avoiding walks of G. Using this mapping our first result is a local upper bound for Z(-p), similar in spirit to the local lower bound for Z(-p) provided by the LLL. Next, using this mapping, we show that when G is chordal, Z_G can be computed exactly and in linear time on the entire complex plane, implying perfect sampling for the hard-core model on chordal graphs. We also revisit the task of bounding Z(-p) from below, i.e., the LLL setting, and derive four new lower bounds of increasing sophistication. Already our simplest (and weakest) bound yields a strict improvement of the famous asymmetric LLL, i.e., a strict relaxation of the inequalities of the asymmetric LLL without any further assumptions. This new asymmetric local lemma is sharp enough to recover Shearer’s optimal bound in terms of the maximum degree Δ(G). We also apply our more sophisticated bounds to estimate the zero-free region of the hard-core model on the triangular lattice (hard hexagons model).
Cite as
Dimitris Achlioptas and Kostas Zampetakis. Local Approximations of the Independent Set Polynomial. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{achlioptas_et_al:LIPIcs.ICALP.2021.8,
author = {Achlioptas, Dimitris and Zampetakis, Kostas},
title = {{Local Approximations of the Independent Set Polynomial}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {8:1--8:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.8},
URN = {urn:nbn:de:0030-drops-140773},
doi = {10.4230/LIPIcs.ICALP.2021.8},
annote = {Keywords: Independent Set Polynomial, Lov\'{a}sz Local Lemma, Self-avoiding Walks}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva
Abstract
We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (𝓁²₂ + 𝓁^p_p) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit 𝓁_p weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find (1+2^{-poly(log n)}) approximations for weighted 𝓁_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = ω(1), which is almost-linear for graphs that are slightly dense (m ≥ n^{4/3 + o(1)}).
Cite as
Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva. Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{adil_et_al:LIPIcs.ICALP.2021.9,
author = {Adil, Deeksha and Bullins, Brian and Kyng, Rasmus and Sachdeva, Sushant},
title = {{Almost-Linear-Time Weighted 𝓁\underlinep-Norm Solvers in Slightly Dense Graphs via Sparsification}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {9:1--9:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.9},
URN = {urn:nbn:de:0030-drops-140782},
doi = {10.4230/LIPIcs.ICALP.2021.9},
annote = {Keywords: Weighted 𝓁\underlinep-norm, Sparsification, Spanners, Iterative Refinement}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Pankaj K. Agarwal and Alex Steiger
Abstract
Let C be a set of n axis-aligned cubes of arbitrary sizes in ℝ³. Let U be their union, and let κ be the number of vertices on ∂U; κ can vary between O(1) and O(n²). We show that U can be computed in O(n log³ n + κ) time if C is in general position. The algorithm also computes the union of a set of fat boxes (i.e., boxes with bounded aspect ratio) within the same time bound. If the cubes in C are congruent or have bounded depth, the running time improves to O(n log² n), and if both conditions hold, the running time improves to O(n log n).
Cite as
Pankaj K. Agarwal and Alex Steiger. An Output-Sensitive Algorithm for Computing the Union of Cubes and Fat Boxes in 3D. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{agarwal_et_al:LIPIcs.ICALP.2021.10,
author = {Agarwal, Pankaj K. and Steiger, Alex},
title = {{An Output-Sensitive Algorithm for Computing the Union of Cubes and Fat Boxes in 3D}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {10:1--10:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.10},
URN = {urn:nbn:de:0030-drops-140790},
doi = {10.4230/LIPIcs.ICALP.2021.10},
annote = {Keywords: union of cubes, fat boxes, plane-sweep}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Pankaj K. Agarwal, Xiao Hu, Stavros Sintos, and Jun Yang
Abstract
This paper considers enumerating answers to similarity-join queries under dynamic updates: Given two sets of n points A,B in ℝ^d, a metric ϕ(⋅), and a distance threshold r > 0, report all pairs of points (a, b) ∈ A × B with ϕ(a,b) ≤ r. Our goal is to store A,B into a dynamic data structure that, whenever asked, can enumerate all result pairs with worst-case delay guarantee, i.e., the time between enumerating two consecutive pairs is bounded. Furthermore, the data structure can be efficiently updated when a point is inserted into or deleted from A or B.
We propose several efficient data structures for answering similarity-join queries in low dimension. For exact enumeration of similarity join, we present near-linear-size data structures for 𝓁₁, 𝓁_∞ metrics with log^{O(1)} n update time and delay. We show that such a data structure is not feasible for the 𝓁₂ metric for d ≥ 4. For approximate enumeration of similarity join, where the distance threshold is a soft constraint, we obtain a unified linear-size data structure for 𝓁_p metric, with log^{O(1)} n delay and update time. In high dimensions, we present an efficient data structure with worst-case delay-guarantee using locality sensitive hashing (LSH).
Cite as
Pankaj K. Agarwal, Xiao Hu, Stavros Sintos, and Jun Yang. Dynamic Enumeration of Similarity Joins. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{agarwal_et_al:LIPIcs.ICALP.2021.11,
author = {Agarwal, Pankaj K. and Hu, Xiao and Sintos, Stavros and Yang, Jun},
title = {{Dynamic Enumeration of Similarity Joins}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {11:1--11:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.11},
URN = {urn:nbn:de:0030-drops-140803},
doi = {10.4230/LIPIcs.ICALP.2021.11},
annote = {Keywords: dynamic enumeration, similarity joins, worst-case delay guarantee}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Shyan Akmal and Ce Jin
Abstract
Tree edit distance is a well-studied measure of dissimilarity between rooted trees with node labels. It can be computed in O(n³) time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007], and fine-grained hardness results suggest that the weighted version of this problem cannot be solved in truly subcubic time unless the APSP conjecture is false [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018].
We consider the unweighted version of tree edit distance, where every insertion, deletion, or relabeling operation has unit cost. Given a parameter k as an upper bound on the distance, the previous fastest algorithm for this problem runs in O(nk³) time [Touzet, CPM 2005], which improves upon the cubic-time algorithm for k≪ n^{2/3}. In this paper, we give a faster algorithm taking O(nk² log n) time, improving both of the previous results for almost the full range of log n ≪ k≪ n/√{log n}.
Cite as
Shyan Akmal and Ce Jin. Faster Algorithms for Bounded Tree Edit Distance. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{akmal_et_al:LIPIcs.ICALP.2021.12,
author = {Akmal, Shyan and Jin, Ce},
title = {{Faster Algorithms for Bounded Tree Edit Distance}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {12:1--12:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.12},
URN = {urn:nbn:de:0030-drops-140819},
doi = {10.4230/LIPIcs.ICALP.2021.12},
annote = {Keywords: tree edit distance, edit distance, dynamic programming}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Shyan Akmal and Virginia Vassilevska Williams
Abstract
This paper investigates the approximability of the Longest Common Subsequence (LCS) problem. The fastest algorithm for solving the LCS problem exactly runs in essentially quadratic time in the length of the input, and it is known that under the Strong Exponential Time Hypothesis the quadratic running time cannot be beaten. There are no such limitations for the approximate computation of the LCS however, except in some limited scenarios. There is also a scarcity of approximation algorithms. When the two given strings are over an alphabet of size k, returning the subsequence formed by the most frequent symbol occurring in both strings achieves a 1/k approximation for the LCS. It is an open problem whether a better than 1/k approximation can be achieved in truly subquadratic time (O(n^{2-δ}) time for constant δ > 0).
A recent result [Rubinstein and Song SODA'2020] showed that a 1/2+ε approximation for the LCS over a binary alphabet is possible in truly subquadratic time, provided the input strings have the same length. In this paper we show that if a 1/2+ε approximation (for ε > 0) is achievable for binary LCS in truly subquadratic time when the input strings can be unequal, then for every constant k, there is a truly subquadratic time algorithm that achieves a 1/k+δ approximation for k-ary alphabet LCS for some δ > 0. Thus the binary case is the hardest. We also show that for every constant k, if one is given two strings of equal length over a k-ary alphabet, one can obtain a 1/k+ε approximation for some constant ε > 0 in truly subquadratic time, thus extending the Rubinstein and Song result to all alphabets of constant size.
Cite as
Shyan Akmal and Virginia Vassilevska Williams. Improved Approximation for Longest Common Subsequence over Small Alphabets. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{akmal_et_al:LIPIcs.ICALP.2021.13,
author = {Akmal, Shyan and Vassilevska Williams, Virginia},
title = {{Improved Approximation for Longest Common Subsequence over Small Alphabets}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {13:1--13:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.13},
URN = {urn:nbn:de:0030-drops-140821},
doi = {10.4230/LIPIcs.ICALP.2021.13},
annote = {Keywords: approximation algorithms, longest common subsequence, subquadratic}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Noga Alon and Andrei Graur
Abstract
We provide efficient approximation algorithms for the Necklace Splitting problem. The input consists of a sequence of beads of n types and an integer k. The objective is to split the necklace, with a small number of cuts made between consecutive beads, and distribute the resulting intervals into k collections so that the discrepancy between the shares of any two collections, according to each type, is at most 1. We also consider an approximate version where each collection should contain at least a (1-ε)/k and at most a (1+ε)/k fraction of the beads of each type. It is known that there is always a solution making at most n(k-1) cuts, and this number of cuts is optimal in general. The proof is topological and provides no efficient procedure for finding these cuts. It is also known that for k = 2, and some fixed positive ε, finding a solution with n cuts is PPAD-hard.
We describe an efficient algorithm that produces an ε-approximate solution for k = 2 making n (2+log (1/ε)) cuts. This is an exponential improvement of a (1/ε)^O(n) bound of Bhatt and Leighton from the 80s. We also present an online algorithm for the problem (in its natural online model), in which the number of cuts made to produce discrepancy at most 1 on each type is Õ(m^{2/3} n), where m is the maximum number of beads of any type. Lastly, we establish a lower bound showing that for the online setup this is tight up to logarithmic factors. Similar results are obtained for k > 2.
Cite as
Noga Alon and Andrei Graur. Efficient Splitting of Necklaces. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{alon_et_al:LIPIcs.ICALP.2021.14,
author = {Alon, Noga and Graur, Andrei},
title = {{Efficient Splitting of Necklaces}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {14:1--14:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.14},
URN = {urn:nbn:de:0030-drops-140832},
doi = {10.4230/LIPIcs.ICALP.2021.14},
annote = {Keywords: necklace splitting, necklace halving, approximation algorithms, online algorithms, discrepancy}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Markus Anders, Pascal Schweitzer, and Florian Wetzels
Abstract
Color refinement is a crucial subroutine in symmetry detection in theory as well as practice. It has further applications in machine learning and in computational problems from linear algebra.
While tight lower bounds for the worst case complexity are known [Berkholz, Bonsma, Grohe, ESA2013] no comparative analysis of design choices for color refinement algorithms is available.
We devise two models within which we can compare color refinement algorithms using formal methods, an online model and an approximation model. We use these to show that no online algorithm is competitive beyond a logarithmic factor and no algorithm can approximate the optimal color refinement splitting scheme beyond a logarithmic factor.
We also directly compare strategies used in practice showing that, on some graphs, queue based strategies outperform stack based ones by a logarithmic factor and vice versa. Similar results hold for strategies based on priority queues.
Cite as
Markus Anders, Pascal Schweitzer, and Florian Wetzels. Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{anders_et_al:LIPIcs.ICALP.2021.15,
author = {Anders, Markus and Schweitzer, Pascal and Wetzels, Florian},
title = {{Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {15:1--15:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.15},
URN = {urn:nbn:de:0030-drops-140846},
doi = {10.4230/LIPIcs.ICALP.2021.15},
annote = {Keywords: Color refinement, Online algorithms, Graph isomorphism, Lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Markus Anders and Pascal Schweitzer
Abstract
We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other.
We describe a randomized algorithm with errors for which the number of visited nodes is quasilinear in the square root of the size of the smaller of the two trees. For inputs of bounded degree, we develop a Las Vegas algorithm with a similar running time.
We prove that these results are optimal up to logarithmic factors. For this, we show a lower bound for randomized algorithms on inputs of bounded degree that is the square root of the tree sizes. For inputs of unbounded degree, we show a linear lower bound for Las Vegas algorithms. For deterministic algorithms we can prove a linear bound even for inputs of bounded degree. This shows why randomized algorithms outperform deterministic ones.
Our results explain why the randomized "breadth-first with intermixed experimental path" search strategy of the isomorphism tool Traces (Piperno 2008) is often superior to the depth-first search strategy of other tools such as nauty (McKay 1977) or bliss (Junttila, Kaski 2007). However, our algorithm also provides a new traversal strategy, which is theoretically near optimal and which has better worst case behavior than traversal strategies that have previously been used.
Cite as
Markus Anders and Pascal Schweitzer. Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{anders_et_al:LIPIcs.ICALP.2021.16,
author = {Anders, Markus and Schweitzer, Pascal},
title = {{Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {16:1--16:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.16},
URN = {urn:nbn:de:0030-drops-140853},
doi = {10.4230/LIPIcs.ICALP.2021.16},
annote = {Keywords: Online algorithms, Graph isomorphism, Lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Antonios Antoniadis, Matthias Englert, Nicolaos Matsakis, and Pavel Veselý
Abstract
We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably rejected. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets.
The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from the back of whichever queue is currently the longest, breaking ties arbitrarily. The LQD algorithm was first introduced in 1991, and is known to be 2-competitive since 2001. Although LQD remains the best known online algorithm for the problem and is of practical interest, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.707-competitive, establishing the first (2-ε) upper bound for the competitive ratio of LQD, for a constant ε > 0.
Cite as
Antonios Antoniadis, Matthias Englert, Nicolaos Matsakis, and Pavel Veselý. Breaking the Barrier Of 2 for the Competitiveness of Longest Queue Drop. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{antoniadis_et_al:LIPIcs.ICALP.2021.17,
author = {Antoniadis, Antonios and Englert, Matthias and Matsakis, Nicolaos and Vesel\'{y}, Pavel},
title = {{Breaking the Barrier Of 2 for the Competitiveness of Longest Queue Drop}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {17:1--17:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.17},
URN = {urn:nbn:de:0030-drops-140864},
doi = {10.4230/LIPIcs.ICALP.2021.17},
annote = {Keywords: buffer management, online scheduling, online algorithms, longest queue drop}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Vahid R. Asadi and Igor Shinkar
Abstract
Locally decodable codes (LDCs) are error-correcting codes C: Σ^k → Σⁿ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important question in this line of research is to understand the optimal trade-off between the query complexity of LDCs and their block length. Despite importance of these objects, the best known constructions of constant query LDCs have super-polynomial length, and there is a significant gap between the best constructions and the known lower bounds in terms of the block length.
For many applications it suffices to consider the weaker notion of relaxed LDCs (RLDCs), which allows the local decoding algorithm to abort if by querying a few bits it detects that the input is not a codeword. This relaxation turned out to allow decoding algorithms with constant query complexity for codes with almost linear length. Specifically, [{Ben-Sasson} et al., 2006] constructed a q-query RLDC that encodes a message of length k using a codeword of block length n = O_q(k^{1+O(1/√q)}) for any sufficiently large q, where O_q(⋅) hides some constant that depends only on q.
In this work we improve the parameters of [{Ben-Sasson} et al., 2006] by constructing a q-query RLDC that encodes a message of length k using a codeword of block length O_q(k^{1+O(1/{q})}) for any sufficiently large q. This construction matches (up to a multiplicative constant factor) the lower bounds of [Jonathan Katz and Trevisan, 2000; Woodruff, 2007] for constant query LDCs, thus making progress toward understanding the gap between LDCs and RLDCs in the constant query regime.
In fact, our construction extends to the stronger notion of relaxed locally correctable codes (RLCCs), introduced in [Tom Gur et al., 2018], where given a noisy codeword the correcting algorithm either recovers each individual bit of the codeword by only reading a small part of the input, or aborts if the input is detected to be corrupt.
Cite as
Vahid R. Asadi and Igor Shinkar. Relaxed Locally Correctable Codes with Improved Parameters. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 18:1-18:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{asadi_et_al:LIPIcs.ICALP.2021.18,
author = {Asadi, Vahid R. and Shinkar, Igor},
title = {{Relaxed Locally Correctable Codes with Improved Parameters}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {18:1--18:12},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.18},
URN = {urn:nbn:de:0030-drops-140878},
doi = {10.4230/LIPIcs.ICALP.2021.18},
annote = {Keywords: Algorithmic coding theory, consistency test using random walk, Reed-Muller code, relaxed locally decodable codes, relaxed locally correctable codes}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Sepehr Assadi and Soheil Behnezhad
Abstract
We study the maximum matching problem in the random-order semi-streaming setting. In this problem, the edges of an arbitrary n-vertex graph G = (V, E) arrive in a stream one by one and in a random order. The goal is to have a single pass over the stream, use O(n ⋅ polylog) space, and output a large matching of G.
We prove that for an absolute constant ε₀ > 0, one can find a (2/3 + ε₀)-approximate maximum matching of G using O(n log n) space with high probability. This breaks the natural boundary of 2/3 for this problem prevalent in the prior work and resolves an open problem of Bernstein [ICALP'20] on whether a (2/3 + Ω(1))-approximation is achievable.
Cite as
Sepehr Assadi and Soheil Behnezhad. Beating Two-Thirds For Random-Order Streaming Matching. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{assadi_et_al:LIPIcs.ICALP.2021.19,
author = {Assadi, Sepehr and Behnezhad, Soheil},
title = {{Beating Two-Thirds For Random-Order Streaming Matching}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {19:1--19:13},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.19},
URN = {urn:nbn:de:0030-drops-140887},
doi = {10.4230/LIPIcs.ICALP.2021.19},
annote = {Keywords: Maximum Matching, Streaming, Random-Order Streaming}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Mitali Bafna and Nikhil Vyas
Abstract
The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A,b), for A ∈ ℝ^{n×n} and b ∈ ℝⁿ, we wish to find a vector x ∈ ℝⁿ such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n^ω). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L₂-norm, that is, given a satisfiable linear system (A ∈ ℝ^{n×n}, b ∈ ℝⁿ), find an x ∈ ℝⁿ such that ||Ax - b||₂ ≤ ε||b||₂. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n^ω) time algorithm for computing the rank of n × O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n^ω) time, even for ε as large as (1 - 1/poly(n)). We also prove (under some modified conjectures for the rank-finding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices and well-conditioned linear systems. At the heart of our results is a novel reduction from the rank problem to a decision version of the approximate linear systems problem. This reduction preserves properties such as matrix sparsity and bit complexity.
Cite as
Mitali Bafna and Nikhil Vyas. Optimal Fine-Grained Hardness of Approximation of Linear Equations. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bafna_et_al:LIPIcs.ICALP.2021.20,
author = {Bafna, Mitali and Vyas, Nikhil},
title = {{Optimal Fine-Grained Hardness of Approximation of Linear Equations}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {20:1--20:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.20},
URN = {urn:nbn:de:0030-drops-140894},
doi = {10.4230/LIPIcs.ICALP.2021.20},
annote = {Keywords: Linear Equations, Fine-Grained Complexity, Hardness of Approximation}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Tanvi Bajpai, Deeparnab Chakrabarty, Chandra Chekuri, and Maryam Negahbani
Abstract
In the Priority k-Center problem, the input consists of a metric space (X,d), an integer k and for each point v ∈ X a priority radius r(v). The goal is to choose k-centers S ⊆ X to minimize max_{v ∈ X} 1/(r(v)) d(v,S). If all r(v)’s were uniform, one obtains the classical k-center problem. Plesník [Ján Plesník, 1987] introduced this problem and gave a 2-approximation algorithm matching the best possible algorithm for vanilla k-center. We show how the Priority k-Center problem is related to two different notions of fair clustering [Harris et al., 2019; Christopher Jung et al., 2020]. Motivated by these developments we revisit the problem and, in our main technical contribution, develop a framework that yields constant factor approximation algorithms for Priority k-Center with outliers. Our framework extends to generalizations of Priority k-Center to matroid and knapsack constraints, and as a corollary, also yields algorithms with fairness guarantees in the lottery model of Harris et al.
Cite as
Tanvi Bajpai, Deeparnab Chakrabarty, Chandra Chekuri, and Maryam Negahbani. Revisiting Priority k-Center: Fairness and Outliers. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bajpai_et_al:LIPIcs.ICALP.2021.21,
author = {Bajpai, Tanvi and Chakrabarty, Deeparnab and Chekuri, Chandra and Negahbani, Maryam},
title = {{Revisiting Priority k-Center: Fairness and Outliers}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {21:1--21:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.21},
URN = {urn:nbn:de:0030-drops-140909},
doi = {10.4230/LIPIcs.ICALP.2021.21},
annote = {Keywords: Fairness, Clustering, Approximation, Outliers}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Etienne Bamas, Paritosh Garg, and Lars Rohwedder
Abstract
The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function f_i. The goal is to maximize min_i f_i(S_i) where S₁,...,S_m is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n^{1/2 +ε})-approximation algorithm.
Since then progress on this problem was limited to the linear case, that is, all f_i are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources Γ_i and the individual valuation functions are defined as f_i(S) = f(S ∩ Γ_i) for a global linear function f. This can also be interpreted as maximizing min_i f(S_i) with additional assignment restrictions, i.e., resources can only be assigned to certain players.
