Volume
LIPIcs, Volume 219
39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
Publication Details
- published at: 2022-03-09
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-222-8
- DBLP: db/conf/stacs/stacs2022
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Complete Volume
LIPIcs, Volume 219, STACS 2022, Complete Volume
Abstract
LIPIcs, Volume 219, STACS 2022, Complete Volume
Cite as
39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 1-1044, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{berenbrink_et_al:LIPIcs.STACS.2022,
title = {{LIPIcs, Volume 219, STACS 2022, Complete Volume}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {1--1044},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022},
URN = {urn:nbn:de:0030-drops-158098},
doi = {10.4230/LIPIcs.STACS.2022},
annote = {Keywords: LIPIcs, Volume 219, STACS 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{berenbrink_et_al:LIPIcs.STACS.2022.0,
author = {Berenbrink, Petra and Monmege, Benjamin},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.0},
URN = {urn:nbn:de:0030-drops-158101},
doi = {10.4230/LIPIcs.STACS.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Local Limit of Random Discrete Surface with (Or Without!) a Statistical Physics Model (Invited Talk)
Abstract
A planar map is an embedding of a planar graph in the sphere, considered up to deformations. A triangulation is a planar map, where all the faces are triangles.
In 2003, in order to define a model of generic planar geometry, Angel and Schramm studied the limit of random triangulations on the sphere, [Angel and Schramm, 2003]. They proved that this model of random maps converges for the Benjamini-Schramm topology (see [Benjamini and Schramm, 2001]), or local topology, towards the now famous Uniform Infinite Planar Triangulation (or UIPT), a probability distribution on infinite triangulations, see Figure 1. Soon after, Angel [Angel, 2003] studied some properties of the UIPT. He established that the volume of the balls the UIPT of radius R scales as R⁴. Similar results (but with quite different proofs) were then obtained for quadrangulations by Chassaing and Durhuus and Krikun.
The results cited above deal with models of maps that fall in the same "universality class", identified in the physics literature as the class of "pure 2D quantum gravity": the generating series all admit the same critical exponent and the volume of the balls of the local limits of several of those models of random maps are known to grow as R⁴. To capture this universal behaviour, a good framework is to consider scaling limits of random maps in the Gromov Hausdorff topology. Indeed, for a wide variety of models the scaling limit exists and is the so-called Brownian map [Le Gall, 2013; Miermont, 2013], see Figure 2.
To escape this pure gravity behaviour, physicists have long ago understood that one should "couple gravity with matter", that is, consider models of random maps endowed with a statistical physics model. I will present in particular the case of triangulations decorated by an Ising model. It consists in colouring in black and white the vertices of a triangulation, and consider probability distribution which are now biased by their number of monochromatic edges. In a recent work, in collaboration with Laurent Ménard and Gilles Schaeffer [Albenque et al., 2020], we proved that the local limit of this model also exists.
In this talk, I will present these results and explain the main ideas underlying their proof, which rely in part on some enumerative formulas obtained by Tutte in the 60s [Tutte, 1962], or their generalization to coloured triangulations by Bernardi and Bousquet-Mélou [Bernardi and Bousquet-Mélou, 2011].
Cite as
Marie Albenque. Local Limit of Random Discrete Surface with (Or Without!) a Statistical Physics Model (Invited Talk). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{albenque:LIPIcs.STACS.2022.1,
author = {Albenque, Marie},
title = {{Local Limit of Random Discrete Surface with (Or Without!) a Statistical Physics Model}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {1:1--1:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.1},
URN = {urn:nbn:de:0030-drops-158118},
doi = {10.4230/LIPIcs.STACS.2022.1},
annote = {Keywords: Random graphs, triangulations, Benjamini-Schramm convergence, Ising model}
}
Document
Invited Talk
Generalization Guarantees for Data-Driven Mechanism Design (Invited Talk)
Abstract
Many mechanisms including pricing mechanisms and auctions typically come with a variety of tunable parameters which impact significantly their desired performance guarantees. Data-driven mechanism design is a powerful approach for designing mechanisms, where these parameters are tuned via machine learning based on data. In this talk I will discuss how techniques from machine learning theory can be adapted and extended to analyze generalization guarantees of data-driven mechanism design.