In this paper we make comparable progress for the submodular variant: If f is a monotone submodular function, we can in polynomial time compute an O(log log(n))-approximate solution.
Cite as
Etienne Bamas, Paritosh Garg, and Lars Rohwedder. The Submodular Santa Claus Problem in the Restricted Assignment Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bamas_et_al:LIPIcs.ICALP.2021.22,
author = {Bamas, Etienne and Garg, Paritosh and Rohwedder, Lars},
title = {{The Submodular Santa Claus Problem in the Restricted Assignment Case}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {22:1--22:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.22},
URN = {urn:nbn:de:0030-drops-140912},
doi = {10.4230/LIPIcs.ICALP.2021.22},
annote = {Keywords: Scheduling, submodularity, approximation algorithm, hypergraph matching}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov
Abstract
Fair clustering is a variant of constrained clustering where the goal is to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS 2017] and became widely popular. This paper proposes a new construction of coresets for fair k-means and k-median clustering for Euclidean and general metrics based on random sampling. For the Euclidean space ℝ^d, we provide the first coresets whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt, Schwiegelshohn, and Sohler [WAOA 2019] and Huang, Jiang, and Vishnoi [NeurIPS 2019]. For general metric, our construction provides the first coreset for fair k-means and k-median.
New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. In particular, we obtain
- the first fixed-parameter tractable (FPT) PTAS for fair k-means and k-median clustering in ℝ^d. The near-linear time of our PTAS improves over the previous scheme of Böhm, Fazzone, Leonardi, and Schwiegelshohn [ArXiv 2020] with running time n^{poly(k/ε)};
- FPT "true" constant-approximation for metric fair clustering. All previous algorithms for fair k-means and k-median in general metric are bicriteria and violate the fairness constraints;
- FPT 3-approximation for lower-bounded k-median improving the best-known 3.736 factor of Bera, Chakrabarty, and Negahbani [ArXiv 2019];
- the first FPT constant-approximations for metric chromatic clustering and 𝓁-Diversity clustering;
- near linear-time (in n) PTAS for capacitated and lower-bounded clustering improving over PTAS of Bhattacharya, Jaiswal, and Kumar [TOCS 2018] with super-quadratic running time;
- a streaming (1+ε)-approximation for fair k-means and k-median of space complexity polynomial in k, d, ε and log{n} (the previous algorithms have exponential space complexity on either d or k).
Cite as
Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov. On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bandyapadhyay_et_al:LIPIcs.ICALP.2021.23,
author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Simonov, Kirill},
title = {{On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {23:1--23:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.23},
URN = {urn:nbn:de:0030-drops-140923},
doi = {10.4230/LIPIcs.ICALP.2021.23},
annote = {Keywords: fair clustering, coresets, approximation algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Eleni Batziou, Kristoffer Arnsfelt Hansen, and Kasper Høgh
Abstract
In the consensus halving problem we are given n agents with valuations over the interval [0,1]. The goal is to divide the interval into at most n+1 pieces (by placing at most n cuts), which may be combined to give a partition of [0,1] into two sets valued equally by all agents. The existence of a solution may be established by the Borsuk-Ulam theorem. We consider the task of computing an approximation of an exact solution of the consensus halving problem, where the valuations are given by distribution functions computed by algebraic circuits. Here approximation refers to computing a point that is ε-close to an exact solution, also called strong approximation. We show that this task is polynomial time equivalent to computing an approximation to an exact solution of the Borsuk-Ulam search problem defined by a continuous function that is computed by an algebraic circuit.
The Borsuk-Ulam search problem is the defining problem of the complexity class BU. We introduce a new complexity class BBU to also capture an alternative formulation of the Borsuk-Ulam theorem from a computational point of view. We investigate their relationship and prove several structural results for these classes as well as for the complexity class FIXP.
Cite as
Eleni Batziou, Kristoffer Arnsfelt Hansen, and Kasper Høgh. Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{batziou_et_al:LIPIcs.ICALP.2021.24,
author = {Batziou, Eleni and Hansen, Kristoffer Arnsfelt and H{\o}gh, Kasper},
title = {{Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {24:1--24:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.24},
URN = {urn:nbn:de:0030-drops-140939},
doi = {10.4230/LIPIcs.ICALP.2021.24},
annote = {Keywords: Consensus halving, Computational Complexity, Borsuk-Ulam}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ran Ben Basat, Michael Mitzenmacher, and Shay Vargaftik
Abstract
We consider the fundamental problem of communicating an estimate of a real number x ∈ [0,1] using a single bit. A sender that knows x chooses a value X ∈ {0,1} to transmit. In turn, a receiver estimates x based on the value of X. The goal is to minimize the cost, defined as the worst-case (over the choice of x) expected squared error.
We first overview common biased and unbiased estimation approaches and prove their optimality when no shared randomness is allowed. We then show how a small amount of shared randomness, which can be as low as a single bit, reduces the cost in both cases. Specifically, we derive lower bounds on the cost attainable by any algorithm with unrestricted use of shared randomness and propose optimal and near-optimal solutions that use a small number of shared random bits. Finally, we discuss open problems and future directions.
Cite as
Ran Ben Basat, Michael Mitzenmacher, and Shay Vargaftik. How to Send a Real Number Using a Single Bit (And Some Shared Randomness). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{benbasat_et_al:LIPIcs.ICALP.2021.25,
author = {Ben Basat, Ran and Mitzenmacher, Michael and Vargaftik, Shay},
title = {{How to Send a Real Number Using a Single Bit (And Some Shared Randomness)}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {25:1--25:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.25},
URN = {urn:nbn:de:0030-drops-140942},
doi = {10.4230/LIPIcs.ICALP.2021.25},
annote = {Keywords: Randomized Algorithms, Approximation Algorithms, Shared Randomness, Distributed Protocols, Estimation, Subtractive Dithering}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Matthias Bentert, André Nichterlein, Malte Renken, and Philipp Zschoche
Abstract
Given an undirected n-vertex graph and k pairs of terminal vertices (s_1,t_1), …, (s_k,t_k), the k-Disjoint Shortest Paths (k-DSP) problem asks whether there are k pairwise vertex-disjoint paths P_1, …, P_k such that P_i is a shortest s_i-t_i-path for each i ∈ [k]. Recently, Lochet [SODA '21] provided an algorithm that solves k-DSP in n^{O(k^{5^k})} time, answering a 20-year old question about the computational complexity of k-DSP for constant k. On the one hand, we present an improved O(kn^{16k ⋅ k! + k + 1})-time algorithm based on a novel geometric view on this problem. For the special case k = 2, we show that the running time can be further reduced to O(nm) by small modifications of the algorithm and a further refined analysis. On the other hand, we show that k-DSP is W[1]-hard with respect to k, showing that the dependency of the degree of the polynomial running time on the parameter k is presumably unavoidable.
Cite as
Matthias Bentert, André Nichterlein, Malte Renken, and Philipp Zschoche. Using a Geometric Lens to Find k Disjoint Shortest Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bentert_et_al:LIPIcs.ICALP.2021.26,
author = {Bentert, Matthias and Nichterlein, Andr\'{e} and Renken, Malte and Zschoche, Philipp},
title = {{Using a Geometric Lens to Find k Disjoint Shortest Paths}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {26:1--26:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.26},
URN = {urn:nbn:de:0030-drops-140954},
doi = {10.4230/LIPIcs.ICALP.2021.26},
annote = {Keywords: graph algorithms, dynamic programming, W\lbrack1\rbrack-hardness, geometry}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Sayan Bhattacharya and Peter Kiss
Abstract
We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a (2-δ)-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a (1+δ)-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a (2+δ)-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, O(log⁴ n)) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19].
Previously, there were two known general-purpose rounding schemes for dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and Wajc [STOC'20], were randomized.
Our rounding scheme works by maintaining a good matching-sparsifier with bounded arboricity, and then applying the algorithm of Peleg and Solomon [SODA'16] to maintain a near-optimal matching in this low arboricity graph. To the best of our knowledge, this is the first dynamic matching algorithm that works on general graphs by using an algorithm for low-arboricity graphs as a black-box subroutine. This feature of our rounding scheme might be of independent interest.
Cite as
Sayan Bhattacharya and Peter Kiss. Deterministic Rounding of Dynamic Fractional Matchings. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bhattacharya_et_al:LIPIcs.ICALP.2021.27,
author = {Bhattacharya, Sayan and Kiss, Peter},
title = {{Deterministic Rounding of Dynamic Fractional Matchings}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {27:1--27:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.27},
URN = {urn:nbn:de:0030-drops-140960},
doi = {10.4230/LIPIcs.ICALP.2021.27},
annote = {Keywords: Matching, Dynamic Algorithms, Data Structures}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Marcin Bienkowski, Artur Kraska, and Hsiang-Hsuan Liu
Abstract
We present a unified framework for minimizing average completion time for many seemingly disparate online scheduling problems, such as the traveling repairperson problems (TRP), dial-a-ride problems (DARP), and scheduling on unrelated machines.
We construct a simple algorithm that handles all these scheduling problems, by computing and later executing auxiliary schedules, each optimizing a certain function on already seen prefix of the input. The optimized function resembles a prize-collecting variant of the original scheduling problem. By a careful analysis of the interplay between these auxiliary schedules, and later employing the resulting inequalities in a factor-revealing linear program, we obtain improved bounds on the competitive ratio for all these scheduling problems.
In particular, our techniques yield a 4-competitive deterministic algorithm for all previously studied variants of online TRP and DARP, and a 3-competitive one for the scheduling on unrelated machines (also with precedence constraints). This improves over currently best ratios for these problems that are 5.14 and 4, respectively. We also show how to use randomization to further reduce the competitive ratios to 1+2/ln 3 < 2.821 and 1+1/ln 2 < 2.443, respectively. The randomized bounds also substantially improve the current state of the art. Our upper bound for DARP contradicts the lower bound of 3 given by Fink et al. (Inf. Process. Lett. 2009); we pinpoint a flaw in their proof.
Cite as
Marcin Bienkowski, Artur Kraska, and Hsiang-Hsuan Liu. Traveling Repairperson, Unrelated Machines, and Other Stories About Average Completion Times. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bienkowski_et_al:LIPIcs.ICALP.2021.28,
author = {Bienkowski, Marcin and Kraska, Artur and Liu, Hsiang-Hsuan},
title = {{Traveling Repairperson, Unrelated Machines, and Other Stories About Average Completion Times}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {28:1--28:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.28},
URN = {urn:nbn:de:0030-drops-140977},
doi = {10.4230/LIPIcs.ICALP.2021.28},
annote = {Keywords: traveling repairperson problem, dial-a-ride, machine scheduling, unrelated machines, minimizing completion time, competitive analysis, factor-revealing LP}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Andreas Björklund and Petteri Kaski
Abstract
Given as input two n-element sets A, B ⊆ {0,1}^d with d = clog n ≤ (log n)²/(log log n)⁴ and a target t ∈ {0,1,…,d}, we show how to count the number of pairs (x,y) ∈ A× B with integer inner product ⟨ x,y ⟩ = t deterministically, in n²/2^{Ω(√{log nlog log n/(clog² c)})} time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to log² n dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm.
Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, or modular tomography, which can be seen as an additive analog of the Chinese Remainder Theorem.
As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.
Cite as
Andreas Björklund and Petteri Kaski. Counting Short Vector Pairs by Inner Product and Relations to the Permanent. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 29:1-29:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bjorklund_et_al:LIPIcs.ICALP.2021.29,
author = {Bj\"{o}rklund, Andreas and Kaski, Petteri},
title = {{Counting Short Vector Pairs by Inner Product and Relations to the Permanent}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {29:1--29:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.29},
URN = {urn:nbn:de:0030-drops-140988},
doi = {10.4230/LIPIcs.ICALP.2021.29},
annote = {Keywords: additive reconstruction, Chinese Remainder Theorem, counting, inner product, modular tomography, orthogonal vectors, permanent}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Guy Blanc, Jane Lange, and Li-Yang Tan
Abstract
We give a quasipolynomial-time algorithm for learning stochastic decision trees that is optimally resilient to adversarial noise. Given an η-corrupted set of uniform random samples labeled by a size-s stochastic decision tree, our algorithm runs in time n^{O(log(s/ε)/ε²)} and returns a hypothesis with error within an additive 2η + ε of the Bayes optimal. An additive 2η is the information-theoretic minimum.
Previously no non-trivial algorithm with a guarantee of O(η) + ε was known, even for weaker noise models. Our algorithm is furthermore proper, returning a hypothesis that is itself a decision tree; previously no such algorithm was known even in the noiseless setting.
Cite as
Guy Blanc, Jane Lange, and Li-Yang Tan. Learning Stochastic Decision Trees. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{blanc_et_al:LIPIcs.ICALP.2021.30,
author = {Blanc, Guy and Lange, Jane and Tan, Li-Yang},
title = {{Learning Stochastic Decision Trees}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {30:1--30:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.30},
URN = {urn:nbn:de:0030-drops-140994},
doi = {10.4230/LIPIcs.ICALP.2021.30},
annote = {Keywords: Learning theory, decision trees, proper learning algorithms, adversarial noise}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Joakim Blikstad
Abstract
We present algorithms that break the Õ(nr)-independence-query bound for the Matroid Intersection problem for the full range of r; where n is the size of the ground set and r ≤ n is the size of the largest common independent set. The Õ(nr) bound was due to the efficient implementations [CLSSW FOCS'19; Nguyên 2019] of the classic algorithm of Cunningham [SICOMP'86]. It was recently broken for large r (r = ω(√n)), first by the Õ(n^{1.5}/ε^{1.5})-query (1-ε)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the Õ(n^{6/5}r^{3/5})-query exact algorithm of BvdBMN [STOC'21]. No algorithm - even an approximation one - was known to break the Õ(nr) bound for the full range of r. We present an Õ(n√r/ε)-query (1-ε)-approximation algorithm and an Õ(nr^{3/4})-query exact algorithm. Our algorithms improve the Õ(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r.
Cite as
Joakim Blikstad. Breaking O(nr) for Matroid Intersection. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{blikstad:LIPIcs.ICALP.2021.31,
author = {Blikstad, Joakim},
title = {{Breaking O(nr) for Matroid Intersection}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {31:1--31:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.31},
URN = {urn:nbn:de:0030-drops-141004},
doi = {10.4230/LIPIcs.ICALP.2021.31},
annote = {Keywords: Matroid Intersection, Combinatorial Optimization, Approximation Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Jan Böker
Abstract
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization based on a natural system of linear equations whose integer solutions correspond to graph isomorphism. By results of Tinhofer (1986, 1991) and Dvořák (2010), two graphs G and H are fractionally isomorphic if and only if, for every tree T, the number of homomorphisms from T to G equals the corresponding number from T to H, which means that the tree distance of G and H is zero. Our main result is that this correspondence between the equivalence relations "fractional isomorphism" and "equal tree homomorphism densities" can be extended to a correspondence between the associated distance measures. Our result is inspired by a similar result due to Lovász and Szegedy (2006) and Borgs, Chayes, Lovász, Sós, and Vesztergombi (2008) that connects the cut distance of graphs to their homomorphism densities (over all graphs), which is a fundamental theorem in the theory of graph limits. We also introduce the path distance of graphs and take the corresponding result of Dell, Grohe, and Rattan (2018) for exact path homomorphism counts to an approximate level. Our results answer an open question of Grohe (2020) and help to build a theoretical understanding of vector embeddings of graphs.
The distance measures we define turn out be closely related to the cut distance. We establish our main results by generalizing our definitions to graphons, which are limit objects of sequences of graphs, as this allows us to apply techniques from functional analysis. We prove the fairly general statement that, for every "reasonably" defined graphon pseudometric, an exact correspondence to homomorphism densities can be turned into an approximate one. We also provide an example of a distance measure that violates this reasonableness condition. This incidentally answers an open question of Grebík and Rocha (2021).
Cite as
Jan Böker. Graph Similarity and Homomorphism Densities. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{boker:LIPIcs.ICALP.2021.32,
author = {B\"{o}ker, Jan},
title = {{Graph Similarity and Homomorphism Densities}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {32:1--32:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.32},
URN = {urn:nbn:de:0030-drops-141014},
doi = {10.4230/LIPIcs.ICALP.2021.32},
annote = {Keywords: graph similarity, homomorphism densities, cut distance}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Andrej Bogdanov and Gautam Prakriya
Abstract
A function f(x₁, … , x_n) from a product domain 𝒟₁ × ⋯ × 𝒟_n to an abelian group 𝒢 is a direct sum if it is of the form f₁(x₁) + ⋯ + f_n(x_n). We present a new 4-query direct sum test with optimal (up to constant factors) soundness error. This generalizes a result of Dinur and Golubev (RANDOM 2019) which is tailored to the target group 𝒢 = ℤ₂. As a special case, we obtain an optimal affinity test for 𝒢-valued functions on domain {0, 1}ⁿ under product measure. Our analysis relies on the hypercontractivity of the binary erasure channel.
We also study the testability of function partitionability over product domains into disjoint components. A 𝒢-valued f(x₁, … , x_n) is k-direct sum partitionable if it can be written as a sum of functions over k nonempty disjoint sets of inputs. A function f(x₁, … , x_n) with unstructured product range ℛ^k is direct product partitionable if its outputs depend on disjoint sets of inputs.
We show that direct sum partitionability and direct product partitionability are one-sided error testable with O((n - k)(log n + 1/ε) + 1/ε) adaptive queries and O((n/ε) log²(n/ε)) nonadaptive queries, respectively. Both bounds are tight up to the logarithmic factors for constant ε even with respect to adaptive, two-sided error testers. We also give a non-adaptive one-sided error tester for direct sum partitionability with query complexity O(kn² (log n)² / ε).
Cite as
Andrej Bogdanov and Gautam Prakriya. Direct Sum and Partitionability Testing over General Groups. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bogdanov_et_al:LIPIcs.ICALP.2021.33,
author = {Bogdanov, Andrej and Prakriya, Gautam},
title = {{Direct Sum and Partitionability Testing over General Groups}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {33:1--33:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.33},
URN = {urn:nbn:de:0030-drops-141028},
doi = {10.4230/LIPIcs.ICALP.2021.33},
annote = {Keywords: Direct Sum Test, Function Partitionability, Hypercontractive Inequality, Property Testing}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Édouard Bonnet
Abstract
We show, assuming the Strong Exponential Time Hypothesis, that for every ε > 0, approximating undirected unweighted Diameter on n-vertex m-edge graphs within ratio 7/4 - ε requires m^{4/3 - o(1)} time, even when m = Õ(n). This is the first result that conditionally rules out a near-linear time 5/3-approximation for undirected Diameter.
Cite as
Édouard Bonnet. 4 vs 7 Sparse Undirected Unweighted Diameter is SETH-Hard at Time n^{4/3}. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bonnet:LIPIcs.ICALP.2021.34,
author = {Bonnet, \'{E}douard},
title = {{4 vs 7 Sparse Undirected Unweighted Diameter is SETH-Hard at Time n^\{4/3\}}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {34:1--34:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.34},
URN = {urn:nbn:de:0030-drops-141034},
doi = {10.4230/LIPIcs.ICALP.2021.34},
annote = {Keywords: Diameter, inapproximability, SETH lower bounds, k-Orthogonal Vectors}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant
Abstract
We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time f(d,k)n for n-vertex graphs given with a witness that the twin-width is at most d, called d-contraction sequence or d-sequence, and formulas of size k [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that f is a tower of exponentials of height roughly k. In this paper, we show that algorithms based on twin-width need not be impractical. We present 2^{O(k)}n-time algorithms for k-Independent Set, r-Scattered Set, k-Clique, and k-Dominating Set when an O(1)-sequence of the graph is given in input. We further show how to solve the weighted version of k-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in the slightly worse running time 2^{O(k log k)}n. Up to logarithmic factors in the exponent, all these running times are optimal, unless the Exponential Time Hypothesis fails. Like our FO model checking algorithm, these new algorithms are based on a dynamic programming scheme following the sequence of contractions forward.
We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of such a reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, thereby establishing that bounded twin-width classes are χ-bounded. This significantly extends the χ-boundedness of bounded rank-width classes, and does so with a very concise proof. It readily yields a constant approximation for Max Independent Set on K_t-free graphs of bounded twin-width, and a 2^{O(OPT)}-approximation for Min Coloring on bounded twin-width graphs. We further observe that a constant approximation for Max Independent Set on bounded twin-width graphs (but arbitrarily large clique number) would actually imply a PTAS.
The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. This property is trivially shared with graphs of bounded average degree. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in time O(n log n) and time O(n² log n), respectively. In sharp contrast, even Diameter does not admit a truly subquadratic algorithm on bounded twin-width graphs, unless the Strong Exponential Time Hypothesis fails.
The fourth algorithmic use of twin-width builds on the so-called versatile tree of contractions [Bonnet et al., SODA '21], a branching and more robust witness of low twin-width. We present constant-approximation algorithms for Min Dominating Set and related problems, on bounded twin-width graphs, by showing that the integrality gap is constant. This is done by going down the versatile tree and stopping accordingly to a problem-dependent criterion. At the reached node, a greedy approach yields the desired approximation.
Cite as
Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: Max Independent Set, Min Dominating Set, and Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bonnet_et_al:LIPIcs.ICALP.2021.35,
author = {Bonnet, \'{E}douard and Geniet, Colin and Kim, Eun Jung and Thomass\'{e}, St\'{e}phan and Watrigant, R\'{e}mi},
title = {{Twin-width III: Max Independent Set, Min Dominating Set, and Coloring}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {35:1--35:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.35},
URN = {urn:nbn:de:0030-drops-141044},
doi = {10.4230/LIPIcs.ICALP.2021.35},
annote = {Keywords: Twin-width, Max Independent Set, Min Dominating Set, Coloring, Parameterized Algorithms, Approximation Algorithms, Exact Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Sébastien Bouchard, Yoann Dieudonné, Arnaud Labourel, and Andrzej Pelc
Abstract
A mobile agent navigating along edges of a simple connected graph, either finite or countably infinite, has to find an inert target (treasure) hidden in one of the nodes. This task is known as treasure hunt. The agent has no a priori knowledge of the graph, of the location of the treasure or of the initial distance to it. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until finding the treasure. Awerbuch, Betke, Rivest and Singh [Baruch Awerbuch et al., 1999] considered graph exploration and treasure hunt for finite graphs in a restricted model where the agent has a fuel tank that can be replenished only at the starting node s. The size of the tank is B = 2(1+α)r, for some positive real constant α, where r, called the radius of the graph, is the maximum distance from s to any other node. The tank of size B allows the agent to make at most {⌊ B⌋} edge traversals between two consecutive visits at node s.
Let e(d) be the number of edges whose at least one extremity is at distance less than d from s. Awerbuch, Betke, Rivest and Singh [Baruch Awerbuch et al., 1999] conjectured that it is impossible to find a treasure hidden in a node at distance at most d at cost nearly linear in e(d). We first design a deterministic treasure hunt algorithm working in the model without any restrictions on the moves of the agent at cost 𝒪(e(d) log d), and then show how to modify this algorithm to work in the model from [Baruch Awerbuch et al., 1999] with the same complexity. Thus we refute the above twenty-year-old conjecture. We observe that no treasure hunt algorithm can beat cost Θ(e(d)) for all graphs and thus our algorithms are also almost optimal.
Cite as
Sébastien Bouchard, Yoann Dieudonné, Arnaud Labourel, and Andrzej Pelc. Almost-Optimal Deterministic Treasure Hunt in Arbitrary Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bouchard_et_al:LIPIcs.ICALP.2021.36,
author = {Bouchard, S\'{e}bastien and Dieudonn\'{e}, Yoann and Labourel, Arnaud and Pelc, Andrzej},
title = {{Almost-Optimal Deterministic Treasure Hunt in Arbitrary Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {36:1--36:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.36},
URN = {urn:nbn:de:0030-drops-141051},
doi = {10.4230/LIPIcs.ICALP.2021.36},
annote = {Keywords: treasure hunt, graph, mobile agent}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep
Abstract
Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Håstad [Per Austrin et al., 2017], there has been a flurry of works on PCSPs, including recent breakthroughs in approximate graph coloring [Barto et al., 2018; Andrei A. Krokhin and Jakub Opršal, 2019; Marcin Wrochna and Stanislav Zivný, 2020]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as polymorphisms are analyzed.
The polymorphisms of PCSPs are significantly richer than CSPs - even in the Boolean case, we still do not know if there exists a dichotomy result for PCSPs analogous to Schaefer’s dichotomy result [Thomas J. Schaefer, 1978] for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate x ≤ y. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [Mark Braverman et al., 2021] which is a perfect completeness surrogate of the Unique Games Conjecture.
In particular, assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every ε > 0, it has polymorphisms where each coordinate has Shapley value at most ε, else it is NP-hard. The algorithmic part of our dichotomy result is based on a structural lemma showing that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. As a structural result of independent interest, we construct an example to show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.
Cite as
Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. Conditional Dichotomy of Boolean Ordered Promise CSPs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2021.37,
author = {Brakensiek, Joshua and Guruswami, Venkatesan and Sandeep, Sai},
title = {{Conditional Dichotomy of Boolean Ordered Promise CSPs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {37:1--37:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.37},
URN = {urn:nbn:de:0030-drops-141060},
doi = {10.4230/LIPIcs.ICALP.2021.37},
annote = {Keywords: promise constraint satisfaction, Boolean ordered PCSP, Shapley value, rich 2-to-1 conjecture, random minor}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Cornelius Brand and Kevin Pratt
Abstract
We study the following problem and its applications: given a homogeneous degree-d polynomial g as an arithmetic circuit C, and a d × d matrix X whose entries are homogeneous linear polynomials, compute g(∂/∂ x₁, …, ∂/∂ x_n) det X. We show that this quantity can be computed using 2^{ω d}|C|poly(n,d) arithmetic operations, where ω is the exponent of matrix multiplication. In the case that C is skew, we improve this to 4^d|C| poly(n,d) operations, and if furthermore X is a Hankel matrix, to φ^{2d}|C| poly(n,d) operations, where φ = (1+√5)/2 is the golden ratio.
Using these observations we give faster parameterized algorithms for the matroid k-parity and k-matroid intersection problems for linear matroids, and faster deterministic algorithms for several problems, including the first deterministic polynomial time algorithm for testing if a linear space of matrices of logarithmic dimension contains an invertible matrix. We also match the runtime of the fastest deterministic algorithm for detecting subgraphs of bounded pathwidth with a new and simple approach. Our approach generalizes several previous methods in parameterized algorithms and can be seen as a relaxation of Waring rank based methods [Pratt, FOCS19].
Cite as
Cornelius Brand and Kevin Pratt. Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{brand_et_al:LIPIcs.ICALP.2021.38,
author = {Brand, Cornelius and Pratt, Kevin},
title = {{Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {38:1--38:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.38},
URN = {urn:nbn:de:0030-drops-141079},
doi = {10.4230/LIPIcs.ICALP.2021.38},
annote = {Keywords: Parameterized Algorithms, Algebraic Algorithms, Longest Cycle, Matroid Parity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Karl Bringmann and Debarati Das
Abstract
We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. While a simple quadratic algorithm has been known for the problem for more than 40 years, no faster algorithm has been found despite an extensive effort. The lack of progress on the problem has recently been explained by Abboud, Backurs, and Vassilevska Williams [FOCS'15] and Bringmann and Künnemann [FOCS'15] who proved that there is no subquadratic algorithm unless the Strong Exponential Time Hypothesis fails. This major roadblock for getting faster exact algorithms has led the community to look for subquadratic approximation algorithms for the problem.
Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting (where we make no assumption on the length of the optimum solution or the alphabet size), only a naive O(n^{ε/2})-approximation algorithm with running time Õ(n^{2-ε}) has been known, for any constant 0 < ε ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA'19] provided a linear-time algorithm that yields a O(n^{0.497956})-approximation in expectation; improving upon the naive O(√n)-approximation for the first time.
In this paper, we provide an algorithm that in time O(n^{2-ε}) computes an Õ(n^{2ε/5})-approximation with high probability, for any 0 < ε ≤ 1. Our result (1) gives an Õ(n^{0.4})-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n^{2-ε}), improving upon the naive bound of O(n^{ε/2}) for any ε, and (3) instead of only in expectation, succeeds with high probability.
Cite as
Karl Bringmann and Debarati Das. A Linear-Time n^{0.4}-Approximation for Longest Common Subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bringmann_et_al:LIPIcs.ICALP.2021.39,
author = {Bringmann, Karl and Das, Debarati},
title = {{A Linear-Time n^\{0.4\}-Approximation for Longest Common Subsequence}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {39:1--39:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.39},
URN = {urn:nbn:de:0030-drops-141082},
doi = {10.4230/LIPIcs.ICALP.2021.39},
annote = {Keywords: approximation algorithm, longest common subsequence, string algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Karl Bringmann and Jasper Slusallek
Abstract
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time O(n^{tw(H)+1}) [Alon, Yuster, Zwick'95], where n is the number of vertices of the host graph G. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of O(n^{tw(H)+1-ε}) or even faster (e.g. for k-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time n^{o(tw(H) / log(tw(H)))} for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx'07].
In this paper, we demonstrate the existence of maximally hard pattern graphs H that require time n^{tw(H)+1-o(1)}. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth t:
For any ε > 0 there exists t ≥ 3 and a pattern graph H of treewidth t such that Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-ε}).
Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth t ≥ 3:
For any t ≥ 3 there exists a pattern graph H of treewidth t such that for any ε > 0 Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-ε}).
In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for tw < 3, (3) Subgraph Isomorphism parameterized by pathwidth instead of treewidth, and (4) a weighted variant that we call Exact Weight Subgraph Isomorphism, for which we examine pseudo-polynomial time algorithms. For many of these settings we obtain similarly tight upper and lower bounds.
Cite as
Karl Bringmann and Jasper Slusallek. Current Algorithms for Detecting Subgraphs of Bounded Treewidth Are Probably Optimal. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bringmann_et_al:LIPIcs.ICALP.2021.40,
author = {Bringmann, Karl and Slusallek, Jasper},
title = {{Current Algorithms for Detecting Subgraphs of Bounded Treewidth Are Probably Optimal}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {40:1--40:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.40},
URN = {urn:nbn:de:0030-drops-141095},
doi = {10.4230/LIPIcs.ICALP.2021.40},
annote = {Keywords: subgraph isomorphism, treewidth, fine-grained complexity, hyperclique}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Karl Bringmann and Vasileios Nakos
Abstract
We consider the problem of computing the Boolean convolution (with wraparound) of n vectors of dimension m, or, equivalently, the problem of computing the sumset A₁+A₂+…+A_n for A₁,…,A_n ⊆ ℤ_m. Boolean convolution formalizes the frequent task of combining two subproblems, where the whole problem has a solution of size k if for some i the first subproblem has a solution of size i and the second subproblem has a solution of size k-i. Our problem formalizes a natural generalization, namely combining solutions of n subproblems subject to a modular constraint. This simultaneously generalises Modular Subset Sum and Boolean Convolution (Sumset Computation). Although nearly optimal algorithms are known for special cases of this problem, not even tiny improvements are known for the general case.
We almost resolve the computational complexity of this problem, shaving essentially a factor of n from the running time of previous algorithms. Specifically, we present a deterministic algorithm running in almost linear time with respect to the input plus output size k. We also present a Las Vegas algorithm running in nearly linear expected time with respect to the input plus output size k. Previously, no deterministic or randomized o(nk) algorithm was known.
At the heart of our approach lies a careful usage of Kneser’s theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution. In total, our work builds a solid toolbox that could be of independent interest.
Cite as
Karl Bringmann and Vasileios Nakos. Fast n-Fold Boolean Convolution via Additive Combinatorics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bringmann_et_al:LIPIcs.ICALP.2021.41,
author = {Bringmann, Karl and Nakos, Vasileios},
title = {{Fast n-Fold Boolean Convolution via Additive Combinatorics}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {41:1--41:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.41},
URN = {urn:nbn:de:0030-drops-141108},
doi = {10.4230/LIPIcs.ICALP.2021.41},
annote = {Keywords: convolution, sumset computation, modular subset sum, output sensitive}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Moritz Buchem, Lars Rohwedder, Tjark Vredeveld, and Andreas Wiese
Abstract
We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ε h for some suitable parameter h and any given ε > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs.
Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ε⋅ p_{max}.
Cite as
Moritz Buchem, Lars Rohwedder, Tjark Vredeveld, and Andreas Wiese. Additive Approximation Schemes for Load Balancing Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{buchem_et_al:LIPIcs.ICALP.2021.42,
author = {Buchem, Moritz and Rohwedder, Lars and Vredeveld, Tjark and Wiese, Andreas},
title = {{Additive Approximation Schemes for Load Balancing Problems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {42:1--42:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.42},
URN = {urn:nbn:de:0030-drops-141116},
doi = {10.4230/LIPIcs.ICALP.2021.42},
annote = {Keywords: Load balancing, Approximation schemes, Parallel machine scheduling}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Massimo Cairo, Romeo Rizzi, Alexandru I. Tomescu, and Elia C. Zirondelli
Abstract
Genome assembly asks to reconstruct an unknown string from many shorter substrings of it. Even though it is one of the key problems in Bioinformatics, it is generally lacking major theoretical advances. Its hardness stems both from practical issues (size and errors of real data), and from the fact that problem formulations inherently admit multiple solutions. Given these, at their core, most state-of-the-art assemblers are based on finding non-branching paths (unitigs) in an assembly graph. While such paths constitute only partial assemblies, they are likely to be correct. More precisely, if one defines a genome assembly solution as a closed arc-covering walk of the graph, then unitigs appear in all solutions, being thus safe partial solutions. Until recently, it was open what are all the safe walks of an assembly graph. Tomescu and Medvedev (RECOMB 2016) characterized all such safe walks (omnitigs), thus giving the first safe and complete genome assembly algorithm. Even though omnitig finding was later improved to quadratic time, it remained open whether the crucial linear-time feature of finding unitigs can be attained with omnitigs.
We answer this question affirmatively, by describing a surprising O(m)-time algorithm to identify all maximal omnitigs of a graph with n nodes and m arcs, notwithstanding the existence of families of graphs with Θ(mn) total maximal omnitig size. This is based on the discovery of a family of walks (macrotigs) with the property that all the non-trivial omnitigs are univocal extensions of subwalks of a macrotig. This has two consequences: (1) A linear-time output-sensitive algorithm enumerating all maximal omnitigs. (2) A compact O(m) representation of all maximal omnitigs, which allows, e.g., for O(m)-time computation of various statistics on them. Our results close a long-standing theoretical question inspired by practical genome assemblers, originating with the use of unitigs in 1995. We envision our results to be at the core of a reverse transfer from theory to practical and complete genome assembly programs, as has been the case for other key Bioinformatics problems.
Cite as
Massimo Cairo, Romeo Rizzi, Alexandru I. Tomescu, and Elia C. Zirondelli. Genome Assembly, from Practice to Theory: Safe, Complete and Linear-Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cairo_et_al:LIPIcs.ICALP.2021.43,
author = {Cairo, Massimo and Rizzi, Romeo and Tomescu, Alexandru I. and Zirondelli, Elia C.},
title = {{Genome Assembly, from Practice to Theory: Safe, Complete and Linear-Time}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {43:1--43:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.43},
URN = {urn:nbn:de:0030-drops-141122},
doi = {10.4230/LIPIcs.ICALP.2021.43},
annote = {Keywords: Graph algorithm, strong connectivity, reachability under failures}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova
Abstract
Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas.
We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest.
We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.
Cite as
Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Lifting for Constant-Depth Circuits and Applications to MCSP. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{carmosino_et_al:LIPIcs.ICALP.2021.44,
author = {Carmosino, Marco and Hoover, Kenneth and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina},
title = {{Lifting for Constant-Depth Circuits and Applications to MCSP}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {44:1--44:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.44},
URN = {urn:nbn:de:0030-drops-141135},
doi = {10.4230/LIPIcs.ICALP.2021.44},
annote = {Keywords: circuit complexity, constant-depth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ruoxu Cen, Yu Cheng, Debmalya Panigrahi, and Kevin Sun
Abstract
In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczúr and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).
Cite as
Ruoxu Cen, Yu Cheng, Debmalya Panigrahi, and Kevin Sun. Sparsification of Directed Graphs via Cut Balance. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 45:1-45:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cen_et_al:LIPIcs.ICALP.2021.45,
author = {Cen, Ruoxu and Cheng, Yu and Panigrahi, Debmalya and Sun, Kevin},
title = {{Sparsification of Directed Graphs via Cut Balance}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {45:1--45:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.45},
URN = {urn:nbn:de:0030-drops-141143},
doi = {10.4230/LIPIcs.ICALP.2021.45},
annote = {Keywords: Graph sparsification, directed graphs, cut sketches, space complexity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan
Abstract
In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ⊆ V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-ε) against an adaptive adversary and of α_{GW}≈ 0.8786 against an oblivious adversary (here α_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of α_{GW} against both types of adversaries, rendering our results (virtually) tight.
The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.
Cite as
Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan. Fault Tolerant Max-Cut. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2021.46,
author = {Censor-Hillel, Keren and Marelly, Noa and Schwartz, Roy and Tonoyan, Tigran},
title = {{Fault Tolerant Max-Cut}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {46:1--46:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.46},
URN = {urn:nbn:de:0030-drops-141158},
doi = {10.4230/LIPIcs.ICALP.2021.46},
annote = {Keywords: fault-tolerance, max-cut, approximation}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Timothy M. Chan, Virginia Vassilevska Williams, and Yinzhan Xu
Abstract
All-Pairs Shortest Paths (APSP) is one of the most well studied problems in graph algorithms. This paper studies several variants of APSP in unweighted graphs or graphs with small integer weights.
APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an Õ(n^ω) time algorithm, where ω < 2.373 is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be Ω(n^{2.5}) even if ω = 2 [Zwick'02]. To understand this n^{2.5} bottleneck, we build a web of reductions around directed unweighted APSP . We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on the value of ω. As a consequence, we establish an equivalence between APSP in directed unweighted graphs, APSP in directed graphs with small (Õ(1)) integer weights, All-Pairs Longest Paths in DAGs with small weights, cRed-APSP in undirected graphs with small weights, for any c ≥ 2 (computing all-pairs shortest path distances among paths that use at most c red edges), #_{≤ c}APSP in directed graphs with small weights (counting the number of shortest paths for each vertex pair, up to c), and approximate APSP with additive error c in directed graphs with small weights, for c ≤ Õ(1).
We also provide fine-grained reductions from directed unweighted APSP to All-Pairs Shortest Lightest Paths (APSLP) in undirected graphs with {0,1} weights and #_{mod c}APSP in directed unweighted graphs (computing counts mod c), thus showing that unless the current algorithms for APSP in directed unweighted graphs can be improved substantially, these problems need at least Ω(n^{2.528}) time.
We complement our hardness results with new algorithms. We improve the known algorithms for APSLP in directed graphs with small integer weights (previously studied by Zwick [STOC'99]) and for approximate APSP with sublinear additive error in directed unweighted graphs (previously studied by Roditty and Shapira [ICALP'08]). Our algorithm for approximate APSP with sublinear additive error is optimal, when viewed as a reduction to Min-Plus product. We also give new algorithms for variants of #APSP (such as #_{≤ U}APSP and #_{mod U}APSP for U ≤ n^{Õ(1)}) in unweighted graphs, as well as a near-optimal Õ(n³)-time algorithm for the original #APSP problem in unweighted graphs (when counts may be exponentially large). This also implies an Õ(n³)-time algorithm for Betweenness Centrality, improving on the previous Õ(n⁴) running time for the problem. Our techniques also lead to a simpler alternative to Shoshan and Zwick’s algorithm [FOCS'99] for the original APSP problem in undirected graphs with small integer weights.
Cite as
Timothy M. Chan, Virginia Vassilevska Williams, and Yinzhan Xu. Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 47:1-47:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chan_et_al:LIPIcs.ICALP.2021.47,
author = {Chan, Timothy M. and Vassilevska Williams, Virginia and Xu, Yinzhan},
title = {{Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {47:1--47:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.47},
URN = {urn:nbn:de:0030-drops-141166},
doi = {10.4230/LIPIcs.ICALP.2021.47},
annote = {Keywords: All-Pairs Shortest Paths, Fine-Grained Complexity, Graph Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann
Abstract
We consider the problem of preprocessing two strings S and T, of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i,j in S and positions a,b in T, return the optimal alignment score of S[i..j] and T[a..b]. Let N = mn. We present an oracle with preprocessing time N^{1+o(1)} and space N^{1+o(1)} that answers queries in log^{2+o(1)}N time. In other words, we show that we can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS).
The best previously known oracle with construction time and size 𝒪(N) has slow Ω(√N) query time [Sakai, TCS 2019], and the one with size N^{1+o(1)} and query time log^{2+o(1)}N (using a planar graph distance oracle) has slow Ω(N^{3/2}) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.
Cite as
Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann. An Almost Optimal Edit Distance Oracle. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{charalampopoulos_et_al:LIPIcs.ICALP.2021.48,
author = {Charalampopoulos, Panagiotis and Gawrychowski, Pawe{\l} and Mozes, Shay and Weimann, Oren},
title = {{An Almost Optimal Edit Distance Oracle}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {48:1--48:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.48},
URN = {urn:nbn:de:0030-drops-141175},
doi = {10.4230/LIPIcs.ICALP.2021.48},
annote = {Keywords: longest common subsequence, edit distance, planar graphs, Voronoi diagrams}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Chandra Chekuri and Kent Quanrud
Abstract
We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time Õ(n²). For rooted edge connectivity this is the first algorithm to improve on the Ω(n³) time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems.
We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a (1+ε)-approximation for rooted vertex connectivity in Õ(nW/ε) time where W is the total vertex weight (assuming integral vertex weights); in particular this yields an Õ(n²/ε) time randomized algorithm for unweighted graphs. This translates to a Õ(KnW) time exact algorithm where K is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity.
Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [Harold N. Gabow, 1995] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [Nanongkai et al., 2019] and Forster et al. [Sebastian Forster et al., 2020] for vertex connectivity.