Cite as
Maria-Florina Balcan. Generalization Guarantees for Data-Driven Mechanism Design (Invited Talk). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{balcan:LIPIcs.STACS.2022.2,
author = {Balcan, Maria-Florina},
title = {{Generalization Guarantees for Data-Driven Mechanism Design}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.2},
URN = {urn:nbn:de:0030-drops-158127},
doi = {10.4230/LIPIcs.STACS.2022.2},
annote = {Keywords: mechanism configuration, algorithm configuration, machine learning, generalization guarantees}
}
Document
Invited Talk
Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring (Invited Talk)
Abstract
The problem of obtaining fast deterministic algorithms for distributed symmetry breaking problems in graphs has long been considered one of the most challenging problems in the area of distributed graph algorithms. Consider for example the distributed coloring problem, where a (computer) network is modeled by an arbitrary graph G = (V,E) and the objective is to compute a vertex coloring of G by running a distributed algorithm on the graph G. It is maybe not surprising that randomization can be a helpful tool to efficiently compute such a coloring. In fact, as long as each node v ∈ V can choose among deg(v)+1 different colors, even an almost trivial algorithm in which all nodes keep trying a random available color allows to color all nodes in O(log n) parallel steps. How to obtain a similarly efficient deterministic distributed coloring algorithm is far less obvious. In fact, for a long time, there has been an exponential gap between the time complexities of the best randomized and the best deterministic distributed algorithms for various graph coloring variants and for many other basic graph problems. In the last few years, there however has been substantial progress on deterministic distributed graph algorithms that are nearly as fast as randomized algorithms for the same tasks. In particular, in a recent breakthrough, Rozhoň and Ghaffari managed to reduce the gap between the randomized and deterministic complexities of locally checkable graph problems to at most polylog n.
In the talk, we give a brief overview of the history of the problem of finding fast deterministic algorithms for distributed symmetry breaking problems and of what we know about the relation between deterministic and randomized distributed algorithms for such problems. Together with some additional recent developments, the result of Rozhoň and Ghaffari provides a generic, somewhat brute-force way to efficiently derandomize randomized distributed algorithms. Apart from this, there has also been substantial progress on more direct, problem-specific algorithms. In the talk, we in particular discuss some novel deterministic distributed graph coloring algorithms. The algorithms are signficantly faster and we believe also simpler than previous algorithms for the same coloring problems.
Cite as
Fabian Kuhn. Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring (Invited Talk). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kuhn:LIPIcs.STACS.2022.3,
author = {Kuhn, Fabian},
title = {{Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.3},
URN = {urn:nbn:de:0030-drops-158131},
doi = {10.4230/LIPIcs.STACS.2022.3},
annote = {Keywords: distributed graph algorithms, derandomization, distributed coloring}
}
Document
Mapping Networks via Parallel kth-Hop Traceroute Queries
Abstract
For a source node, v, and target node, w, the traceroute command iteratively issues "kth-hop" queries, for k = 1, 2, … , δ(v,w), which return the name of the kth vertex on a shortest path from v to w, where δ(v,w) is the distance between v and w, that is, the number of edges in a shortest-path from v to w. The traceroute command is often used for network mapping applications, the study of the connectivity of networks, and it has been studied theoretically with respect to biases it introduces for network mapping when only a subset of nodes in the network can be the source of traceroute queries. In this paper, we provide efficient network mapping algorithms, that are based on kth-hop traceroute queries. Our results include an algorithm that runs in a constant number of parallel rounds with a subquadratic number of queries under reasonable assumptions about the sampling coverage of the nodes that may issue kth-hop traceroute queries. In addition, we introduce a number of new algorithmic techniques, including a high-probability parametric parallelization of a graph clustering technique of Thorup and Zwick, which may be of independent interest.
Cite as
Ramtin Afshar, Michael T. Goodrich, Pedro Matias, and Martha C. Osegueda. Mapping Networks via Parallel kth-Hop Traceroute Queries. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{afshar_et_al:LIPIcs.STACS.2022.4,
author = {Afshar, Ramtin and Goodrich, Michael T. and Matias, Pedro and Osegueda, Martha C.},
title = {{Mapping Networks via Parallel kth-Hop Traceroute Queries}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {4:1--4:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.4},
URN = {urn:nbn:de:0030-drops-158142},
doi = {10.4230/LIPIcs.STACS.2022.4},
annote = {Keywords: Network mapping, graph algorithms, parallel algorithms, distributed computing, query complexity, kth-hop queries}
}
Document
On Robustness for the Skolem and Positivity Problems
Abstract
The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: The best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006).
In this paper, we consider problems that lie between the initialized and uninitialized variant. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighborhood is given as part of the input, then robust Skolem and robust positivity are Diophantine-hard, i.e., solving either would entail major breakthrough in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine-hardness result on a variant of the Skolem problem, to the best of our knowledge. On the other hand, if one asks whether such a neighborhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations.