Cite as
Chandra Chekuri and Kent Quanrud. Faster Algorithms for Rooted Connectivity in Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2021.49,
author = {Chekuri, Chandra and Quanrud, Kent},
title = {{Faster Algorithms for Rooted Connectivity in Directed Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {49:1--49:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.49},
URN = {urn:nbn:de:0030-drops-141187},
doi = {10.4230/LIPIcs.ICALP.2021.49},
annote = {Keywords: rooted connectivity, directed graph, fast algorithm, sparsification}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Chandra Chekuri and Kent Quanrud
Abstract
Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations.
In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.
Cite as
Chandra Chekuri and Kent Quanrud. Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2021.50,
author = {Chekuri, Chandra and Quanrud, Kent},
title = {{Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {50:1--50:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.50},
URN = {urn:nbn:de:0030-drops-141199},
doi = {10.4230/LIPIcs.ICALP.2021.50},
annote = {Keywords: cuts, vertex connectivity, hypergraphs, fast algorithms, submodularity, bisumodularity, lattices, isolating cuts, element connectivity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Lijie Chen, Zhenjian Lu, Xin Lyu, and Igor C. Oliveira
Abstract
We develop a general framework that characterizes strong average-case lower bounds against circuit classes 𝒞 contained in NC¹, such as AC⁰[⊕] and ACC⁰. We apply this framework to show:
- Generic seed reduction: Pseudorandom generators (PRGs) against 𝒞 of seed length ≤ n -1 and error ε(n) = n^{-ω(1)} can be converted into PRGs of sub-polynomial seed length.
- Hardness under natural distributions: If 𝖤 (deterministic exponential time) is average-case hard against 𝒞 under some distribution, then 𝖤 is average-case hard against 𝒞 under the uniform distribution.
- Equivalence between worst-case and average-case hardness: Worst-case lower bounds against MAJ∘𝒞 for problems in 𝖤 are equivalent to strong average-case lower bounds against 𝒞. This can be seen as a certain converse to the Discriminator Lemma [Hajnal et al., JCSS'93].
These results were not known to hold for circuit classes that do not compute majority. Additionally, we prove that classical and recent approaches to worst-case lower bounds against ACC⁰ via communication lower bounds for NOF multi-party protocols [Håstad and Goldmann, CC'91; Razborov and Wigderson, IPL'93] and Torus polynomials degree lower bounds [Bhrushundi et al., ITCS'19] also imply strong average-case hardness against ACC⁰ under the uniform distribution.
Crucial to these results is the use of non-black-box hardness amplification techniques and the interplay between Majority (MAJ) and Approximate Linear Sum (SUM̃) gates. Roughly speaking, while a MAJ gate outputs 1 when the sum of the m input bits is at least m/2, a SUM̃ gate computes a real-valued bounded weighted sum of the input bits and outputs 1 (resp. 0) if the sum is close to 1 (resp. close to 0), with the promise that one of the two cases always holds. As part of our framework, we explore ideas introduced in [Chen and Ren, STOC'20] to show that, for the purpose of proving lower bounds, a top layer MAJ gate is equivalent to a (weaker) SUM̃ gate. Motivated by this result, we extend the algorithmic method and establish stronger lower bounds against bounded-depth circuits with layers of MAJ and SUM̃ gates. Among them, we prove that:
- Lower bound: NQP does not admit fixed quasi-polynomial size MAJ∘SUM̃∘ACC⁰∘THR circuits.
This is the first explicit lower bound against circuits with distinct layers of MAJ, SUM̃, and THR gates. Consequently, if the aforementioned equivalence between MAJ and SUM̃ as a top gate can be extended to intermediate layers, long sought-after lower bounds against the class THR∘THR of depth-2 polynomial-size threshold circuits would follow.
Cite as
Lijie Chen, Zhenjian Lu, Xin Lyu, and Igor C. Oliveira. Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC¹. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 51:1-51:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2021.51,
author = {Chen, Lijie and Lu, Zhenjian and Lyu, Xin and Oliveira, Igor C.},
title = {{Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC¹}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {51:1--51:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.51},
URN = {urn:nbn:de:0030-drops-141202},
doi = {10.4230/LIPIcs.ICALP.2021.51},
annote = {Keywords: circuit complexity, average-case hardness, complexity lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu
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.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
Track A: Algorithms, Complexity and Games
Authors:
Yu Chen, Sanjeev Khanna, and Ansh Nagda
Abstract
The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph.
Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries:
1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and
2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges.
We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient.
Cite as
Yu Chen, Sanjeev Khanna, and Ansh Nagda. Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2021.53,
author = {Chen, Yu and Khanna, Sanjeev and Nagda, Ansh},
title = {{Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {53:1--53:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.53},
URN = {urn:nbn:de:0030-drops-141227},
doi = {10.4230/LIPIcs.ICALP.2021.53},
annote = {Keywords: hypergraphs, graph sparsification, cut queries}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng
Abstract
The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we first consider these problems in the asymmetric streaming model introduced by Andoni, Krauthgamer and Onak [Andoni et al., 2010] (FOCS'10) and Saks and Seshadhri [Saks and Seshadhri, 2013] (SODA'13). In this model we have random access to one string and streaming access the other one. Our main contribution is a constant factor approximation algorithm for ED with memory Õ(n^δ) for any constant δ > 0. In addition to this, we present an upper bound of Õ _ε(√n) on the memory needed to approximate ED or LCS within a factor 1±ε. All our algorithms are deterministic and run in polynomial time in a single pass.
We further study small-space approximation algorithms for ED, LCS, and longest increasing sequence (LIS) in the non-streaming setting. Here, we design algorithms that achieve 1 ± ε approximation for all three problems, where ε > 0 can be any constant and even slightly sub-constant. Our algorithms only use poly-logarithmic space while maintaining a polynomial running time. This significantly improves previous results in terms of space complexity, where all known results need to use space at least Ω(√n). Our algorithms make novel use of triangle inequality and carefully designed recursions to save space, which can be of independent interest.
Cite as
Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng. Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cheng_et_al:LIPIcs.ICALP.2021.54,
author = {Cheng, Kuan and Farhadi, Alireza and Hajiaghayi, MohammadTaghi and Jin, Zhengzhong and Li, Xin and Rubinstein, Aviad and Seddighin, Saeed and Zheng, Yu},
title = {{Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {54:1--54:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.54},
URN = {urn:nbn:de:0030-drops-141236},
doi = {10.4230/LIPIcs.ICALP.2021.54},
annote = {Keywords: Edit Distance, Longest Common Subsequence, Longest Increasing Subsequence, Space Efficient Algorithm, Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Andrew M. Childs, Shih-Han Hung, and Tongyang Li
Abstract
We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.
Cite as
Andrew M. Childs, Shih-Han Hung, and Tongyang Li. Quantum Query Complexity with Matrix-Vector Products. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{childs_et_al:LIPIcs.ICALP.2021.55,
author = {Childs, Andrew M. and Hung, Shih-Han and Li, Tongyang},
title = {{Quantum Query Complexity with Matrix-Vector Products}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {55:1--55:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.55},
URN = {urn:nbn:de:0030-drops-141242},
doi = {10.4230/LIPIcs.ICALP.2021.55},
annote = {Keywords: Quantum algorithms, quantum query complexity, matrix-vector products}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
George Christodoulou, Elias Koutsoupias, and Annamária Kovács
Abstract
We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.
Cite as
George Christodoulou, Elias Koutsoupias, and Annamária Kovács. Truthful Allocation in Graphs and Hypergraphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{christodoulou_et_al:LIPIcs.ICALP.2021.56,
author = {Christodoulou, George and Koutsoupias, Elias and Kov\'{a}cs, Annam\'{a}ria},
title = {{Truthful Allocation in Graphs and Hypergraphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {56:1--56:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.56},
URN = {urn:nbn:de:0030-drops-141256},
doi = {10.4230/LIPIcs.ICALP.2021.56},
annote = {Keywords: Algorithmic Game Theory, Scheduling Unrelated Machines, Mechanism Design}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Christian Coester and Elias Koutsoupias
Abstract
The k-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a k-competitive deterministic algorithm for the k-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: k = 2, (k+1)-point metrics, (k+2)-point metrics, the line metric, weighted star metrics, and k = 3 in the Manhattan plane.
The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving k-competitiveness of WFA for all these cases. We also use this potential to show k-competitiveness of WFA on multiray spaces and for k = 3 on trees. While the DoubleCoverage algorithm was known to be k-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the k-server conjecture.
To our major surprise, this is not the case, as we show (using connections to the k-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the k-server conjecture. On the other hand, if the k-server conjecture holds, then we believe it can be proved by a variant of our potential.
Cite as
Christian Coester and Elias Koutsoupias. Towards the k-Server Conjecture: A Unifying Potential, Pushing the Frontier to the Circle. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 57:1-57:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{coester_et_al:LIPIcs.ICALP.2021.57,
author = {Coester, Christian and Koutsoupias, Elias},
title = {{Towards the k-Server Conjecture: A Unifying Potential, Pushing the Frontier to the Circle}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {57:1--57:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.57},
URN = {urn:nbn:de:0030-drops-141263},
doi = {10.4230/LIPIcs.ICALP.2021.57},
annote = {Keywords: Online algorithms, competitive analysis, k-server, work function algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Artur Czumaj, George Kontogeorgiou, and Mike Paterson
Abstract
We want to efficiently find a specific object in a large unstructured set, which we model by a random n-permutation, and we have to do it by revealing just a single element. Clearly, without any help this task is hopeless and the best one can do is to select the element at random, and achieve the success probability 1/n. Can we do better with some small amount of advice about the permutation, even without knowing the object sought? We show that by providing advice of just one integer in {0,1,… ,n-1}, one can improve the success probability considerably, by a Θ((log n)/(log log n)) factor.
We study this and related problems, and show asymptotically matching upper and lower bounds for their optimal probability of success. Our analysis relies on a close relationship of such problems to some intrinsic properties of random permutations related to the rencontres number.
Cite as
Artur Czumaj, George Kontogeorgiou, and Mike Paterson. Haystack Hunting Hints and Locker Room Communication. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{czumaj_et_al:LIPIcs.ICALP.2021.58,
author = {Czumaj, Artur and Kontogeorgiou, George and Paterson, Mike},
title = {{Haystack Hunting Hints and Locker Room Communication}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {58:1--58:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.58},
URN = {urn:nbn:de:0030-drops-141270},
doi = {10.4230/LIPIcs.ICALP.2021.58},
annote = {Keywords: Random permutations, Search, Communication complexity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Gianlorenzo D'Angelo, Debashmita Poddar, and Cosimo Vinci
Abstract
In the adaptive influence maximization problem, we are given a social network and a budget k, and we iteratively select k nodes, called seeds, in order to maximize the expected number of nodes that are reached by an influence cascade that they generate according to a stochastic model for influence diffusion. The decision on the next seed to select is based on the observed cascade of previously selected seeds. We focus on the myopic feedback model, in which we can only observe which neighbors of previously selected seeds have been influenced and on the independent cascade model, where each edge is associated with an independent probability of diffusing influence. While adaptive policies are strictly stronger than non-adaptive ones, in which all the seeds are selected beforehand, the latter are much easier to design and implement and they provide good approximation factors if the adaptivity gap, the ratio between the adaptive and the non-adaptive optima, is small. Previous works showed that the adaptivity gap is at most 4, and that simple adaptive or non-adaptive greedy algorithms guarantee an approximation of 1/4 (1-1/e) ≈ 0.158 for the adaptive optimum. This is the best approximation factor known so far for the adaptive influence maximization problem with myopic feedback.
In this paper, we directly analyze the approximation factor of the non-adaptive greedy algorithm, without passing through the adaptivity gap, and show an improved bound of 1/2 (1-1/e) ≈ 0.316. Therefore, the adaptivity gap is at most 2e/e-1 ≈ 3.164. To prove these bounds, we introduce a new approach to relate the greedy non-adaptive algorithm to the adaptive optimum. The new approach does not rely on multi-linear extensions or random walks on optimal decision trees, which are commonly used techniques in the field. We believe that it is of independent interest and may be used to analyze other adaptive optimization problems. Finally, we also analyze the adaptive greedy algorithm, and show that guarantees an improved approximation factor of 1-1/(√{e)}≈ 0.393.
Cite as
Gianlorenzo D'Angelo, Debashmita Poddar, and Cosimo Vinci. Improved Approximation Factor for Adaptive Influence Maximization via Simple Greedy Strategies. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 59:1-59:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dangelo_et_al:LIPIcs.ICALP.2021.59,
author = {D'Angelo, Gianlorenzo and Poddar, Debashmita and Vinci, Cosimo},
title = {{Improved Approximation Factor for Adaptive Influence Maximization via Simple Greedy Strategies}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {59:1--59:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.59},
URN = {urn:nbn:de:0030-drops-141282},
doi = {10.4230/LIPIcs.ICALP.2021.59},
annote = {Keywords: Adaptive Optimization, Influence Maximization, Submodular Optimization, Stochastic Optimization}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Mina Dalirrooyfard and Jenny Kaufmann
Abstract
Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no path between them in one of the two directions. So it is natural to consider the distance between two nodes as the length of the shortest path in the direction in which this path exists, motivating the definition of the min-distance. The min-distance between two nodes u and v is the minimum of the shortest path distances from u to v and from v to u.
As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in Õ(mn) time. So it is natural to resort to approximation algorithms in Õ(mn^{1-ε}) time for some positive ε. Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, and Dalirrooyfard et al [ICALP 2019] gave the first constant factor approximation algorithms on general graphs for min-diameter, min-radius and min-eccentricities. Abboud et al obtained a 3-approximation algorithm for min-radius on DAGs which works in Õ(m√n) time, and showed that any (2-δ)-approximation requires n^{2-o(1)} time for any δ > 0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in Õ(m√n) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time 2-approximation algorithm for min-diameter along with a lower bound stating that any (3/2-δ)-approximation algorithm for sparse DAGs requires n^{2-o(1)} time under SETH. We close this gap for dense DAGs by obtaining a 3/2-approximation algorithm which works in O(n^{2.350}) time and showing that the approximation factor is unlikely to be improved within O(n^{ω - o(1)}) time under the high dimensional Orthogonal Vectors Conjecture, where ω is the matrix multiplication exponent.
Cite as
Mina Dalirrooyfard and Jenny Kaufmann. Approximation Algorithms for Min-Distance Problems in DAGs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dalirrooyfard_et_al:LIPIcs.ICALP.2021.60,
author = {Dalirrooyfard, Mina and Kaufmann, Jenny},
title = {{Approximation Algorithms for Min-Distance Problems in DAGs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {60:1--60:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.60},
URN = {urn:nbn:de:0030-drops-141293},
doi = {10.4230/LIPIcs.ICALP.2021.60},
annote = {Keywords: Fine-grained complexity, Graph algorithms, Diameter, Radius, Eccentricities}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Christoph Damerius, Dominik Kaaser, Peter Kling, and Florian Schneider
Abstract
Consider a set P of points in the unit square U = [1,0), one of them being the origin. For each point p ∈ P you may draw an axis-aligned rectangle in U with its lower-left corner being p. What is the maximum area such rectangles can cover without overlapping each other?
Freedman posed this problem in 1969, asking whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu and Tóth [Adrian Dumitrescu and Csaba D. Tóth, 2015] achieved the first constant coverage of 9.1%; since then, no significant progress was made. While 9.1% might seem low, the authors could not find any instance where their algorithm covers less than 50%, nourishing the hope to eventually prove a 50% bound. While we indeed significantly raise the algorithm’s coverage to 39%, we extinguish the hope of reaching 50% by giving points for which its coverage stays below 43.3%.
Our analysis studies the algorithm’s average and worst-case density of so-called tiles, which represent the staircase polygons in which a point can freely choose its maximum-area rectangle. Our approach is comparatively general and may potentially help in analyzing related algorithms.
Cite as
Christoph Damerius, Dominik Kaaser, Peter Kling, and Florian Schneider. On Greedily Packing Anchored Rectangles. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{damerius_et_al:LIPIcs.ICALP.2021.61,
author = {Damerius, Christoph and Kaaser, Dominik and Kling, Peter and Schneider, Florian},
title = {{On Greedily Packing Anchored Rectangles}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {61:1--61:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.61},
URN = {urn:nbn:de:0030-drops-141306},
doi = {10.4230/LIPIcs.ICALP.2021.61},
annote = {Keywords: lower-left anchored rectangle packing, greedy algorithm, charging scheme}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ewan Davies and Will Perkins
Abstract
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density α_c(Δ) and provide (i) for α < α_c(Δ) randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most α n in n-vertex graphs of maximum degree Δ; and (ii) a proof that unless NP=RP, no such algorithms exist for α > α_c(Δ). The critical density is the occupancy fraction of hard core model on the clique K_{Δ+1} at the uniqueness threshold on the infinite Δ-regular tree, giving α_c(Δ) ~ e/(1+e)1/(Δ) as Δ → ∞.
Cite as
Ewan Davies and Will Perkins. Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{davies_et_al:LIPIcs.ICALP.2021.62,
author = {Davies, Ewan and Perkins, Will},
title = {{Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {62:1--62:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.62},
URN = {urn:nbn:de:0030-drops-141310},
doi = {10.4230/LIPIcs.ICALP.2021.62},
annote = {Keywords: approximate counting, independent sets, Markov chains}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Jonas Ellert and Johannes Fischer
Abstract
A run in a string is a maximal periodic substring. For example, the string bananatree contains the runs anana = (an)^{5/2} and ee = e². There are less than n runs in any length-n string, and computing all runs for a string over a linearly-sortable alphabet takes 𝒪(n) time (Bannai et al., SIAM J. Comput. 2017). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an 𝒪(nα(n)) time algorithm (where α(n) is the extremely slowly growing inverse Ackermann function). We show how to achieve 𝒪(n) time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov’s conjecture. This also positively answers the at least 29-year-old question whether square-freeness can be tested in linear time over general ordered alphabets (Breslauer, PhD thesis, Columbia University 1992).
Cite as
Jonas Ellert and Johannes Fischer. Linear Time Runs Over General Ordered Alphabets. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ellert_et_al:LIPIcs.ICALP.2021.63,
author = {Ellert, Jonas and Fischer, Johannes},
title = {{Linear Time Runs Over General Ordered Alphabets}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {63:1--63:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.63},
URN = {urn:nbn:de:0030-drops-141322},
doi = {10.4230/LIPIcs.ICALP.2021.63},
annote = {Keywords: String algorithms, Lyndon array, runs, squares, longest common extension, general ordered alphabets, combinatorics on words}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen
Abstract
Given an unweighted digraph G = (V,E), undergoing a sequence of edge deletions, with m = |E|, n = |V|, we consider the problem of maintaining all-pairs shortest paths (APSP).
Whilst this problem has been studied in a long line of research [ACM'81, FOCS'99, FOCS'01, STOC'02, STOC'03, SWAT'04, STOC'13] and the problem of (1+ε)-approximate, weighted APSP was solved to near-optimal update time Õ(mn) by Bernstein [STOC'13], the problem has mainly been studied in the context of an oblivious adversary which fixes the update sequence before the algorithm is started. In this paper, we make significant progress on the problem for an adaptive adversary which can perform updates based on answers to previous queries:
- We first present a deterministic data structure that maintains the exact distances with total update time Õ(n³).
- We also present a deterministic data structure that maintains (1+ε)-approximate distance estimates with total update time Õ(√m n²/ε) which for sparse graphs is Õ(n^{2+1/2}/ε).
- Finally, we present a randomized (1+ε)-approximate data structure which works against an adaptive adversary; its total update time is Õ(m^{2/3}n^{5/3} + n^{8/3}/(m^{1/3}ε²)) which for sparse graphs is Õ(n^{2+1/3}/ε²). Our exact data structure matches the total update time of the best randomized data structure by Baswana et al. [STOC'02] and maintains the distance matrix in near-optimal time. Our approximate data structures improve upon the best data structures against an adaptive adversary which have Õ(mn²) total update time [JACM'81, STOC'03].
Cite as
Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen. Decremental APSP in Unweighted Digraphs Versus an Adaptive Adversary. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 64:1-64:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{evald_et_al:LIPIcs.ICALP.2021.64,
author = {Evald, Jacob and Fredslund-Hansen, Viktor and Gutenberg, Maximilian Probst and Wulff-Nilsen, Christian},
title = {{Decremental APSP in Unweighted Digraphs Versus an Adaptive Adversary}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {64:1--64:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.64},
URN = {urn:nbn:de:0030-drops-141337},
doi = {10.4230/LIPIcs.ICALP.2021.64},
annote = {Keywords: Dynamic Graph Algorithm, Data Structure, Shortest Paths}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Dimitris Fotakis, Panagiotis Kostopanagiotis, Vasileios Nakos, Georgios Piliouras, and Stratis Skoulakis
Abstract
We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover (Mult-MSSC), a natural and intriguing generalization of the classical List Update problem. In Mult-MSSC, we maintain a sequence of permutations (π⁰, π¹, …, π^T) on n elements, based on a sequence of requests ℛ = (R¹, …, R^T). We aim to minimize the total cost of updating π^{t-1} to π^{t}, quantified by the Kendall tau distance d_{KT}(π^{t-1}, π^t), plus the total cost of covering each request R^t with the current permutation π^t, quantified by the position of the first element of R^t in π^t.