Cite as
S. Akshay, Hugo Bazille, Blaise Genest, and Mihir Vahanwala. On Robustness for the Skolem and Positivity Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{akshay_et_al:LIPIcs.STACS.2022.5,
author = {Akshay, S. and Bazille, Hugo and Genest, Blaise and Vahanwala, Mihir},
title = {{On Robustness for the Skolem and Positivity Problems}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {5:1--5:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.5},
URN = {urn:nbn:de:0030-drops-158156},
doi = {10.4230/LIPIcs.STACS.2022.5},
annote = {Keywords: Skolem problem, verification, dynamical systems, robustness}
}
Document
Approximability of Robust Network Design: The Directed Case
Abstract
We consider robust network design problems where an uncertain traffic vector belonging to a polytope has to be dynamically routed to minimize either the network congestion or some linear reservation cost. We focus on the variant in which the underlying graph is directed. We prove that an O(√k) = O(n)-approximation can be obtained by solving the problem under static routing, where k is the number of commodities and n is the number of nodes. This improves previous results of Hajiaghayi et al. [SODA'2005] and matches the Ω(n) lower bound of Ene et al. [STOC'2016] and the Ω(√k) lower bound of Azar et al. [STOC'2003]. Finally, we introduce a slightly more general problem version where some flow restrictions can be added. We show that it cannot be approximated within a ratio of k^{c/(log log k)} (resp. n^{c/(log log n)}) for some constant c. Making use of a weaker complexity assumption, we prove that there is no approximation within a factor of 2^{log^{1- ε} k} (resp. 2^{log^{1- ε} n}) for any ε > 0.
Cite as
Yacine Al-Najjar, Walid Ben-Ameur, and Jérémie Leguay. Approximability of Robust Network Design: The Directed Case. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{alnajjar_et_al:LIPIcs.STACS.2022.6,
author = {Al-Najjar, Yacine and Ben-Ameur, Walid and Leguay, J\'{e}r\'{e}mie},
title = {{Approximability of Robust Network Design: The Directed Case}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {6:1--6:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.6},
URN = {urn:nbn:de:0030-drops-158165},
doi = {10.4230/LIPIcs.STACS.2022.6},
annote = {Keywords: Robust Optimization, Network Design, Approximation, Inapproximability, Competitive Ratio of Oblivious Routing}
}
Document
Existential Definability over the Subword Ordering
Abstract
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the Σ₁ (i.e., existential) fragment is undecidable, already for binary alphabets A.
However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable.
We show that if |A| ≥ 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments Σ_i: If |A| ≥ 3, then a relation is definable in Σ_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the Σ_i-fragments for i ≥ 2 of the pure logic, where the words of A^* are not available as constants.
Cite as
Pascal Baumann, Moses Ganardi, Ramanathan S. Thinniyam, and Georg Zetzsche. Existential Definability over the Subword Ordering. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{baumann_et_al:LIPIcs.STACS.2022.7,
author = {Baumann, Pascal and Ganardi, Moses and Thinniyam, Ramanathan S. and Zetzsche, Georg},
title = {{Existential Definability over the Subword Ordering}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.7},
URN = {urn:nbn:de:0030-drops-158178},
doi = {10.4230/LIPIcs.STACS.2022.7},
annote = {Keywords: subword, subsequence, definability, expressiveness, first order logic, existential fragment, quantifier alternation}
}
Document
Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order
Abstract
The intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate the intrinsic complexity of unary total recursive functions on nonnegative integers with standard order. According to existing results, the possible spectra of such functions include three sets consisting of precisely: the computable degree, all c.e. degrees and all Δ₂ degrees. These results, however, fall far short of the full classification. In this paper, we obtain a more complete picture by giving a few criteria for a function to have intrinsic complexity equal to one of the three candidate sets of degrees. Our investigations are based on the notion of block functions and a broader class of quasi-block functions beyond which all functions of interest have intrinsic complexity equal to the c.e. degrees. We also answer the questions raised by Wright [Wright, 2018] and Harrison-Trainor [Harrison-Trainor, 2018] by showing that the division between computable, c.e. and Δ₂ degrees is insufficient in this context as there is a unary total recursive function whose spectrum contains all c.e. degrees but is strictly contained in the Δ₂ degrees.
Cite as
Nikolay Bazhenov, Dariusz Kalociński, and Michał Wrocławski. Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bazhenov_et_al:LIPIcs.STACS.2022.8,
author = {Bazhenov, Nikolay and Kaloci\'{n}ski, Dariusz and Wroc{\l}awski, Micha{\l}},
title = {{Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.8},
URN = {urn:nbn:de:0030-drops-158185},
doi = {10.4230/LIPIcs.STACS.2022.8},
annote = {Keywords: Computable Structure Theory, Degree Spectra, \omega-Type Order, c.e. Degrees, d.c.e. Degrees, \Delta₂ Degrees, Learnability}
}
Document
Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs
Abstract
On sparse graphs, Roditty and Williams [2013] proved that no O(n^{2-ε})-time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier?