Using a reduction from Set Cover, we show that Mult-MSSC does not admit an O(1)-approximation, unless P = NP, and that any o(log n) (resp. o(r)) approximation to Mult-MSSC implies a sublogarithmic (resp. o(r)) approximation to Set Cover (resp. where each element appears at most r times). Our main technical contribution is to show that Mult-MSSC can be approximated in polynomial-time within a factor of O(log² n) in general instances, by randomized rounding, and within a factor of O(r²), if all requests have cardinality at most r, by deterministic rounding.
Cite as
Dimitris Fotakis, Panagiotis Kostopanagiotis, Vasileios Nakos, Georgios Piliouras, and Stratis Skoulakis. On the Approximability of Multistage Min-Sum Set Cover. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{fotakis_et_al:LIPIcs.ICALP.2021.65,
author = {Fotakis, Dimitris and Kostopanagiotis, Panagiotis and Nakos, Vasileios and Piliouras, Georgios and Skoulakis, Stratis},
title = {{On the Approximability of Multistage Min-Sum Set Cover}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {65:1--65:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.65},
URN = {urn:nbn:de:0030-drops-141341},
doi = {10.4230/LIPIcs.ICALP.2021.65},
annote = {Keywords: Approximation Algorithms, Multistage Min-Sum Set Cover, Multistage Optimization Problems}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Tobias Friedrich, Andreas Göbel, Martin S. Krejca, and Marcus Pappik
Abstract
The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in d dimensions. Up to a fugacity of λ < e/2^d, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure.
Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.
Cite as
Tobias Friedrich, Andreas Göbel, Martin S. Krejca, and Marcus Pappik. A Spectral Independence View on Hard Spheres via Block Dynamics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{friedrich_et_al:LIPIcs.ICALP.2021.66,
author = {Friedrich, Tobias and G\"{o}bel, Andreas and Krejca, Martin S. and Pappik, Marcus},
title = {{A Spectral Independence View on Hard Spheres via Block Dynamics}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {66:1--66:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.66},
URN = {urn:nbn:de:0030-drops-141353},
doi = {10.4230/LIPIcs.ICALP.2021.66},
annote = {Keywords: Hard-sphere Model, Markov Chain, Partition Function, Gibbs Distribution, Approximate Counting, Spectral Independence}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Zachary Friggstad and Chaitanya Swamy
Abstract
We investigate a genre of vehicle-routing problems (VRPs), that we call max-reward VRPs, wherein nodes located in a metric space have associated rewards that depend on their visiting times, and we seek a path that earns maximum reward. A prominent problem in this genre is deadline TSP, where nodes have deadlines and we seek a path that visits all nodes by their deadlines and earns maximum reward. Our main result is a constant-factor approximation for deadline TSP running in time O(n^O(log(nΔ))) in metric spaces with integer distances at most Δ. This is the first improvement over the approximation factor of O(log n) due to Bansal et al. [N. Bansal et al., 2004] in over 15 years (but is achieved in super-polynomial time). Our result provides the first concrete indication that log n is unlikely to be a real inapproximability barrier for deadline TSP, and raises the exciting possibility that deadline TSP might admit a polytime constant-factor approximation.
At a high level, we obtain our result by carefully guessing an appropriate sequence of O(log (nΔ)) nodes appearing on the optimal path, and finding suitable paths between any two consecutive guessed nodes. We argue that the problem of finding a path between two consecutive guessed nodes can be relaxed to an instance of a special case of deadline TSP called point-to-point (P2P) orienteering. Any approximation algorithm for P2P orienteering can then be utilized in conjunction with either a greedy approach, or an LP-rounding approach, to find a good set of paths overall between every pair of guessed nodes. While concatenating these paths does not immediately yield a feasible solution, we argue that it can be covered by a constant number of feasible solutions. Overall our result therefore provides a novel reduction showing that any α-approximation for P2P orienteering can be leveraged to obtain an O(α)-approximation for deadline TSP in O(n^O(log nΔ)) time.
Our results extend to yield the same guarantees (in approximation ratio and running time) for a substantial generalization of deadline TSP, where the reward obtained by a client is given by an arbitrary non-increasing function (specified by a value oracle) of its visiting time. Finally, we discuss applications of our results to variants of deadline TSP, including settings where both end-nodes are specified, nodes have release dates, and orienteering with time windows.
Cite as
Zachary Friggstad and Chaitanya Swamy. Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 67:1-67:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{friggstad_et_al:LIPIcs.ICALP.2021.67,
author = {Friggstad, Zachary and Swamy, Chaitanya},
title = {{Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {67:1--67:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.67},
URN = {urn:nbn:de:0030-drops-141369},
doi = {10.4230/LIPIcs.ICALP.2021.67},
annote = {Keywords: Approximation algorithms, Vehicle routing problems, Deadline TSP, Orienteering}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Hu Fu, Zhihao Gavin Tang, Hongxun Wu, Jinzhao Wu, and Qianfan Zhang
Abstract
With a wide range of applications, stochastic matching problems have been studied in different models, including prophet inequality, Query-Commit, and Price-of-Information. While there have been recent breakthroughs in all these settings for bipartite graphs, few non-trivial results are known for general graphs.
In this paper, we study the random order vertex arrival contention resolution scheme for matching in general graphs, which is inspired by the recent work of Ezra et al. (EC 2020). We design an 8/15-selectable batched RCRS for matching and apply it to achieve 8/15-competitive/approximate algorithms for all the three models. Our results are the first non-trivial results for random order prophet matching and Price-of-Information matching in general graphs. For the Query-Commit model, our result substantially improves upon the 0.501 approximation ratio by Tang et al. (STOC 2020). We also show that no batched RCRS for matching can be better than 1/2+1/(2e²) ≈ 0.567-selectable.
Cite as
Hu Fu, Zhihao Gavin Tang, Hongxun Wu, Jinzhao Wu, and Qianfan Zhang. Random Order Vertex Arrival Contention Resolution Schemes for Matching, with Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 68:1-68:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{fu_et_al:LIPIcs.ICALP.2021.68,
author = {Fu, Hu and Tang, Zhihao Gavin and Wu, Hongxun and Wu, Jinzhao and Zhang, Qianfan},
title = {{Random Order Vertex Arrival Contention Resolution Schemes for Matching, with Applications}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {68:1--68:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.68},
URN = {urn:nbn:de:0030-drops-141376},
doi = {10.4230/LIPIcs.ICALP.2021.68},
annote = {Keywords: Matching, Contention Resolution Scheme, Price of Information, Query-Commit}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang
Abstract
The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in NP∩coNP but not known to be in 𝖯. The currently best algorithms have runtime 2^Θ(n) where n is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime 2^O(√nlog n). We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.
Cite as
Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang. A Subexponential Algorithm for ARRIVAL. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gartner_et_al:LIPIcs.ICALP.2021.69,
author = {G\"{a}rtner, Bernd and Haslebacher, Sebastian and Hoang, Hung P.},
title = {{A Subexponential Algorithm for ARRIVAL}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {69:1--69:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.69},
URN = {urn:nbn:de:0030-drops-141387},
doi = {10.4230/LIPIcs.ICALP.2021.69},
annote = {Keywords: Pseudorandom walks, reachability, graph games, switching systems}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi
Abstract
This paper presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution sol for a clustering problem is said to be an (α, β)-approximation if for all subsets of clients C', it satisfies sol(C') ≤ α ⋅ opt(C') + β ⋅ mr, where opt(C') is the cost of the optimal solution for clients C' and mr is the minimum regret achievable by any solution.
Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other 𝓁_p-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.
Cite as
Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi. Universal Algorithms for Clustering Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 70:1-70:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ganesh_et_al:LIPIcs.ICALP.2021.70,
author = {Ganesh, Arun and Maggs, Bruce M. and Panigrahi, Debmalya},
title = {{Universal Algorithms for Clustering Problems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {70:1--70:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.70},
URN = {urn:nbn:de:0030-drops-141397},
doi = {10.4230/LIPIcs.ICALP.2021.70},
annote = {Keywords: universal algorithms, clustering, k-median, k-means, k-center}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Arnab Ganguly, Dhrumil Patel, Rahul Shah, and Sharma V. Thankachan
Abstract
Two strings are order isomorphic iff the relative ordering of their characters is the same at all positions. For a given text T[1,n] over an ordered alphabet of size σ, we can maintain an order-isomorphic suffix tree/array in O(nlog n) bits and support (order-isomorphic) pattern/substring matching queries efficiently. It is interesting to know if we can encode these structures in space close to the text’s size of nlogσ bits. We answer this question positively by presenting an O(nlog σ)-bit index that allows access to any entry in order-isomorphic suffix array (and its inverse array) in t_{SA} = {O}(log²n/logσ) time. For any pattern P given as a query, this index can count the number of substrings of T that are order-isomorphic to P (denoted by occ) in {O}((|P|logσ+t_{SA})log n) time using standard techniques. Also, it can report the locations of those substrings in additional O(occ ⋅ t_{SA}) time.
Cite as
Arnab Ganguly, Dhrumil Patel, Rahul Shah, and Sharma V. Thankachan. LF Successor: Compact Space Indexing for Order-Isomorphic Pattern Matching. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ganguly_et_al:LIPIcs.ICALP.2021.71,
author = {Ganguly, Arnab and Patel, Dhrumil and Shah, Rahul and Thankachan, Sharma V.},
title = {{LF Successor: Compact Space Indexing for Order-Isomorphic Pattern Matching}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {71:1--71:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.71},
URN = {urn:nbn:de:0030-drops-141400},
doi = {10.4230/LIPIcs.ICALP.2021.71},
annote = {Keywords: Succinct data structures, Pattern Matching}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber
Abstract
In extension problems of partial graph drawings one is given an incomplete drawing of an input graph G and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting a single edge.
In this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for k-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing.
Cite as
Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber. Crossing-Optimal Extension of Simple Drawings. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ganian_et_al:LIPIcs.ICALP.2021.72,
author = {Ganian, Robert and Hamm, Thekla and Klute, Fabian and Parada, Irene and Vogtenhuber, Birgit},
title = {{Crossing-Optimal Extension of Simple Drawings}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {72:1--72:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.72},
URN = {urn:nbn:de:0030-drops-141412},
doi = {10.4230/LIPIcs.ICALP.2021.72},
annote = {Keywords: Simple drawings, Extension problems, Crossing minimization, FPT-algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Uma Girish, Ran Raz, and Wei Zhan
Abstract
We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary n× n contraction matrix A, and a parameter T ≤ poly(n) and outputs the entries of A^T, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result:
First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space O(S + log T) that takes as an input the description of a quantum algorithm with quantum space S and time T, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements.
Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace [Lange et al., 2000].
Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.
Cite as
Uma Girish, Ran Raz, and Wei Zhan. Quantum Logspace Algorithm for Powering Matrices with Bounded Norm. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{girish_et_al:LIPIcs.ICALP.2021.73,
author = {Girish, Uma and Raz, Ran and Zhan, Wei},
title = {{Quantum Logspace Algorithm for Powering Matrices with Bounded Norm}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {73:1--73:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.73},
URN = {urn:nbn:de:0030-drops-141426},
doi = {10.4230/LIPIcs.ICALP.2021.73},
annote = {Keywords: BQL, Matrix Powering, Quantum Circuit, Reversible Computation}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Nick Gravin, Zhihao Gavin Tang, and Kangning Wang
Abstract
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by Gamlath et al., who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than 0.5. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution.
We focus on a natural class of prune & greedy online policies motivated by practical considerations from a multitude of online matching platforms. Any prune & greedy algorithm consists of two stages: first, it decreases the probabilities of some edges in the stochastic instance and then runs greedy algorithm on the pruned graph. We propose prune & greedy algorithms that are 0.552-competitive on the instances that can be pruned to a 2-regular stochastic bipartite graph, and 0.503-competitive on arbitrary stochastic bipartite graphs. The algorithms and our analysis significantly deviate from the prior work. We first obtain analytically manageable lower bound on the size of the matching, which leads to a non-linear optimization problem. We further reduce this problem to a continuous optimization with a constant number of parameters that can be solved using standard software tools.
Cite as
Nick Gravin, Zhihao Gavin Tang, and Kangning Wang. Online Stochastic Matching with Edge Arrivals. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 74:1-74:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gravin_et_al:LIPIcs.ICALP.2021.74,
author = {Gravin, Nick and Tang, Zhihao Gavin and Wang, Kangning},
title = {{Online Stochastic Matching with Edge Arrivals}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {74:1--74:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.74},
URN = {urn:nbn:de:0030-drops-141438},
doi = {10.4230/LIPIcs.ICALP.2021.74},
annote = {Keywords: online matching, graph algorithms, prophet inequality}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Yuzhou Gu, Adam Polak, Virginia Vassilevska Williams, and Yinzhan Xu
Abstract
One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n^{3-o(1)} time, and it is widely open whether it has an O(n^{3-ε}) time algorithm for ε > 0. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product.
A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an n-vertex graph with potentially negative edge weights in {-M, …, M}.
SSRP with positive integer edge weights bounded by M can be solved in Õ(Mn^ω) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in O(M^{0.7519} n^{2.5286}) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M^{0.8043} n^{2.4957}) time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require n^{2.5-o(1)} time.
Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an Õ(M n^{ω}) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.
Cite as
Yuzhou Gu, Adam Polak, Virginia Vassilevska Williams, and Yinzhan Xu. Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gu_et_al:LIPIcs.ICALP.2021.75,
author = {Gu, Yuzhou and Polak, Adam and Vassilevska Williams, Virginia and Xu, Yinzhan},
title = {{Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {75:1--75:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.75},
URN = {urn:nbn:de:0030-drops-141440},
doi = {10.4230/LIPIcs.ICALP.2021.75},
annote = {Keywords: APSP, Min-Plus Product, Range Mode, Single-Source Replacement Paths}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Yong Gu and Hanlin Ren
Abstract
We continue the study of distance sensitivity oracles (DSOs). Given a directed graph G with n vertices and edge weights in {1, 2, … , M}, we want to build a data structure such that given any source vertex u, any target vertex v, and any failure f (which is either a vertex or an edge), it outputs the length of the shortest path from u to v not going through f. Our main result is a DSO with preprocessing time O(n^2.5794 M) and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is O(n^2.7233 M), and even in the easier case of undirected graphs, the best preprocessing time is O(n^2.6865 M) [Ren, ESA 2020]. One drawback of our DSOs, though, is that it only supports distance queries but not path queries.
Our main technical ingredient is an algorithm that computes the inverse of a degree-d polynomial matrix (i.e. a matrix whose entries are degree-d univariate polynomials) modulo x^r. The algorithm is adapted from [Zhou, Labahn and Storjohann, Journal of Complexity, 2015], and we replace some of its intermediate steps with faster rectangular matrix multiplication algorithms.
We also show how to compute unique shortest paths in a directed graph with edge weights in {1, 2, … , M}, in O(n^2.5286 M) time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the O(n^2.6865 M) time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths.
Cite as
Yong Gu and Hanlin Ren. Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 76:1-76:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gu_et_al:LIPIcs.ICALP.2021.76,
author = {Gu, Yong and Ren, Hanlin},
title = {{Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {76:1--76:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.76},
URN = {urn:nbn:de:0030-drops-141450},
doi = {10.4230/LIPIcs.ICALP.2021.76},
annote = {Keywords: graph theory, shortest paths, distance sensitivity oracles}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Anupam Gupta, Benjamin Moseley, and Rudy Zhou
Abstract
This paper considers approximation algorithms for generalized k-median problems. This class of problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution with a constant number of fractional variables. The algorithm is based on iteratively rounding linear programs, and the main technical innovation comes from understanding the rich structure of the resulting extreme points.
Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios 6.994 + ε and 6.387 + ε for k-median with outliers and knapsack median, respectively. These both improve on the best known approximations.
Cite as
Anupam Gupta, Benjamin Moseley, and Rudy Zhou. Structural Iterative Rounding for Generalized k-Median Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 77:1-77:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gupta_et_al:LIPIcs.ICALP.2021.77,
author = {Gupta, Anupam and Moseley, Benjamin and Zhou, Rudy},
title = {{Structural Iterative Rounding for Generalized k-Median Problems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {77:1--77:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.77},
URN = {urn:nbn:de:0030-drops-141465},
doi = {10.4230/LIPIcs.ICALP.2021.77},
annote = {Keywords: approximation algorithms, clustering, linear programming}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Bernhard Haeupler, D. Ellis Hershkowitz, and David Wajc
Abstract
We study the store-and-forward packet routing problem for simultaneous multicasts, in which multiple packets have to be forwarded along given trees as fast as possible.
This is a natural generalization of the seminal work of Leighton, Maggs and Rao, which solved this problem for unicasts, i.e. the case where all trees are paths. They showed the existence of asymptotically optimal O(C + D)-length schedules, where the congestion C is the maximum number of packets sent over an edge and the dilation D is the maximum depth of a tree. This improves over the trivial O(CD) length schedules.
We prove a lower bound for multicasts, which shows that there do not always exist schedules of non-trivial length, o(CD). On the positive side, we construct O(C+D+log² n)-length schedules in any n-node network. These schedules are near-optimal, since our lower bound shows that this length cannot be improved to O(C+D) + o(log n).
Cite as
Bernhard Haeupler, D. Ellis Hershkowitz, and David Wajc. Near-Optimal Schedules for Simultaneous Multicasts. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 78:1-78:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{haeupler_et_al:LIPIcs.ICALP.2021.78,
author = {Haeupler, Bernhard and Hershkowitz, D. Ellis and Wajc, David},
title = {{Near-Optimal Schedules for Simultaneous Multicasts}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {78:1--78:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.78},
URN = {urn:nbn:de:0030-drops-141471},
doi = {10.4230/LIPIcs.ICALP.2021.78},
annote = {Keywords: Packet routing, multicast, scheduling algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Maria Hartmann, László Kozma, Corwin Sinnamon, and Robert E. Tarjan
Abstract
The smooth heap is a recently introduced self-adjusting heap [Kozma, Saranurak, 2018] similar to the pairing heap [Fredman, Sedgewick, Sleator, Tarjan, 1986]. The smooth heap was obtained as a heap-counterpart of Greedy BST, a binary search tree updating strategy conjectured to be instance-optimal [Lucas, 1988], [Munro, 2000]. Several adaptive properties of smooth heaps follow from this connection; moreover, the smooth heap itself has been conjectured to be instance-optimal within a certain class of heaps. Nevertheless, no general analysis of smooth heaps has existed until now, the only previous analysis showing that, when used in sorting mode (n insertions followed by n delete-min operations), smooth heaps sort n numbers in O(nlg n) time.
In this paper we describe a simpler variant of the smooth heap we call the slim heap. We give a new, self-contained analysis of smooth heaps and slim heaps in unrestricted operation, obtaining amortized bounds that match the best bounds known for self-adjusting heaps. Previous experimental work has found the pairing heap to dominate other data structures in this class in various settings. Our tests show that smooth heaps and slim heaps are competitive with pairing heaps, outperforming them in some cases, while being comparably easy to implement.
Cite as
Maria Hartmann, László Kozma, Corwin Sinnamon, and Robert E. Tarjan. Analysis of Smooth Heaps and Slim Heaps. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{hartmann_et_al:LIPIcs.ICALP.2021.79,
author = {Hartmann, Maria and Kozma, L\'{a}szl\'{o} and Sinnamon, Corwin and Tarjan, Robert E.},
title = {{Analysis of Smooth Heaps and Slim Heaps}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {79:1--79:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.79},
URN = {urn:nbn:de:0030-drops-141482},
doi = {10.4230/LIPIcs.ICALP.2021.79},
annote = {Keywords: data structure, heap, priority queue, amortized analysis, self-adjusting}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Chien-Chung Huang, Mathieu Mari, Claire Mathieu, and Jens Vygen
Abstract
We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.
Cite as
Chien-Chung Huang, Mathieu Mari, Claire Mathieu, and Jens Vygen. Approximating Maximum Integral Multiflows on Bounded Genus Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 80:1-80:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{huang_et_al:LIPIcs.ICALP.2021.80,
author = {Huang, Chien-Chung and Mari, Mathieu and Mathieu, Claire and Vygen, Jens},
title = {{Approximating Maximum Integral Multiflows on Bounded Genus Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {80:1--80:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.80},
URN = {urn:nbn:de:0030-drops-141491},
doi = {10.4230/LIPIcs.ICALP.2021.80},
annote = {Keywords: Multi-commodity flows, approximation algorithms, bounded genus graphs}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Sharat Ibrahimpur and Chaitanya Swamy
Abstract
We consider the minimum-norm load-balancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:ℝ^m → ℝ_{≥ 0}. We seek an assignment σ of jobs to machines that minimizes the f-norm of the induced load vector load->_σ ∈ ℝ_{≥ 0}^m, where load_σ(i) = ∑_{j:σ(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the current-best result for MinNormLB is a (4+ε)-approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment σ that minimizes the expected f-norm of the induced random load vector.
We obtain results that (essentially) match the best-known guarantees for deterministic makespan minimization (MinNormLB with 𝓁_∞ norm). For MinNormLB, we obtain a (2+ε)-approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any α-approximation algorithm for MinNormLB in that machine environment yields a randomized α(1+ε)-approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+ε)-approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020].
Cite as
Sharat Ibrahimpur and Chaitanya Swamy. Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 81:1-81:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ibrahimpur_et_al:LIPIcs.ICALP.2021.81,
author = {Ibrahimpur, Sharat and Swamy, Chaitanya},
title = {{Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {81:1--81:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.81},
URN = {urn:nbn:de:0030-drops-141504},
doi = {10.4230/LIPIcs.ICALP.2021.81},
annote = {Keywords: Approximation algorithms, Load balancing, Minimum-norm optimization, LP rounding}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta
Abstract
In a recent landmark result [Ji et al., arXiv:2001.04383 (2020)], it was shown that approximating the value of a two-player game is undecidable when the players are allowed to share quantum states of unbounded dimension. In this paper, we study the computational complexity of two-player games when the dimension of the quantum systems is bounded by T. More specifically, we give a semidefinite program of size exp(𝒪(T^{12}(log²(AT)+log(Q)log(AT))/ε²)) to compute additive ε-approximations on the value of two-player free games with T× T-dimensional quantum entanglement, where A and Q denote the number of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints that we derive via quantum entropy inequalities.
Cite as
Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta. Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{jee_et_al:LIPIcs.ICALP.2021.82,
author = {Jee, Hyejung H. and Sparaciari, Carlo and Fawzi, Omar and Berta, Mario},
title = {{Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {82:1--82:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.82},
URN = {urn:nbn:de:0030-drops-141514},
doi = {10.4230/LIPIcs.ICALP.2021.82},
annote = {Keywords: non-local game, semidefinite programming, quantum correlation, approximation algorithm, Lasserre hierarchy, de Finetti theorem}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Adam Karczmarz
Abstract
We consider the directed minimum weight cycle problem in the fully dynamic setting. To the best of our knowledge, so far no fully dynamic algorithms have been designed specifically for the minimum weight cycle problem in general digraphs. One can achieve Õ(n²) amortized update time by simply invoking the fully dynamic APSP algorithm of Demetrescu and Italiano [J. ACM '04]. This bound, however, yields no improvement over the trivial recompute-from-scratch algorithm for sparse graphs.
Our first contribution is a very simple deterministic (1+ε)-approximate algorithm supporting vertex updates (i.e., changing all edges incident to a specified vertex) in conditionally near-optimal Õ(mlog{(W)}/ε) amortized time for digraphs with real edge weights in [1,W]. Using known techniques, the algorithm can be implemented on planar graphs and also gives some new sublinear fully dynamic algorithms maintaining approximate cuts and flows in planar digraphs.
Additionally, we show a Monte Carlo randomized exact fully dynamic minimum weight cycle algorithm with Õ(mn^{2/3}) worst-case update that works for real edge weights. To this end, we generalize the exact fully dynamic APSP data structure of Abraham et al. [SODA'17] to solve the multiple-pairs shortest paths problem, where one is interested in computing distances for some k (instead of all n²) fixed source-target pairs after each update. We show that in such a scenario, Õ((m+k)n^{2/3}) worst-case update time is possible.
Cite as
Adam Karczmarz. Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{karczmarz:LIPIcs.ICALP.2021.83,
author = {Karczmarz, Adam},
title = {{Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {83:1--83:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.83},
URN = {urn:nbn:de:0030-drops-141521},
doi = {10.4230/LIPIcs.ICALP.2021.83},
annote = {Keywords: Dynamic graph algorithms, minimum weight cycle, dynamic shortest paths}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Tali Kaufman and Izhar Oppenheim
Abstract
Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders were derived from the theory of buildings that is far from being elementary.
In this work we study high dimensional complexes which are strongly symmetric. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion.
It was an open question whether the recent elementary construction of bounded degree spectral high dimensional expanders based on coset complexes give rise to bounded degree cosystolic expanders. In this work we answer this question affirmatively. We show that these complexes give rise to bounded degree cosystolic expanders in dimension two, and that their links are (two-dimensional) coboundary expanders. We do so by exploiting the strong symmetry properties of the links of these complexes using a new machinery developed in this work.
Previous works have shown a way to bound the co-boundary expansion using strong symmetry in the special situation of "building like" complexes. Our new machinery shows how to get coboundary expansion for general strongly symmetric coset complexes, which are not necessarily "building like", via studying the (Dehn function of the) presentation of the symmetry group of these complexes.
Cite as
Tali Kaufman and Izhar Oppenheim. Coboundary and Cosystolic Expansion from Strong Symmetry. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kaufman_et_al:LIPIcs.ICALP.2021.84,
author = {Kaufman, Tali and Oppenheim, Izhar},
title = {{Coboundary and Cosystolic Expansion from Strong Symmetry}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {84:1--84:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.84},
URN = {urn:nbn:de:0030-drops-141539},
doi = {10.4230/LIPIcs.ICALP.2021.84},
annote = {Keywords: High dimensional expanders, Cosystolic expansion, Coboundary expansion}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Telikepalli Kavitha
Abstract
Let G be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching M in G is a popular max-matching if for any maximum matching N in G, the number of nodes that prefer M is at least the number that prefer N. Popular max-matchings always exist in G and they are relevant in applications where the size of the matching is of higher priority than node preferences. Here we assume there is also a cost function on the edge set. So what we seek is a min-cost popular max-matching. Our main result is that such a matching can be computed in polynomial time.
We show a compact extended formulation for the popular max-matching polytope and the algorithmic result follows from this. In contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard.
Cite as
Telikepalli Kavitha. Maximum Matchings and Popularity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 85:1-85:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kavitha:LIPIcs.ICALP.2021.85,
author = {Kavitha, Telikepalli},
title = {{Maximum Matchings and Popularity}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {85:1--85:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.85},
URN = {urn:nbn:de:0030-drops-141548},
doi = {10.4230/LIPIcs.ICALP.2021.85},
annote = {Keywords: Bipartite graphs, Popular matchings, Stable matchings, Dual certificates}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, and Peter Zeman
Abstract
A map is a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.
Cite as
Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, and Peter Zeman. Automorphisms and Isomorphisms of Maps in Linear Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 86:1-86:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kawarabayashi_et_al:LIPIcs.ICALP.2021.86,
author = {Kawarabayashi, Ken-ichi and Mohar, Bojan and Nedela, Roman and Zeman, Peter},
title = {{Automorphisms and Isomorphisms of Maps in Linear Time}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {86:1--86:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.86},
URN = {urn:nbn:de:0030-drops-141558},
doi = {10.4230/LIPIcs.ICALP.2021.86},
annote = {Keywords: maps on surfaces, automorphisms, isomorphisms, algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Tuukka Korhonen
Abstract
We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2^{Ω(w/d)} gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2^{Ω(w)} gates. An MWIS-formula is an MWIS-circuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2^{Ω(t/d)} gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.
Cite as
Tuukka Korhonen. Lower Bounds on Dynamic Programming for Maximum Weight Independent Set. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 87:1-87:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{korhonen:LIPIcs.ICALP.2021.87,
author = {Korhonen, Tuukka},
title = {{Lower Bounds on Dynamic Programming for Maximum Weight Independent Set}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {87:1--87:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.87},
URN = {urn:nbn:de:0030-drops-141562},
doi = {10.4230/LIPIcs.ICALP.2021.87},
annote = {Keywords: Maximum weight independent set, Treewidth, Tropical circuits, Dynamic programming, Treedepth, Monotone circuit complexity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Michal Koucký and Karel Král
Abstract
We build boolean circuits of size 𝒪(nm²) and depth 𝒪(log(n) + m log(m)) for sorting n integers each of m-bits. We build also circuits that sort n integers each of m-bits according to their first k bits that are of size 𝒪(nmk (1 + log^*(n) - log^*(m))) and depth 𝒪(log³(n)). This improves on the results of Asharov et al. [Asharov et al., 2021] and resolves some of their open questions.
Cite as
Michal Koucký and Karel Král. Sorting Short Integers. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 88:1-88:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{koucky_et_al:LIPIcs.ICALP.2021.88,
author = {Kouck\'{y}, Michal and Kr\'{a}l, Karel},
title = {{Sorting Short Integers}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {88:1--88:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.88},
URN = {urn:nbn:de:0030-drops-141577},
doi = {10.4230/LIPIcs.ICALP.2021.88},
annote = {Keywords: sorting, small integers, boolean circuits}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Jakub Kozik
Abstract
In 1964 Erdős proved, by randomized construction, that the minimum number of edges in a k-graph that is not two colorable is O(k² 2^k). To this day, it is not known whether there exist such k-graphs with smaller number of edges. Known deterministic constructions use much larger number of edges. The most recent one by Gebauer requires 2^{k+Θ(k^{2/3})} edges. Applying a derandomization technique we reduce that number to 2^{k+Θ̃(k^{1/2})}.
Cite as
Jakub Kozik. Improving Gebauer’s Construction of 3-Chromatic Hypergraphs with Few Edges. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 89:1-89:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kozik:LIPIcs.ICALP.2021.89,
author = {Kozik, Jakub},
title = {{Improving Gebauer’s Construction of 3-Chromatic Hypergraphs with Few Edges}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {89:1--89:9},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.89},
URN = {urn:nbn:de:0030-drops-141587},
doi = {10.4230/LIPIcs.ICALP.2021.89},
annote = {Keywords: Property B, Hypergraph Coloring, Deterministic Constructions}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth
Abstract
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√{k(n-k)}) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function.
Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf-Yuen conjecture and prove that the SoS rank for SQFs is at most O(√{nk}log(n)).
We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√nlog(n)) SoS certificate for this problem.
Cite as
Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth. SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 90:1-90:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kurpisz_et_al:LIPIcs.ICALP.2021.90,
author = {Kurpisz, Adam and Potechin, Aaron and Wirth, Elias Samuel},
title = {{SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {90:1--90:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.90},
URN = {urn:nbn:de:0030-drops-141595},
doi = {10.4230/LIPIcs.ICALP.2021.90},
annote = {Keywords: symmetric quadratic functions, SoS certificate, hypercube optimization, semidefinite programming}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, and Tobias Friedrich
Abstract
We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold.
First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.
Cite as
J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, and Tobias Friedrich. On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 91:1-91:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lagodzinski_et_al:LIPIcs.ICALP.2021.91,
author = {Lagodzinski, J. A. Gregor and G\"{o}bel, Andreas and Casel, Katrin and Friedrich, Tobias},
title = {{On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {91:1--91:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.91},
URN = {urn:nbn:de:0030-drops-141608},
doi = {10.4230/LIPIcs.ICALP.2021.91},
annote = {Keywords: Algorithms, Theory, Quantum Graphs, Bipartite Graphs, Graph Homomorphisms, Modular Counting, Complexity Dichotomy}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Michael Lampis
Abstract
A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory.
In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (Δ⋅ W)^{O(tw)}n^{O(1)}, where tw is the treewidth, Δ the maximum degree, and W the maximum weight. On the other hand, bounding Δ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and Δ and obtain an FPT algorithm running in time 2^{O(Δtw)}(n+log W)^{O(1)}. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW)^{o(pw)} or 2^{o(Δpw)}(n+log W)^{O(1)}, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1+ε).
Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time Δ^{O(tw)}n^{O(1)}. We show that this is also probably essentially optimal: an algorithm running in n^{o(pw)} would contradict the ETH.
Cite as
Michael Lampis. Minimum Stable Cut and Treewidth. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 92:1-92:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lampis:LIPIcs.ICALP.2021.92,
author = {Lampis, Michael},
title = {{Minimum Stable Cut and Treewidth}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {92:1--92:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.92},
URN = {urn:nbn:de:0030-drops-141616},
doi = {10.4230/LIPIcs.ICALP.2021.92},
annote = {Keywords: Treewidth, Local Max-Cut, Nash Stability}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Reut Levi
Abstract
We study the problem of testing triangle freeness in the general graph model. This problem was first studied in the general graph model by Alon et al. (SIAM J. Discret. Math. 2008) who provided both lower bounds and upper bounds that depend on the number of vertices and the average degree of the graph. Their bounds are tight only when d_max = O(d) and ̄{d} ≤ √n or when ̄{d} = Θ(1), where d_max denotes the maximum degree and ̄{d} denotes the average degree of the graph. In this paper we provide bounds that depend on the arboricity of the graph and the average degree. As in Alon et al., the parameters of our tester is the number of vertices, n, the number of edges, m, and the proximity parameter ε (the arboricity of the graph is not a parameter of the algorithm). The query complexity of our tester is Õ(Γ/ ̄{d} + Γ)⋅ poly(1/ε) on expectation, where Γ denotes the arboricity of the input graph (we use Õ(⋅) to suppress O(log log n) factors). We show that for graphs with arboricity O(√n) this upper bound is tight in the following sense. For any Γ ∈ [s] where s = Θ(√n) there exists a family of graphs with arboricity Γ and average degree ̄{d} such that Ω(Γ/ ̄{d} + Γ) queries are required for testing triangle freeness on this family of graphs. Moreover, this lower bound holds for any such Γ and for a large range of feasible average degrees .
Cite as
Reut Levi. Testing Triangle Freeness in the General Model in Graphs with Arboricity O(√n). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 93:1-93:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{levi:LIPIcs.ICALP.2021.93,
author = {Levi, Reut},
title = {{Testing Triangle Freeness in the General Model in Graphs with Arboricity O(√n)}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {93:1--93:13},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.93},
URN = {urn:nbn:de:0030-drops-141626},
doi = {10.4230/LIPIcs.ICALP.2021.93},
annote = {Keywords: Property Testing, Triangle-Freeness}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Zhenjian Lu and Igor C. Oliveira
Abstract
A probabilistic representation of a string x ∈ {0,1}ⁿ is given by the code of a randomized algorithm that outputs x with high probability [Igor C. Oliveira, 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ensemble 𝒟_m can be uniformly sampled in time T(m) and generates a string x ∈ {0,1}^* with probability at least δ, then x admits a time-bounded probabilistic representation of complexity O(log(1/δ) + log (T) + log(m)). Under mild assumptions, a representation of this form can be computed from x and the code of the sampler in time polynomial in n = |x|.
We derive consequences of this result relevant to the study of data compression, pseudodeterministic algorithms, time hierarchies for sampling distributions, and complexity lower bounds. In particular, we describe an instance-based search-to-decision reduction for Levin’s Kt complexity [Leonid A. Levin, 1984] and its probabilistic analogue rKt [Igor C. Oliveira, 2019]. As a consequence, if a string x admits a succinct time-bounded representation, then a near-optimal representation can be generated from x with high probability in polynomial time. This partially addresses in a time-bounded setting a question from [Leonid A. Levin, 1984] on the efficiency of computing an optimal encoding of a string.
Cite as
Zhenjian Lu and Igor C. Oliveira. An Efficient Coding Theorem via Probabilistic Representations and Its Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 94:1-94:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lu_et_al:LIPIcs.ICALP.2021.94,
author = {Lu, Zhenjian and Oliveira, Igor C.},
title = {{An Efficient Coding Theorem via Probabilistic Representations and Its Applications}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {94:1--94:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.94},
URN = {urn:nbn:de:0030-drops-141630},
doi = {10.4230/LIPIcs.ICALP.2021.94},
annote = {Keywords: computational complexity, randomized algorithms, Kolmogorov complexity}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Dániel Marx, Govind S. Sankar, and Philipp Schepper
Abstract
In the General Factor problem, we are given an undirected graph G and for each vertex v ∈ V(G) a finite set B_v of non-negative integers. The task is to decide if there is a subset S ⊆ E(G) such that deg_S(v) ∈ B_v for all vertices v of G. Define the max-gap of a finite integer set B to be the largest d ≥ 0 such that there is an a ≥ 0 with [a,a+d+1] ∩ B = {a,a+d+1}. Cornuéjols showed in 1988 that if the max-gap of all sets B_v is at most 1, then the decision version of General Factor is polynomial-time solvable. This result was extended 2018 by Dudycz and Paluch for the optimization (i.e. minimization and maximization) versions. We present a general algorithm counting the number of solutions of a certain size in time #2 (M+1)^{tw}^{𝒪(1)}, given a tree decomposition of width tw, where M is the maximum integer over all B_v. By using convolution techniques from van Rooij (2020), we improve upon the previous (M+1)^{3tw}^𝒪(1) time algorithm by Arulselvan et al. from 2018.
We prove that this algorithm is essentially optimal for all cases that are not trivial or polynomial time solvable for the decision, minimization or maximization versions. Our lower bounds show that such an improvement is not even possible for B-Factor, which is General Factor on graphs where all sets B_v agree with the fixed set B. We show that for every fixed B where the problem is NP-hard, our (max B+1)^tw^𝒪(1) algorithm cannot be significantly improved: assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can solve B-Factor in time (max B+1-ε)^tw^𝒪(1) for any ε > 0. We extend this bound to the counting version of B-Factor for arbitrary, non-trivial sets B, assuming #SETH.
We also investigate the parameterization of the problem by cutwidth. Unlike for treewidth, having a larger set B does not appear to make the problem harder: we give a 2^cutw^𝒪(1) algorithm for any B and provide a matching lower bound that this is optimal for the NP-hard cases.
Cite as
Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{marx_et_al:LIPIcs.ICALP.2021.95,
author = {Marx, D\'{a}niel and Sankar, Govind S. and Schepper, Philipp},
title = {{Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {95:1--95:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.95},
URN = {urn:nbn:de:0030-drops-141647},
doi = {10.4230/LIPIcs.ICALP.2021.95},
annote = {Keywords: General Factor, General Matching, Treewidth, Cutwidth}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Theo McKenzie and Sidhanth Mohanty
Abstract
Kahale proved that linear sized sets in d-regular Ramanujan graphs have vertex expansion at least d/2 and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than d/2. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether the vertex expansion of high-girth Ramanujan graphs breaks past the d/2 bound. Our results are two-fold:
1) For every d = p+1 for prime p ≥ 3 and infinitely many n, we exhibit an n-vertex d-regular graph with girth Ω(log_{d-1} n) and vertex expansion of sublinear sized sets bounded by (d+1)/2 whose nontrivial eigenvalues are bounded in magnitude by 2√{d-1}+O(1/(log_{d-1} n)).
2) In any Ramanujan graph with girth Clog_{d-1} n, all sets of size bounded by n^{0.99C/4} have near-lossless vertex expansion (1-o_d(1))d. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara-Bass formula, a trace moment method inspired by Bordenave’s proof of Friedman’s theorem [Bordenave, 2019], and a method of Kahale [Kahale, 1995] to study dispersion of eigenvalues of perturbed graphs.
Cite as
Theo McKenzie and Sidhanth Mohanty. High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 96:1-96:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{mckenzie_et_al:LIPIcs.ICALP.2021.96,
author = {McKenzie, Theo and Mohanty, Sidhanth},
title = {{High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {96:1--96:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.96},
URN = {urn:nbn:de:0030-drops-141655},
doi = {10.4230/LIPIcs.ICALP.2021.96},
annote = {Keywords: expander graphs, Ramanujan graphs, vertex expansion, girth}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang
Abstract
This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows represent the data points. The running time is potentially exponentially smaller than N, the number of data points to be clustered that the relational database represents.
Few relational algorithms are known and this paper offers techniques for designing relational algorithms as well as characterizing their limitations. We show that given two data points as cluster centers, if we cluster points according to their closest centers, it is NP-Hard to approximate the number of points in the clusters on a general relational input. This is trivial for conventional data inputs and this result exemplifies that standard algorithmic techniques may not be directly applied when designing an efficient relational algorithm. This paper then introduces a new method that leverages rejection sampling and the k-means++ algorithm to construct a O(1)-approximate k-means solution.
Cite as
Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang. Relational Algorithms for k-Means Clustering. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 97:1-97:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{moseley_et_al:LIPIcs.ICALP.2021.97,
author = {Moseley, Benjamin and Pruhs, Kirk and Samadian, Alireza and Wang, Yuyan},
title = {{Relational Algorithms for k-Means Clustering}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {97:1--97:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.97},
URN = {urn:nbn:de:0030-drops-141668},
doi = {10.4230/LIPIcs.ICALP.2021.97},
annote = {Keywords: k-means, clustering, approximation, big-data, databases}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Yonatan Nakar and Dana Ron
Abstract
We continue the line of work initiated by Goldreich and Ron (Journal of the ACM, 2017) on testing dynamic environments and propose to pursue a systematic study of the complexity of testing basic dynamic environments and local rules. As a first step, in this work we focus on dynamic environments that correspond to elementary cellular automata that evolve according to threshold rules.
Our main result is the identification of a set of conditions on local rules, and a meta-algorithm that tests evolution according to local rules that satisfy the conditions. The meta-algorithm has query complexity poly(1/ε), is non-adaptive and has one-sided error. We show that all the threshold rules satisfy the set of conditions, and therefore are poly(1/ε)-testable. We believe that this is a rich area of research and suggest a variety of open problems and natural research directions that may extend and expand our results.
Cite as
Yonatan Nakar and Dana Ron. Testing Dynamic Environments: Back to Basics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 98:1-98:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{nakar_et_al:LIPIcs.ICALP.2021.98,
author = {Nakar, Yonatan and Ron, Dana},
title = {{Testing Dynamic Environments: Back to Basics}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {98:1--98:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.98},
URN = {urn:nbn:de:0030-drops-141672},
doi = {10.4230/LIPIcs.ICALP.2021.98},
annote = {Keywords: Property Testing}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Eike Neumann, Joël Ouaknine, and James Worrell
Abstract
We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.