We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n^{1.6456}log^{O(1)} n).
Cite as
Pierre Bergé, Guillaume Ducoffe, and Michel Habib. Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{berge_et_al:LIPIcs.STACS.2022.9,
author = {Berg\'{e}, Pierre and Ducoffe, Guillaume and Habib, Michel},
title = {{Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {9:1--9:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.9},
URN = {urn:nbn:de:0030-drops-158192},
doi = {10.4230/LIPIcs.STACS.2022.9},
annote = {Keywords: Diameter, Eccentricities, Metric graph theory, Median graphs, Hypercubes}
}
Document
Faster Counting and Sampling Algorithms Using Colorful Decision Oracle
Abstract
In this work, we consider d-Hyperedge Estimation and d-Hyperedge Sample problem in a hypergraph H(U(H),F(H)) in the query complexity framework, where U(H) denotes the set of vertices and F(H) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle (CID), which takes d (non-empty) pairwise disjoint subsets of vertices A₁,…, A_d ⊆ U(ℋ) as input, and answers whether there exists a hyperedge in H having (exactly) one vertex in each A_i, i ∈ {1,2,…,d}. The problem of d-Hyperedge Estimation and d-Hyperedge Sample with CID oracle access is important in its own right as a combinatorial problem. Also, Dell et al. [SODA '20] established that decision vs counting complexities of a number of combinatorial optimization problems can be abstracted out as d-Hyperedge Estimation problems with a CID oracle access.
The main technical contribution of the paper is an algorithm that estimates m = |F(H)| with m̂ such that
1/(C_{d)log^{d-1} n) ≤ m̂/m ≤ C_{d} log ^{d-1} n.
by using at most C_{d}log ^{d+2} n many CID queries, where n denotes the number of vertices in the hypergraph H and C_d is a constant that depends only on d}. Our result coupled with the framework of Dell et al. [SODA '21] implies improved bounds for the following fundamental problems:
Edge Estimation using the Bipartite Independent Set (BIS). We improve the bound obtained by Beame et al. [ITCS '18, TALG '20].
Triangle Estimation using the Tripartite Independent Set (TIS). The previous best bound for the case of graphs with low co-degree (Co-degree for an edge in the graph is the number of triangles incident to that edge in the graph) was due to Bhattacharya et al. [ISAAC '19, TOCS '21], and Dell {et al.}’s result gives the best bound for the case of general graphs [SODA '21]. We improve both of these bounds.
Hyperedge Estimation & Sampling using Colorful Independence Oracle (CID). We give an improvement over the bounds obtained by Dell et al. [SODA '21].
Cite as
Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Faster Counting and Sampling Algorithms Using Colorful Decision Oracle. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhattacharya_et_al:LIPIcs.STACS.2022.10,
author = {Bhattacharya, Anup and Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath},
title = {{Faster Counting and Sampling Algorithms Using Colorful Decision Oracle}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {10:1--10:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.10},
URN = {urn:nbn:de:0030-drops-158205},
doi = {10.4230/LIPIcs.STACS.2022.10},
annote = {Keywords: Query Complexity, Subset Query, Hyperedge Estimation, and Colorful Independent Set oracle}
}
Document
Probabilistic vs Deterministic Gamblers
Abstract
Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion - almost everywhere computable randomness. A binary sequence X is a.e. computably random if there is no probabilistic computable strategy which is total and succeeds on X for positive measure of oracles. Using the fireworks technique we construct a sequence which is partial computably random but not a.e. computably random. We also prove the separation between a.e. computable randomness and partial computable randomness, which happens exactly in the uniformly almost everywhere dominating Turing degrees.
Cite as
Laurent Bienvenu, Valentino Delle Rose, and Tomasz Steifer. Probabilistic vs Deterministic Gamblers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bienvenu_et_al:LIPIcs.STACS.2022.11,
author = {Bienvenu, Laurent and Delle Rose, Valentino and Steifer, Tomasz},
title = {{Probabilistic vs Deterministic Gamblers}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {11:1--11:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.11},
URN = {urn:nbn:de:0030-drops-158210},
doi = {10.4230/LIPIcs.STACS.2022.11},
annote = {Keywords: Algorithmic randomness, Martingales, Probabilistic computation, Almost everywhere domination}
}
Document
Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers
Abstract
Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.
Cite as
Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12,
author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko},
title = {{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {12:1--12:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.12},
URN = {urn:nbn:de:0030-drops-158221},
doi = {10.4230/LIPIcs.STACS.2022.12},
annote = {Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow}
}
Document
A 10-Approximation of the π/2-MST
Abstract
Bounded-angle spanning trees of points in the plane have received considerable attention in the context of wireless networks with directional antennas. For a point set P in the plane and an angle α, an α-spanning tree (α-ST) is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ∈ P lie in a wedge of angle α centered at p. The α-minimum spanning tree (α-MST) problem asks for an α-ST of minimum total edge length. The seminal work of Anscher and Katz (ICALP 2014) shows the NP-hardness of the α-MST problem for α = 2π/3, π and presents approximation algorithms for α = π/2, 2π/3, π.
In this paper we study the α-MST problem for α = π/2 which is also known to be NP-hard. We present a 10-approximation algorithm for this problem. This improves the previous best known approximation ratio of 16.
Cite as
Ahmad Biniaz, Majid Daliri, and Amir Hossein Moradpour. A 10-Approximation of the π/2-MST. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{biniaz_et_al:LIPIcs.STACS.2022.13,
author = {Biniaz, Ahmad and Daliri, Majid and Moradpour, Amir Hossein},
title = {{A 10-Approximation of the \pi/2-MST}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.13},
URN = {urn:nbn:de:0030-drops-158232},
doi = {10.4230/LIPIcs.STACS.2022.13},
annote = {Keywords: Euclidean spanning trees, approximation algorithms, bounded-angle visibility}
}
Document
On Explicit Constructions of Extremely Depth Robust Graphs
Abstract
A directed acyclic graph G = (V,E) is said to be (e,d)-depth robust if for every subset S ⊆ V of |S| ≤ e nodes the graph G-S still contains a directed path of length d. If the graph is (e,d)-depth-robust for any e,d such that e+d ≤ (1-ε)|V| then the graph is said to be ε-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents running in time polylog|V| which takes as input a node v ∈ V and returns the set of v’s parents. We give the first explicit construction of locally navigable ε-extreme depth-robust graphs with indegree O(log |V|). Previous constructions of ε-extreme depth-robust graphs either had indegree ω̃(log² |V|) or were not explicit.
Cite as
Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee, and Jin Young Son. On Explicit Constructions of Extremely Depth Robust Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blocki_et_al:LIPIcs.STACS.2022.14,
author = {Blocki, Jeremiah and Cinkoske, Mike and Lee, Seunghoon and Son, Jin Young},
title = {{On Explicit Constructions of Extremely Depth Robust Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {14:1--14:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.14},
URN = {urn:nbn:de:0030-drops-158241},
doi = {10.4230/LIPIcs.STACS.2022.14},
annote = {Keywords: Depth-Robust Graphs, Explicit Constructions, Data-Independent Memory Hard Functions, Proofs of Space and Replication}
}
Document
Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint
Abstract
We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.
Cite as
Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bousquet_et_al:LIPIcs.STACS.2022.15,
author = {Bousquet, Nicolas and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and Ouvrard, Paul and Suzuki, Akira and Wasa, Kunihiro},
title = {{Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {15:1--15:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.15},
URN = {urn:nbn:de:0030-drops-158253},
doi = {10.4230/LIPIcs.STACS.2022.15},
annote = {Keywords: combinatorial reconfiguration, spanning trees, PSPACE, polynomial-time algorithms}
}
Document
Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs
Abstract
We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ω-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ω-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ω-regular. This provides a game-theoretic characterization of ω-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ω-regularity depends on the value of some parameters.
Cite as
Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bouyer_et_al:LIPIcs.STACS.2022.16,
author = {Bouyer, Patricia and Randour, Mickael and Vandenhove, Pierre},
title = {{Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {16:1--16:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.16},
URN = {urn:nbn:de:0030-drops-158262},
doi = {10.4230/LIPIcs.STACS.2022.16},
annote = {Keywords: two-player games on graphs, infinite arenas, finite-memory determinacy, optimal strategies, \omega-regular languages}
}
Document
On Testing Decision Tree
Abstract
In this paper, we study testing decision tree of size and depth that are significantly smaller than the number of attributes n.
Our main result addresses the problem of poly(n,1/ε) time algorithms with poly(s,1/ε) query complexity (independent of n) that distinguish between functions that are decision trees of size s from functions that are ε-far from any decision tree of size ϕ(s,1/ε), for some function ϕ > s. The best known result is the recent one that follows from Blanc, Lange and Tan, [Guy Blanc et al., 2020], that gives ϕ(s,1/ε) = 2^{O((log³s)/ε³)}. In this paper, we give a new algorithm that achieves ϕ(s,1/ε) = 2^{O(log² (s/ε))}.
Moreover, we study the testability of depth-d decision tree and give a distribution free tester that distinguishes between depth-d decision tree and functions that are ε-far from depth-d² decision tree.