Cite as
Eike Neumann, Joël Ouaknine, and James Worrell. Decision Problems for Second-Order Holonomic Recurrences. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 99:1-99:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{neumann_et_al:LIPIcs.ICALP.2021.99,
author = {Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James},
title = {{Decision Problems for Second-Order Holonomic Recurrences}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {99:1--99:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.99},
URN = {urn:nbn:de:0030-drops-141682},
doi = {10.4230/LIPIcs.ICALP.2021.99},
annote = {Keywords: holonomic sequences, Positivity Problem, Skolem Problem}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ilan Newman and Nithin Varma
Abstract
Estimating the length of the longest increasing subsequence (LIS) in an array is a problem of fundamental importance. Despite the significance of the LIS estimation problem and the amount of attention it has received, there are important aspects of the problem that are not yet fully understood. There are no better lower bounds for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, and also provide novel algorithms that complement our lower bound.
Specifically, we show that for every ε ∈ (0,1), every nonadaptive algorithm that outputs an estimate of the LIS length in an array of length n to within an additive error of ε n has to make log^{Ω(log (1/ε))} n queries. Next, we design nonadaptive LIS estimation algorithms whose complexity decreases as the number of distinct values, r, in the array decreases. We first present a simple algorithm that makes Õ(r/ε³) queries and approximates the LIS length with an additive error bounded by ε n. This algorithm has better complexity than the best previously known adaptive algorithm (Saks and Seshadhri; 2017) for the same problem when r ≪ polylog (n). We use our algorithm to construct a nonadaptive algorithm with query complexity Õ(√r⋅ poly(1/λ)) that, when the LIS is of length at least λ n, outputs a multiplicative Ω(λ)-approximation to the LIS length. Our algorithm improves upon the state of the art nonadaptive LIS estimation algorithm (Rubinstein, Seddighin, Song, and Sun; 2019) in terms of the approximation guarantee.
Finally, we present a O(log n)-query nonadaptive erasure-resilient tester for monotonicity. Our result implies that lower bounds on erasure-resilient testing of monotonicity does not give good lower bounds for LIS estimation. It also implies that nonadaptive tolerant testing is strictly harder than nonadaptive erasure-resilient testing for the natural property of monotonicity.
Cite as
Ilan Newman and Nithin Varma. New Sublinear Algorithms and Lower Bounds for LIS Estimation. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 100:1-100:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{newman_et_al:LIPIcs.ICALP.2021.100,
author = {Newman, Ilan and Varma, Nithin},
title = {{New Sublinear Algorithms and Lower Bounds for LIS Estimation}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {100:1--100:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.100},
URN = {urn:nbn:de:0030-drops-141699},
doi = {10.4230/LIPIcs.ICALP.2021.100},
annote = {Keywords: longest increasing subsequence, monotonicity, distance estimation, sublinear algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Takaaki Nishimoto and Yasuo Tabei
Abstract
Indexing highly repetitive strings (i.e., strings with many repetitions) for fast queries has become a central research topic in string processing, because it has a wide variety of applications in bioinformatics and natural language processing. Although a substantial number of indexes for highly repetitive strings have been proposed thus far, developing compressed indexes that support various queries remains a challenge. The run-length Burrows-Wheeler transform (RLBWT) is a lossless data compression by a reversible permutation of an input string and run-length encoding, and it has received interest for indexing highly repetitive strings. LF and ϕ^{-1} are two key functions for building indexes on RLBWT, and the best previous result computes LF and ϕ^{-1} in O(log log n) time with O(r) words of space for the string length n and the number r of runs in RLBWT. In this paper, we improve LF and ϕ^{-1} so that they can be computed in a constant time with O(r) words of space. Subsequently, we present OptBWTR (optimal-time queries on BWT-runs compressed indexes), the first string index that supports various queries including locate, count, extract queries in optimal time and O(r) words of space.
Cite as
Takaaki Nishimoto and Yasuo Tabei. Optimal-Time Queries on BWT-Runs Compressed Indexes. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 101:1-101:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{nishimoto_et_al:LIPIcs.ICALP.2021.101,
author = {Nishimoto, Takaaki and Tabei, Yasuo},
title = {{Optimal-Time Queries on BWT-Runs Compressed Indexes}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {101:1--101:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.101},
URN = {urn:nbn:de:0030-drops-141702},
doi = {10.4230/LIPIcs.ICALP.2021.101},
annote = {Keywords: Compressed text indexes, Burrows-Wheeler transform, highly repetitive text collections}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Ojas Parekh and Kevin Thompson
Abstract
The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. Here we provide the first ever use of the level-2 hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-2 hierarchy satisfies many physically-motivated constraints that the level-1 does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.
Cite as
Ojas Parekh and Kevin Thompson. Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 102:1-102:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{parekh_et_al:LIPIcs.ICALP.2021.102,
author = {Parekh, Ojas and Thompson, Kevin},
title = {{Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {102:1--102:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.102},
URN = {urn:nbn:de:0030-drops-141718},
doi = {10.4230/LIPIcs.ICALP.2021.102},
annote = {Keywords: Quantum Max Cut, Quantum Approximation Algorithms, Lasserre Hierarchy, Local Hamiltonian, Heisenberg model}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Enoch Peserico and Michele Scquizzato
Abstract
We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds √{log₂(n +1)}/15 for all n = 2ⁱ-1: i ∈ ℕ, settling a 25-year-old open question.
Cite as
Enoch Peserico and Michele Scquizzato. Matching on the Line Admits No o(√log n)-Competitive Algorithm. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 103:1-103:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{peserico_et_al:LIPIcs.ICALP.2021.103,
author = {Peserico, Enoch and Scquizzato, Michele},
title = {{Matching on the Line Admits No o(√log n)-Competitive Algorithm}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {103:1--103:3},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.103},
URN = {urn:nbn:de:0030-drops-141720},
doi = {10.4230/LIPIcs.ICALP.2021.103},
annote = {Keywords: Metric matching, online algorithms, competitive analysis}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Seth Pettie, Dingyu Wang, and Longhui Yin
Abstract
Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance.
We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies 5m bits and whose relative variance is 2/m (standard error √{2/m}) has an MVP of 10. Our contributions are as follows.
- Cohen [Edith Cohen, 2015] and Ting [Daniel Ting, 2014] independently discovered what we call the {Martingale transform} for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches.
- Pettie and Wang proved that the Fishmonger sketch [Seth Pettie and Dingyu Wang, 2021] has the best MVP, H₀/I₀ ≈ 1.98, among a class of mergeable sketches called "linearizable" sketches. (H₀ and I₀ are precisely defined constants.) We prove that the Martingale transform is optimal in the non-mergeable world, and that Martingale Fishmonger in particular is optimal among linearizable sketches, with an MVP of H₀/2 ≈ 1.63. E.g., this is circumstantial evidence that to achieve 1% standard error, we cannot do better than a 2 kilobyte sketch.
- Martingale Fishmonger is neither simple nor practical. We develop a new mergeable sketch called Curtain that strikes a nice balance between simplicity and efficiency, and prove that Martingale Curtain has limiting MVP≈ 2.31. It can be updated with O(1) memory accesses and it has lower empirical variance than Martingale LogLog, a practical non-mergeable version of HyperLogLog.
Cite as
Seth Pettie, Dingyu Wang, and Longhui Yin. Non-Mergeable Sketching for Cardinality Estimation. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{pettie_et_al:LIPIcs.ICALP.2021.104,
author = {Pettie, Seth and Wang, Dingyu and Yin, Longhui},
title = {{Non-Mergeable Sketching for Cardinality Estimation}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {104:1--104:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.104},
URN = {urn:nbn:de:0030-drops-141731},
doi = {10.4230/LIPIcs.ICALP.2021.104},
annote = {Keywords: Cardinality Estimation, Sketching}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Seth Pettie and Longhui Yin
Abstract
In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts.
As a consequence of these investigations, we exhibit a simple O(κ n)-space data structure that can quickly answer pairwise (κ+1)-connectivity queries in a κ-connected graph. We also show how to compute the "closest" κ-cut to every vertex in near linear Õ(m+poly(κ)n) time.
Cite as
Seth Pettie and Longhui Yin. The Structure of Minimum Vertex Cuts. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 105:1-105:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{pettie_et_al:LIPIcs.ICALP.2021.105,
author = {Pettie, Seth and Yin, Longhui},
title = {{The Structure of Minimum Vertex Cuts}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {105:1--105:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.105},
URN = {urn:nbn:de:0030-drops-141746},
doi = {10.4230/LIPIcs.ICALP.2021.105},
annote = {Keywords: Graph theory, vertex connectivity, data structures}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Adam Polak, Lars Rohwedder, and Karol Węgrzycki
Abstract
Knapsack and Subset Sum are fundamental NP-hard problems in combinatorial optimization. Recently there has been a growing interest in understanding the best possible pseudopolynomial running times for these problems with respect to various parameters.
In this paper we focus on the maximum item size s and the maximum item value v. We give algorithms that run in time O(n + s³) and O(n + v³) for the Knapsack problem, and in time Õ(n + s^{5/3}) for the Subset Sum problem.
Our algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance. In these variants n denotes the (possibly much smaller) number of distinct items.
Our results follow from combining and optimizing several diverse lines of research, notably proximity arguments for integer programming due to Eisenbrand and Weismantel (TALG 2019), fast structured (min,+)-convolution by Kellerer and Pferschy (J. Comb. Optim. 2004), and additive combinatorics methods originating from Galil and Margalit (SICOMP 1991).
Cite as
Adam Polak, Lars Rohwedder, and Karol Węgrzycki. Knapsack and Subset Sum with Small Items. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{polak_et_al:LIPIcs.ICALP.2021.106,
author = {Polak, Adam and Rohwedder, Lars and W\k{e}grzycki, Karol},
title = {{Knapsack and Subset Sum with Small Items}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {106:1--106:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.106},
URN = {urn:nbn:de:0030-drops-141752},
doi = {10.4230/LIPIcs.ICALP.2021.106},
annote = {Keywords: Knapsack, Subset Sum, Proximity, Additive Combinatorics, Multiset}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Nicolás Rivera, Thomas Sauerwald, and John Sylvester
Abstract
Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running k multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle in 2008) remains an open problem.
First, we improve and tighten various bounds on the stationary cover time when k random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of Ω((n/k) log n) on the stationary cover time, holding for any n-vertex graph G and any 1 ≤ k = o(nlog n). Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.
Cite as
Nicolás Rivera, Thomas Sauerwald, and John Sylvester. Multiple Random Walks on Graphs: Mixing Few to Cover Many. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 107:1-107:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{rivera_et_al:LIPIcs.ICALP.2021.107,
author = {Rivera, Nicol\'{a}s and Sauerwald, Thomas and Sylvester, John},
title = {{Multiple Random Walks on Graphs: Mixing Few to Cover Many}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {107:1--107:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.107},
URN = {urn:nbn:de:0030-drops-141764},
doi = {10.4230/LIPIcs.ICALP.2021.107},
annote = {Keywords: Multiple Random walks, Markov Chains, Random Walks, Cover Time}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Marc Roth, Johannes Schmitt, and Philip Wellnitz
Abstract
Given a graph property Φ, we consider the problem EdgeSub(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies Φ. Specifically, we study the parameterized complexity of EdgeSub(Φ) and of its counting problem #EdgeSub(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ: the decision problem EdgeSub(Φ) always admits an FPT ("fixed-parameter tractable") algorithm and the counting problem #EdgeSub(Φ) always admits an FPTRAS ("fixed-parameter tractable randomized approximation scheme"). For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #W[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #EdgeSub(Φ) that run in time f(k)⋅ |G|^{o(k/log k)} for any computable function f.
As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #EdgeSub(Φ). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). This approach does not only apply to #EdgeSub(Φ) but also to the more general problem of computing weighted linear combinations of subgraph counts. As a special case of such a linear combination, we introduce a parameterized variant of the Tutte Polynomial T^k_G of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T^k_G(2,1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parameterized complexity of computing the evaluation of T^k_G at every pair of rational coordinates (x,y). In particular, our results give a new proof for the #W[1]-hardness of the problem of counting k-forests in a graph.
Cite as
Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 108:1-108:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{roth_et_al:LIPIcs.ICALP.2021.108,
author = {Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
title = {{Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {108:1--108:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.108},
URN = {urn:nbn:de:0030-drops-141778},
doi = {10.4230/LIPIcs.ICALP.2021.108},
annote = {Keywords: Counting complexity, parameterized complexity, Tutte polynomial}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Amin Saberi and David Wajc
Abstract
Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of 2 of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree Δ = O(log n), which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or for random arrival orders, an answer for fully general adversarial arrivals remained elusive.
We resolve this thirty-year-old conjecture in the affirmative, presenting a (1.9+o(1))-competitive online edge coloring algorithm for general graphs of degree Δ = ω(log n) under vertex arrivals. At the core of our results, and of possible independent interest, is a new online algorithm which rounds a fractional bipartite matching x online under vertex arrivals, guaranteeing that each edge e is matched with probability (1/2+c)⋅ x_e, for a constant c > 0.027.
Cite as
Amin Saberi and David Wajc. The Greedy Algorithm Is not Optimal for On-Line Edge Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 109:1-109:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{saberi_et_al:LIPIcs.ICALP.2021.109,
author = {Saberi, Amin and Wajc, David},
title = {{The Greedy Algorithm Is not Optimal for On-Line Edge Coloring}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {109:1--109:18},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.109},
URN = {urn:nbn:de:0030-drops-141786},
doi = {10.4230/LIPIcs.ICALP.2021.109},
annote = {Keywords: Online algorithms, edge coloring, greedy, online matching}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Joran van Apeldoorn, Sander Gribling, Yinan Li, Harold Nieuwboer, Michael Walter, and Ronald de Wolf
Abstract
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.
Cite as
Joran van Apeldoorn, Sander Gribling, Yinan Li, Harold Nieuwboer, Michael Walter, and Ronald de Wolf. Quantum Algorithms for Matrix Scaling and Matrix Balancing. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 110:1-110:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{vanapeldoorn_et_al:LIPIcs.ICALP.2021.110,
author = {van Apeldoorn, Joran and Gribling, Sander and Li, Yinan and Nieuwboer, Harold and Walter, Michael and de Wolf, Ronald},
title = {{Quantum Algorithms for Matrix Scaling and Matrix Balancing}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {110:1--110:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.110},
URN = {urn:nbn:de:0030-drops-141793},
doi = {10.4230/LIPIcs.ICALP.2021.110},
annote = {Keywords: Matrix scaling, matrix balancing, quantum algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Emanuele Viola
Abstract
Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don't, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.
Cite as
Emanuele Viola. Fourier Conjectures, Correlation Bounds, and Majority. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 111:1-111:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{viola:LIPIcs.ICALP.2021.111,
author = {Viola, Emanuele},
title = {{Fourier Conjectures, Correlation Bounds, and Majority}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {111:1--111:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.111},
URN = {urn:nbn:de:0030-drops-141806},
doi = {10.4230/LIPIcs.ICALP.2021.111},
annote = {Keywords: Fourier analysis, polynomials, Majority, correlation, lower bound, conjectures}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
David P. Woodruff and Samson Zhou
Abstract
We study the classical problem of moment estimation of an underlying vector whose n coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order insertion-only model, then there exist space efficient algorithms with improved dependencies on the approximation parameter ε. In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using 𝒪̃(1/(ε^{4/p})⋅ n^{1-2/p}) bits of memory. Our techniques also give algorithms for F_p moment estimation with p > 2 on arbitrary order insertion-only and turnstile streams, using 𝒪̃(1/(ε^{4/p})⋅ n^{1-2/p}) bits of space and two passes, which is the first optimal multi-pass F_p estimation algorithm up to log n factors. Finally, we give an improved lower bound of Ω(1/(ε²)⋅ n^{1-2/p}) for one-pass insertion-only streams. Our results separate the complexity of this problem both between random and non-random orders, as well as one-pass and multi-pass streams.
Cite as
David P. Woodruff and Samson Zhou. Separations for Estimating Large Frequency Moments on Data Streams. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 112:1-112:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{woodruff_et_al:LIPIcs.ICALP.2021.112,
author = {Woodruff, David P. and Zhou, Samson},
title = {{Separations for Estimating Large Frequency Moments on Data Streams}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {112:1--112:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.112},
URN = {urn:nbn:de:0030-drops-141810},
doi = {10.4230/LIPIcs.ICALP.2021.112},
annote = {Keywords: streaming algorithms, frequency moments, random order, lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Or Zamir
Abstract
The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O^*(2ⁿ) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k = 3,4, better algorithms are known for the k-coloring problem. 3-coloring can be solved in O(1.33ⁿ) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73ⁿ) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k > 4 no improvements over the general O^*(2ⁿ) are known. We show that both 5-coloring and 6-coloring can also be solved in O((2-ε) ⁿ) time for some ε > 0. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ,α > 0, the chromatic number of graphs with at least α⋅ n vertices of degree at most Δ can be computed in O((2-ε) ⁿ) time, for some ε = ε_{Δ,α} > 0. This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajlin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case of bounded-degree graphs.
Cite as
Or Zamir. Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 113:1-113:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{zamir:LIPIcs.ICALP.2021.113,
author = {Zamir, Or},
title = {{Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {113:1--113:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.113},
URN = {urn:nbn:de:0030-drops-141825},
doi = {10.4230/LIPIcs.ICALP.2021.113},
annote = {Keywords: Algorithms, Graph Algorithms, Graph Coloring}
}
Document
Track A: Algorithms, Complexity and Games
Authors:
Tianyi Zhang
Abstract
In this paper we are interested in deterministically computing maximum flows in undirected simple graphs where edges have unit capacities. When the input graph has n vertices and m edges, and the maximum flow is known to be upper bounded by τ as prior knowledge, our algorithm has running time Õ(m + n^{5/3}τ^{1/2}); in the extreme case where τ = Θ(n), our algorithm has running time Õ(n^{2.17}). This always improves upon the previous best deterministic upper bound Õ(n^{9/4}τ^{1/8}) by [Duan, 2013]. Furthermore, when τ ≥ n^{0.67} our algorithm is faster than a classical upper bound of O(m + nτ^{3/2}) by [Karger and Levin, 1998].
Cite as
Tianyi Zhang. Deterministic Maximum Flows in Simple Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 114:1-114:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{zhang:LIPIcs.ICALP.2021.114,
author = {Zhang, Tianyi},
title = {{Deterministic Maximum Flows in Simple Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {114:1--114:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.114},
URN = {urn:nbn:de:0030-drops-141832},
doi = {10.4230/LIPIcs.ICALP.2021.114},
annote = {Keywords: graph algorithms, maximum flows, dynamic data structures}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Samson Abramsky and Luca Reggio
Abstract
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a very general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fraïssé and modal bisimulation games recently introduced in [Abramsky et al., 2017; S. Abramsky and N. Shah, 2018; Abramsky and Shah, 2021] are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.
Cite as
Samson Abramsky and Luca Reggio. Arboreal Categories and Resources. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{abramsky_et_al:LIPIcs.ICALP.2021.115,
author = {Abramsky, Samson and Reggio, Luca},
title = {{Arboreal Categories and Resources}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {115:1--115:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.115},
URN = {urn:nbn:de:0030-drops-141845},
doi = {10.4230/LIPIcs.ICALP.2021.115},
annote = {Keywords: factorisation system, embedding, comonad, coalgebra, open maps, bisimulation, game, resources, relational structures, finite model theory}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Antoine Amarilli, Louis Jachiet, and Charles Paperman
Abstract
We study the dynamic membership problem for regular languages: fix a language L, read a word w, build in time O(|w|) a data structure indicating if w is in L, and maintain this structure efficiently under letter substitutions on w. We consider this problem on the unit cost RAM model with logarithmic word length, where the problem always has a solution in O(log|w| / log log|w|) per operation.
We show that the problem is in O(log log|w|) for languages in an algebraically-defined, decidable class QSG, and that it is in O(1) for another such class QLZG. We show that languages not in QSG admit a reduction from the prefix problem for a cyclic group, so that they require Ω(log|w| /log log|w|) operations in the worst case; and that QSG languages not in QLZG admit a reduction from the prefix problem for the multiplicative monoid U₁ = {0, 1}, which we conjecture cannot be maintained in O(1). This yields a conditional trichotomy. We also investigate intermediate cases between O(1) and O(log log|w|).
Our results are shown via the dynamic word problem for monoids and semigroups, for which we also give a classification. We thus solve open problems of the paper of Skovbjerg Frandsen, Miltersen, and Skyum [Skovbjerg Frandsen et al., 1997] on the dynamic word problem, and additionally cover regular languages.
Cite as
Antoine Amarilli, Louis Jachiet, and Charles Paperman. Dynamic Membership for Regular Languages. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 116:1-116:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{amarilli_et_al:LIPIcs.ICALP.2021.116,
author = {Amarilli, Antoine and Jachiet, Louis and Paperman, Charles},
title = {{Dynamic Membership for Regular Languages}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {116:1--116:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.116},
URN = {urn:nbn:de:0030-drops-141850},
doi = {10.4230/LIPIcs.ICALP.2021.116},
annote = {Keywords: regular language, membership, RAM model, updates, dynamic}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Paolo Baldan, Francesco Ranzato, and Linpeng Zhang
Abstract
Classical results in computability theory, notably Rice’s theorem, focus on the extensional content of programs, namely, on the partial recursive functions that programs compute. Later and more recent work investigated intensional generalisations of such results that take into account the way in which functions are computed, thus affected by the specific programs computing them. In this paper, we single out a novel class of program semantics based on abstract domains of program properties that are able to capture nonextensional aspects of program computations, such as their asymptotic complexity or logical invariants, and allow us to generalise some foundational computability results such as Rice’s Theorem and Kleene’s Second Recursion Theorem to these semantics. In particular, it turns out that for this class of abstract program semantics, any nontrivial abstract property is undecidable and every decidable overapproximation necessarily includes an infinite set of false positives which covers all values of the semantic abstract domain.