Cite as
Nader H. Bshouty and Catherine A. Haddad-Zaknoon. On Testing Decision Tree. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bshouty_et_al:LIPIcs.STACS.2022.17,
author = {Bshouty, Nader H. and Haddad-Zaknoon, Catherine A.},
title = {{On Testing Decision Tree}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {17:1--17:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.17},
URN = {urn:nbn:de:0030-drops-158273},
doi = {10.4230/LIPIcs.STACS.2022.17},
annote = {Keywords: Testing decision trees}
}
Document
The Ideal Membership Problem and Abelian Groups
Abstract
Given polynomials f_0, f_1, …, f_k the Ideal Membership Problem, IMP for short, asks if f₀ belongs to the ideal generated by f_1, …, f_k. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications, for instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved.
Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP(Γ). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP(Γ) where Γ is a Boolean constraint language, while Bulatov and Rafiey [arXiv'21] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over "affine" constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [arXiv'21] to systems of linear equations over GF(p), p prime. Here we prove that if Γ is an affine constraint language then IMP(Γ) is solvable in polynomial time assuming the input polynomial has bounded degree.
Cite as
Andrei A. Bulatov and Akbar Rafiey. The Ideal Membership Problem and Abelian Groups. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bulatov_et_al:LIPIcs.STACS.2022.18,
author = {Bulatov, Andrei A. and Rafiey, Akbar},
title = {{The Ideal Membership Problem and Abelian Groups}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {18:1--18:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.18},
URN = {urn:nbn:de:0030-drops-158280},
doi = {10.4230/LIPIcs.STACS.2022.18},
annote = {Keywords: Polynomial Ideal Membership, Constraint Satisfaction Problems, Polymorphisms, Gr\"{o}bner Bases, Abelian Groups}
}
Document
The Aperiodic Domino Problem in Higher Dimension
Abstract
The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift.
[Grandjean et al., 2018] proved that this problem is co-recursively enumerable (Π₀¹-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic (Σ₁¹-complete), in higher dimension: d ≥ 4 in the finite type case, d ≥ 3 for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity.
This complexity jump is surprising for two reasons: first, it separates 2- and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.
Cite as
Antonin Callard and Benjamin Hellouin de Menibus. The Aperiodic Domino Problem in Higher Dimension. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{callard_et_al:LIPIcs.STACS.2022.19,
author = {Callard, Antonin and Hellouin de Menibus, Benjamin},
title = {{The Aperiodic Domino Problem in Higher Dimension}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {19:1--19:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.19},
URN = {urn:nbn:de:0030-drops-158296},
doi = {10.4230/LIPIcs.STACS.2022.19},
annote = {Keywords: Subshift, periodicity, aperiodicity, domino problem, subshift of finite type, sofic subshift, effective subshift, tilings, computability}
}
Document
Symmetry and Quantum Query-To-Communication Simulation
Abstract
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways.
- We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication.
- In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness.
- In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication).
- We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.
Cite as
Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf. Symmetry and Quantum Query-To-Communication Simulation. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2022.20,
author = {Chakraborty, Sourav and Chattopadhyay, Arkadev and H{\o}yer, Peter and Mande, Nikhil S. and Paraashar, Manaswi and de Wolf, Ronald},
title = {{Symmetry and Quantum Query-To-Communication Simulation}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {20:1--20:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.20},
URN = {urn:nbn:de:0030-drops-158309},
doi = {10.4230/LIPIcs.STACS.2022.20},
annote = {Keywords: Classical and quantum communication complexity, query-to-communication-simulation, quantum computing}
}
Document
Near-Optimal Algorithms for Point-Line Covering Problems
Abstract
We study fundamental point-line covering problems in computational geometry, in which the input is a set S of points in the plane. The first is the Rich Lines problem, which asks for the set of all lines that each covers at least λ points from S, for a given integer parameter λ ≥ 2; this problem subsumes the 3-Points-on-Line problem and the Exact Fitting problem, which - the latter - asks for a line containing the maximum number of points. The second is the NP-hard problem Line Cover, which asks for a set of k lines that cover the points of S, for a given parameter k ∈ ℕ. Both problems have been extensively studied. In particular, the Rich Lines problem is a fundamental problem whose solution serves as a building block for several algorithms in computational geometry.
For Rich Lines and Exact Fitting, we present a randomized Monte Carlo algorithm that achieves a lower running time than that of Guibas et al.’s algorithm [Computational Geometry 1996], for a wide range of the parameter λ. We derive lower-bound results showing that, for λ = Ω(√{n log n}), the upper bound on the running time of this randomized algorithm matches the lower bound that we derive on the time complexity of Rich Lines in the algebraic computation trees model.
For Line Cover, we present two kernelization algorithms: a randomized Monte Carlo algorithm and a deterministic algorithm. Both algorithms improve the running time of existing kernelization algorithms for Line Cover. We derive lower-bound results showing that the running time of the randomized algorithm we present comes close to the lower bound we derive on the time complexity of kernelization algorithms for Line Cover in the algebraic computation trees model.
Cite as
Jianer Chen, Qin Huang, Iyad Kanj, and Ge Xia. Near-Optimal Algorithms for Point-Line Covering Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chen_et_al:LIPIcs.STACS.2022.21,
author = {Chen, Jianer and Huang, Qin and Kanj, Iyad and Xia, Ge},
title = {{Near-Optimal Algorithms for Point-Line Covering Problems}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {21:1--21:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.21},
URN = {urn:nbn:de:0030-drops-158312},
doi = {10.4230/LIPIcs.STACS.2022.21},
annote = {Keywords: line cover, rich lines, exact fitting, kernelization, randomized algorithms, complexity lower bounds, algebraic computation trees}
}
Document
Towards Uniform Certification in QBF
Abstract
We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments.
This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege.
Cite as
Leroy Chew and Friedrich Slivovsky. Towards Uniform Certification in QBF. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chew_et_al:LIPIcs.STACS.2022.22,
author = {Chew, Leroy and Slivovsky, Friedrich},
title = {{Towards Uniform Certification in QBF}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {22:1--22:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.22},
URN = {urn:nbn:de:0030-drops-158326},
doi = {10.4230/LIPIcs.STACS.2022.22},
annote = {Keywords: QBF, Proof Complexity, Verification, Frege, Extended Frege, Strategy Extraction}
}
Document
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths
Abstract
For vertices u and v of an n-vertex graph G, a uv-trail of G is an induced uv-path of G that is not a shortest uv-path of G. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in O(n^{18}) time, to either output a uv-trail of G or ensure that G admits no uv-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of n²× n² Boolean matrices, leading to a largely improved O(n^{4.75})-time algorithm.
Cite as
Yung-Chung Chiu and Hsueh-I Lu. Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chiu_et_al:LIPIcs.STACS.2022.23,
author = {Chiu, Yung-Chung and Lu, Hsueh-I},
title = {{Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {23:1--23:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.23},
URN = {urn:nbn:de:0030-drops-158333},
doi = {10.4230/LIPIcs.STACS.2022.23},
annote = {Keywords: Induced subgraph, induced path, non-shortest path, dynamic data structure}
}
Document
Depth Lower Bounds in Stabbing Planes for Combinatorial Principles
Abstract
Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas.
In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz.
We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.
Cite as
Stefan Dantchev, Nicola Galesi, Abdul Ghani, and Barnaby Martin. Depth Lower Bounds in Stabbing Planes for Combinatorial Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dantchev_et_al:LIPIcs.STACS.2022.24,
author = {Dantchev, Stefan and Galesi, Nicola and Ghani, Abdul and Martin, Barnaby},
title = {{Depth Lower Bounds in Stabbing Planes for Combinatorial Principles}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {24:1--24:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.24},
URN = {urn:nbn:de:0030-drops-158349},
doi = {10.4230/LIPIcs.STACS.2022.24},
annote = {Keywords: proof complexity, computational complexity, lower bounds, cutting planes, stabbing planes}
}
Document
Linear Space Data Structures for Finite Groups with Constant Query-Time
Abstract
A finite group of order n can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order n can be stored using O(n²) words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order n that uses o(n²) space but can still answer a multiplication query in constant time.
We design a constant query-time data structure that can store any finite group using O(n) words where n is the order of the group.
Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of Ω(n) on the number of words to store a group of order n. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time.
A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).
Cite as
Bireswar Das, Anant Kumar, Shivdutt Sharma, and Dhara Thakkar. Linear Space Data Structures for Finite Groups with Constant Query-Time. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{das_et_al:LIPIcs.STACS.2022.25,
author = {Das, Bireswar and Kumar, Anant and Sharma, Shivdutt and Thakkar, Dhara},
title = {{Linear Space Data Structures for Finite Groups with Constant Query-Time}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.25},
URN = {urn:nbn:de:0030-drops-158350},
doi = {10.4230/LIPIcs.STACS.2022.25},
annote = {Keywords: Compact Data Structures, Space Efficient Representations, Finite Groups, Simple Groups, Classification Theorem for Finite Simple Groups}
}
Document
The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}
Abstract
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in Σ₃^𝖯. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).