Cite as
Paolo Baldan, Francesco Ranzato, and Linpeng Zhang. A Rice’s Theorem for Abstract Semantics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 117:1-117:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{baldan_et_al:LIPIcs.ICALP.2021.117,
author = {Baldan, Paolo and Ranzato, Francesco and Zhang, Linpeng},
title = {{A Rice’s Theorem for Abstract Semantics}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {117:1--117:19},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.117},
URN = {urn:nbn:de:0030-drops-141860},
doi = {10.4230/LIPIcs.ICALP.2021.117},
annote = {Keywords: Computability Theory, Recursive Function, Rice’s Theorem, Kleene’s Second Recursion Theorem, Program Analysis, Affine Program Invariants}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Borja Balle, Clara Lacroce, Prakash Panangaden, Doina Precup, and Guillaume Rabusseau
Abstract
We address the approximate minimization problem for weighted finite automata (WFAs) with weights in ℝ, over a one-letter alphabet: to compute the best possible approximation of a WFA given a bound on the number of states. This work is grounded in Adamyan-Arov-Krein approximation theory, a remarkable collection of results on the approximation of Hankel operators. In addition to its intrinsic mathematical relevance, this theory has proven to be very effective for model reduction. We adapt these results to the framework of weighted automata over a one-letter alphabet. We provide theoretical guarantees and bounds on the quality of the approximation in the spectral and 𝓁² norm. We develop an algorithm that, based on the properties of Hankel operators, returns the optimal approximation in the spectral norm.
Cite as
Borja Balle, Clara Lacroce, Prakash Panangaden, Doina Precup, and Guillaume Rabusseau. Optimal Spectral-Norm Approximate Minimization of Weighted Finite Automata. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 118:1-118:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{balle_et_al:LIPIcs.ICALP.2021.118,
author = {Balle, Borja and Lacroce, Clara and Panangaden, Prakash and Precup, Doina and Rabusseau, Guillaume},
title = {{Optimal Spectral-Norm Approximate Minimization of Weighted Finite Automata}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {118:1--118:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.118},
URN = {urn:nbn:de:0030-drops-141873},
doi = {10.4230/LIPIcs.ICALP.2021.118},
annote = {Keywords: Weighted finite automata, approximate minimization, Hankel matrices, AAK Theory}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Gabriel Bathie and Tatiana Starikovskaya
Abstract
In this work, we revisit the problem of testing membership in regular languages, first studied by Alon et al. [Alon et al., 2001]. We develop a one-sided error property tester for regular languages under weighted edit distance that makes 𝒪(ε^{-1} log(1/ε)) non-adaptive queries, assuming that the language is described by an automaton of constant size. Moreover, we show a matching lower bound, essentially closing the problem for the edit distance. As an application, we improve the space bound of the current best streaming property testing algorithm for visibly pushdown languages from 𝒪(ε^{-4} log⁶ n) to 𝒪(ε^{-3} log⁵ n log log n), where n is the size of the input. Finally, we provide a Ω(max(ε^{-1}, log n)) lower bound on the memory necessary to test visibly pushdown languages in the streaming model, significantly narrowing the gap between the known bounds.
Cite as
Gabriel Bathie and Tatiana Starikovskaya. Property Testing of Regular Languages with Applications to Streaming Property Testing of Visibly Pushdown Languages. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 119:1-119:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bathie_et_al:LIPIcs.ICALP.2021.119,
author = {Bathie, Gabriel and Starikovskaya, Tatiana},
title = {{Property Testing of Regular Languages with Applications to Streaming Property Testing of Visibly Pushdown Languages}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {119:1--119:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.119},
URN = {urn:nbn:de:0030-drops-141881},
doi = {10.4230/LIPIcs.ICALP.2021.119},
annote = {Keywords: property testing, regular languages, streaming algorithms, visibly pushdown languages}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Manuel Bodirsky, Simon Knäuer, and Sebastian Rudolph
Abstract
We characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all 𝓁,k ∈ , there exists a canonical Datalog program Π of width (𝓁,k), that is, a Datalog program of width (𝓁,k) which is sound for C (i.e., Π only derives the goal predicate on a finite structure 𝔄 if 𝔄 ∈ C) and with the property that Π derives the goal predicate whenever some Datalog program of width (𝓁,k) which is sound for C derives the goal predicate. The same characterisations also hold for Guarded Second-order Logic (GSO), which properly extends MSO. To prove our results, we show that every class C in GSO whose complement is closed under homomorphisms is a finite union of constraint satisfaction problems (CSPs) of ω-categorical structures.
Cite as
Manuel Bodirsky, Simon Knäuer, and Sebastian Rudolph. Datalog-Expressibility for Monadic and Guarded Second-Order Logic. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 120:1-120:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2021.120,
author = {Bodirsky, Manuel and Kn\"{a}uer, Simon and Rudolph, Sebastian},
title = {{Datalog-Expressibility for Monadic and Guarded Second-Order Logic}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {120:1--120:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.120},
URN = {urn:nbn:de:0030-drops-141897},
doi = {10.4230/LIPIcs.ICALP.2021.120},
annote = {Keywords: Monadic Second-order Logic, Guarded Second-order Logic, Datalog, constraint satisfaction, homomorphism-closed, conjunctive query, primitive positive formula, pebble game, \omega-categoricity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Alex Brandts and Stanislav Živný
Abstract
The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs.
1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (𝟏-in-𝟑,NAE) is tractable.
We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (𝐭-in-𝐤, NAE) by either adding tuples to 𝐭-in-𝐤 or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA'20]. For the latter, we classify all templates as either tractable or NP-hard.
Cite as
Alex Brandts and Stanislav Živný. Beyond PCSP(1-in-3, NAE). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 121:1-121:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{brandts_et_al:LIPIcs.ICALP.2021.121,
author = {Brandts, Alex and \v{Z}ivn\'{y}, Stanislav},
title = {{Beyond PCSP(1-in-3, NAE)}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {121:1--121:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.121},
URN = {urn:nbn:de:0030-drops-141902},
doi = {10.4230/LIPIcs.ICALP.2021.121},
annote = {Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Antonin Callard and Pascal Vanier
Abstract
Subshifts of finite type (SFTs) are sets of colorings of the plane that avoid a finite family of forbidden patterns. In this article, we are interested in the behavior of the growth of the number of valid patterns in SFTs. While entropy h corresponds to growths that are squared exponential 2^{hn²}, surface entropy (introduced in Pace’s thesis in 2018) corresponds to the eventual linear term in exponential growths. We give here a characterization of the possible surface entropies of SFTs as the Π₃ real numbers of [0,+∞].
Cite as
Antonin Callard and Pascal Vanier. Computational Characterization of Surface Entropies for ℤ² Subshifts of Finite Type. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 122:1-122:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{callard_et_al:LIPIcs.ICALP.2021.122,
author = {Callard, Antonin and Vanier, Pascal},
title = {{Computational Characterization of Surface Entropies for \mathbb{Z}² Subshifts of Finite Type}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {122:1--122:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.122},
URN = {urn:nbn:de:0030-drops-141914},
doi = {10.4230/LIPIcs.ICALP.2021.122},
annote = {Keywords: surface entropy, arithmetical hierarchy of real numbers, 2D subshifts, symbolic dynamics}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Antonio Casares, Thomas Colcombet, and Nathanaël Fijalkow
Abstract
We consider the following question: given an automaton or a game with a Muller condition, how can we efficiently construct an equivalent one with a parity condition? There are several examples of such transformations in the literature, including in the determinisation of Büchi automata.
We define a new transformation called the alternating cycle decomposition, inspired and extending Zielonka’s construction. Our transformation operates on transition systems, encompassing both automata and games, and preserves semantic properties through the existence of a locally bijective morphism. We show a strong optimality result: the obtained parity transition system is minimal both in number of states and number of priorities with respect to locally bijective morphisms.
We give two applications: the first is related to the determinisation of Büchi automata, and the second is to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions.
Cite as
Antonio Casares, Thomas Colcombet, and Nathanaël Fijalkow. Optimal Transformations of Games and Automata Using Muller Conditions. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 123:1-123:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{casares_et_al:LIPIcs.ICALP.2021.123,
author = {Casares, Antonio and Colcombet, Thomas and Fijalkow, Nathana\"{e}l},
title = {{Optimal Transformations of Games and Automata Using Muller Conditions}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {123:1--123:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.123},
URN = {urn:nbn:de:0030-drops-141928},
doi = {10.4230/LIPIcs.ICALP.2021.123},
annote = {Keywords: Automata over infinite words, Omega regular languages, Determinisation of automata}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, and Alexander Svozil
Abstract
Graphs and games on graphs are fundamental models for the analysis of reactive systems, in particular, for model-checking and the synthesis of reactive systems. The class of ω-regular languages provides a robust specification formalism for the desired properties of reactive systems. In the classical infinitary formulation of the liveness part of an ω-regular specification, a "good" event must happen eventually without any bound between the good events. A stronger notion of liveness is bounded liveness, which requires that good events happen within d transitions. Given a graph or a game graph with n vertices, m edges, and a bounded liveness objective, the previous best-known algorithmic bounds are as follows: (i) O(dm) for graphs, which in the worst-case is O(n³); and (ii) O(n² d²) for games on graphs. Our main contributions improve these long-standing algorithmic bounds. For graphs we present: (i) a randomized algorithm with one-sided error with running time O(n^{2.5} log n) for the bounded liveness objectives; and (ii) a deterministic linear-time algorithm for the complement of bounded liveness objectives. For games on graphs, we present an O(n² d) time algorithm for the bounded liveness objectives.
Cite as
Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, and Alexander Svozil. Faster Algorithms for Bounded Liveness in Graphs and Game Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 124:1-124:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chatterjee_et_al:LIPIcs.ICALP.2021.124,
author = {Chatterjee, Krishnendu and Henzinger, Monika and Kale, Sagar Sudhir and Svozil, Alexander},
title = {{Faster Algorithms for Bounded Liveness in Graphs and Game Graphs}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {124:1--124:21},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.124},
URN = {urn:nbn:de:0030-drops-141930},
doi = {10.4230/LIPIcs.ICALP.2021.124},
annote = {Keywords: Graphs, Game Graphs, B\"{u}chi}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Luca Ciccone and Luca Padovani
Abstract
Many properties of communication protocols stem from the combination of safety and liveness properties. Characterizing such combined properties by means of a single inference system is difficult because of the fundamentally different techniques (coinduction and induction, respectively) usually involved in defining and proving them. In this paper we show that Generalized Inference Systems allow for simple and insightful characterizations of (at least some of) these combined inductive/coinductive properties for dependent session types. In particular, we illustrate the role of corules in characterizing weak termination (the property of protocols that can always eventually terminate), fair compliance (the property of interactions that can always be extended to reach client satisfaction) and also fair subtyping, a liveness-preserving refinement relation for session types.
Cite as
Luca Ciccone and Luca Padovani. Inference Systems with Corules for Fair Subtyping and Liveness Properties of Binary Session Types. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 125:1-125:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ciccone_et_al:LIPIcs.ICALP.2021.125,
author = {Ciccone, Luca and Padovani, Luca},
title = {{Inference Systems with Corules for Fair Subtyping and Liveness Properties of Binary Session Types}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {125:1--125:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.125},
URN = {urn:nbn:de:0030-drops-141941},
doi = {10.4230/LIPIcs.ICALP.2021.125},
annote = {Keywords: Inference systems, session types, safety, liveness, induction, coinduction}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Lorenzo Clemente and Michał Skrzypczak
Abstract
We show that it is decidable whether two regular languages of infinite trees are separable by a deterministic language, resp., a game language. We consider two variants of separability, depending on whether the set of priorities of the separator is fixed, or not. In each case, we show that separability can be decided in EXPTIME, and that separating automata of exponential size suffice. We obtain our results by reducing to infinite duration games with ω-regular winning conditions and applying the finite-memory determinacy theorem of Büchi and Landweber.
Cite as
Lorenzo Clemente and Michał Skrzypczak. Deterministic and Game Separability for Regular Languages of Infinite Trees. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 126:1-126:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{clemente_et_al:LIPIcs.ICALP.2021.126,
author = {Clemente, Lorenzo and Skrzypczak, Micha{\l}},
title = {{Deterministic and Game Separability for Regular Languages of Infinite Trees}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {126:1--126:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.126},
URN = {urn:nbn:de:0030-drops-141952},
doi = {10.4230/LIPIcs.ICALP.2021.126},
annote = {Keywords: separation, infinite trees, regular languages, deterministic automata, game automata}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Thomas Colcombet and Arthur Jaquard
Abstract
In this paper, we initiate a study of the expressive power of tree algebras, and more generally infinitely sorted algebras, based on their asymptotic complexity. We provide a characterization of the expressiveness of tree algebras of bounded complexity.
Tree algebras in many of their forms, such as clones, hyperclones, operads, etc, as well as other kind of algebras, are infinitely sorted: the carrier is a multi sorted set indexed by a parameter that can be interpreted as the number of variables or hole types. Finite such algebras - meaning when all sorts are finite - can be classified depending on the asymptotic size of the carrier sets as a function of the parameter, that we call the complexity of the algebra. This naturally defines the notions of algebras of bounded, linear, polynomial, exponential or doubly exponential complexity...
We initiate in this work a program of analysis of the complexity of infinitely sorted algebras. Our main result precisely characterizes the tree algebras of bounded complexity based on the languages that they recognize as Boolean closures of simple languages. Along the way, we prove that such algebras that are syntactic (minimal for a language) are exactly those in which, as soon as there are sufficiently many variables, the elements are invariant under permutation of the variables.
Cite as
Thomas Colcombet and Arthur Jaquard. A Complexity Approach to Tree Algebras: the Bounded Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 127:1-127:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{colcombet_et_al:LIPIcs.ICALP.2021.127,
author = {Colcombet, Thomas and Jaquard, Arthur},
title = {{A Complexity Approach to Tree Algebras: the Bounded Case}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {127:1--127:13},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.127},
URN = {urn:nbn:de:0030-drops-141966},
doi = {10.4230/LIPIcs.ICALP.2021.127},
annote = {Keywords: Tree algebra, infinite tree, language theory}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Wojciech Czerwiński, Sławomir Lasota, and Łukasz Orlikowski
Abstract
We investigate computational complexity of the reachability problem for vector addition systems (or, equivalently, Petri nets), the central algorithmic problem in verification of concurrent systems. Concerning its complexity, after 40 years of stagnation, a non-elementary lower bound has been shown recently: the problem needs a tower of exponentials of time or space, where the height of tower is linear in the input size. We improve on this lower bound, by increasing the height of tower from linear to exponential. As a side-effect, we obtain better lower bounds for vector addition systems of fixed dimension.
Cite as
Wojciech Czerwiński, Sławomir Lasota, and Łukasz Orlikowski. Improved Lower Bounds for Reachability in Vector Addition Systems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 128:1-128:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{czerwinski_et_al:LIPIcs.ICALP.2021.128,
author = {Czerwi\'{n}ski, Wojciech and Lasota, S{\l}awomir and Orlikowski, {\L}ukasz},
title = {{Improved Lower Bounds for Reachability in Vector Addition Systems}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {128:1--128:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.128},
URN = {urn:nbn:de:0030-drops-141973},
doi = {10.4230/LIPIcs.ICALP.2021.128},
annote = {Keywords: Petri nets, vector addition systems, reachability problem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Wojciech Czerwiński, Antoine Mottet, and Karin Quaas
Abstract
Register automata are finite automata equipped with a finite set of registers ranging over the domain of some relational structure like (ℕ; =) or (ℚ; <). Register automata process words over the domain, and along a run of the automaton, the registers can store data from the input word for later comparisons. It is long known that the universality problem, i.e., the problem to decide whether a given register automaton accepts all words over the domain, is undecidable. Recently, we proved the problem to be decidable in 2-ExpSpace if the register automaton under study is over (ℕ; =) and unambiguous, i.e., every input word has at most one accepting run; this result was shortly after improved to 2-ExpTime by Barloy and Clemente. In this paper, we go one step further and prove that the problem is in ExpSpace, and in PSpace if the number of registers is fixed. Our proof is based on new techniques that additionally allow us to show that the problem is in PSpace for single-register automata over (ℚ; <). As a third technical contribution we prove that the problem is decidable (in ExpSpace) for a more expressive model of unambiguous register automata, where the registers can take values nondeterministically, if defined over (ℕ; =) and only one register is used.
Cite as
Wojciech Czerwiński, Antoine Mottet, and Karin Quaas. New Techniques for Universality in Unambiguous Register Automata. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 129:1-129:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{czerwinski_et_al:LIPIcs.ICALP.2021.129,
author = {Czerwi\'{n}ski, Wojciech and Mottet, Antoine and Quaas, Karin},
title = {{New Techniques for Universality in Unambiguous Register Automata}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {129:1--129:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.129},
URN = {urn:nbn:de:0030-drops-141983},
doi = {10.4230/LIPIcs.ICALP.2021.129},
annote = {Keywords: Register Automata, Data Languages, Unambiguity, Unambiguous, Universality, Containment, Language Inclusion, Equivalence}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Dominik D. Freydenberger and Liat Peterfreund
Abstract
We propose FC, a new logic on words that combines finite model theory with the theory of concatenation - a first-order logic that is based on word equations. Like the theory of concatenation, FC is built around word equations; in contrast to it, its semantics are defined to only allow finite models, by limiting the universe to a word and all its factors. As a consequence of this, FC has many of the desirable properties of FO on finite models, while being far more expressive than FO[<]. Most noteworthy among these desirable properties are sufficient criteria for efficient model checking, and capturing various complexity classes by adding operators for transitive closures or fixed points.
Not only does FC allow us to obtain new insights and techniques for expressive power and efficient evaluation of document spanners, but it also provides a general framework for logic on words that also has potential applications in other areas.
Cite as
Dominik D. Freydenberger and Liat Peterfreund. The Theory of Concatenation over Finite Models. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 130:1-130:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{freydenberger_et_al:LIPIcs.ICALP.2021.130,
author = {Freydenberger, Dominik D. and Peterfreund, Liat},
title = {{The Theory of Concatenation over Finite Models}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {130:1--130:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.130},
URN = {urn:nbn:de:0030-drops-141997},
doi = {10.4230/LIPIcs.ICALP.2021.130},
annote = {Keywords: finite model theory, word equations, descriptive complexity, model checking, document spanners}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Sergey Goncharov
Abstract
Category theory is famous for its innovative way of thinking of concepts by their descriptions, in particular by establishing universal properties. Concepts that can be characterized in a universal way receive a certain quality seal, which makes them easily transferable across application domains. The notion of partiality is however notoriously difficult to characterize in this way, although the importance of it is certain, especially for computer science where entire research areas, such as synthetic and axiomatic domain theory revolve around it. More recently, this issue resurfaced in the context of (constructive) intensional type theory. Here, we provide a generic categorical iteration-based notion of partiality, which is arguably the most basic one. We show that the emerging free structures, which we dub uniform-iteration algebras enjoy various desirable properties, in particular, yield an equational lifting monad. We then study the impact of classicality assumptions and choice principles on this monad, in particular, we establish a suitable categorial formulation of the axiom of countable choice entailing that the monad is an Elgot monad.
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Sergey Goncharov. Uniform Elgot Iteration in Foundations. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 131:1-131:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{goncharov:LIPIcs.ICALP.2021.131,
author = {Goncharov, Sergey},
title = {{Uniform Elgot Iteration in Foundations}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {131:1--131:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.131},
URN = {urn:nbn:de:0030-drops-142007},
doi = {10.4230/LIPIcs.ICALP.2021.131},
annote = {Keywords: Elgot monad, partiality monad, delay monad, restriction category}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Authors:
Alexandre Goy, Daniela Petrişan, and Marc Aiguier
Abstract
The powerset monad on the category of sets does not distribute over itself. Nevertheless a weaker form of distributive law of the powerset monad over itself exists and it essentially stems from the canonical Egli-Milner extension of the powerset to the category of relations. On the other hand, any regular category yields a category of relations, and some regular categories also possess a powerset-like monad, as is the Vietoris monad on compact Hausdorff spaces. We derive the Egli-Milner extension in three different frameworks : sets, toposes, and compact Hausdorff spaces. We prove that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not. We provide an application to coalgebraic determinization of alternating automata.
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Alexandre Goy, Daniela Petrişan, and Marc Aiguier. Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 132:1-132:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{goy_et_al:LIPIcs.ICALP.2021.132,
author = {Goy, Alexandre and Petri\c{s}an, Daniela and Aiguier, Marc},
title = {{Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces}},
booktitle = {48th International Colloquium on Automata, Languages, and Programm