Cite as
Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß. The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dietrich_et_al:LIPIcs.STACS.2022.26,
author = {Dietrich, Heiko and Elder, Murray and Piggott, Adam and Qiao, Youming and Wei{\ss}, Armin},
title = {{The Isomorphism Problem for Plain Groups Is in \Sigma₃^\{𝖯\}}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {26:1--26:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.26},
URN = {urn:nbn:de:0030-drops-158368},
doi = {10.4230/LIPIcs.STACS.2022.26},
annote = {Keywords: plain group, isomorphism problem, polynomial hierarchy, \Sigma₃^\{𝖯\} complexity class, inverse-closed finite convergent length-reducing rewriting system}
}
Document
Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication
Abstract
Consider an undirected weighted graph G = (V,E,w). We study the problem of computing (1+ε)-approximate shortest paths for S × V, for a subset S ⊆ V of |S| = n^r sources, for some 0 < r ≤ 1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time Õ(|E| ⋅ n^{o(1)} + n^{ω(r)}), where n^{ω(r)} is the time required to multiply an n^r × n matrix by an n × n one. Our PRAM algorithm has polylogarithmic time (log n)^{O(1/ρ)}, and its work complexity is Õ(|E| ⋅ n^ρ + n^{ω(r)}), for any arbitrarily small constant ρ > 0.
In particular, for r ≤ 0.313…, our centralized algorithm computes S × V (1+ε)-approximate shortest paths in n^{2 + o(1)} time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E| ⋅ n^ρ + n^{2+o(1)}), for any arbitrarily small constant ρ > 0. Previously existing solutions either require centralized time/parallel work of O(|E| ⋅ |S|) or provide much weaker approximation guarantees.
In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S| = n^r sources, for r ≤ 0.655, while previous state-of-the-art algorithms did so only for r ≤ 1/2. Moreover, it improves previous bounds for all r > 1/2. For unweighted graphs, the running time is improved further to poly(log log n) for r ≤ 0.655. Previously this running time was known for r ≤ 1/2.
Cite as
Michael Elkin and Ofer Neiman. Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{elkin_et_al:LIPIcs.STACS.2022.27,
author = {Elkin, Michael and Neiman, Ofer},
title = {{Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {27:1--27:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.27},
URN = {urn:nbn:de:0030-drops-158379},
doi = {10.4230/LIPIcs.STACS.2022.27},
annote = {Keywords: Shortest paths, matrix multiplication, hopsets}
}
Document
Cardinality Constrained Scheduling in Online Models
Abstract
Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham’s famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most k jobs to each machine where k is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of 2 on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size p, we are allowed to migrate jobs of total size at most a constant times p. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches. More specifically, we show that in both cases, one can get a competitive ratio that is strictly lower than 2, which is the bound from the standard online setting. On the other hand, we prove that no PTAS is possible.
Cite as
Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder. Cardinality Constrained Scheduling in Online Models. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{epstein_et_al:LIPIcs.STACS.2022.28,
author = {Epstein, Leah and Lassota, Alexandra and Levin, Asaf and Maack, Marten and Rohwedder, Lars},
title = {{Cardinality Constrained Scheduling in Online Models}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.28},
URN = {urn:nbn:de:0030-drops-158385},
doi = {10.4230/LIPIcs.STACS.2022.28},
annote = {Keywords: Cardinality Constrained Scheduling, Makespan Minimization, Online Algorithms, Lower Bounds, Pure Online, Migration, Ordinal Algorithms}
}
Document
Detours in Directed Graphs
Abstract
We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. [Ivona Bezáková et al., 2019] proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O(k)}⋅ n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O (k)}· n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs.
In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k(diam(G)denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path above Diameter is solvable in polynomial time for each k ∈ {1,..., 4} and is NP-complete for every k ≥ 5. The parameterized complexity of Longest Detour on general directed graphs remains an interesting open problem.
Cite as
Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov, Kirill Simonov, and Saket Saurabh. Detours in Directed Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fomin_et_al:LIPIcs.STACS.2022.29,
author = {Fomin, Fedor V. and Golovach, Petr A. and Lochet, William and Sagunov, Danil and Simonov, Kirill and Saurabh, Saket},
title = {{Detours in Directed Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {29:1--29:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.29},
URN = {urn:nbn:de:0030-drops-158390},
doi = {10.4230/LIPIcs.STACS.2022.29},
annote = {Keywords: longest path, longest detour, diameter, directed graphs, parameterized complexity}
}
Document
Delay-Robust Routes in Temporal Graphs
Abstract
Most transportation networks are inherently temporal: Connections (e.g. flights, train runs) are only available at certain, scheduled times. When transporting passengers or commodities, this fact must be considered for the the planning of itineraries. This has already led to several well-studied algorithmic problems on temporal graphs. The difficulty of the described task is increased by the fact that connections are often unreliable - in particular, many modes of transportation suffer from occasional delays. If these delays cause subsequent connections to be missed, the consequences can be severe. Thus, it is a vital problem to design itineraries that are robust to (small) delays. We initiate the study of this problem from a parameterized complexity perspective by proving its NP-completeness as well as several hardness and tractability results for natural parameterizations.
Cite as
Eugen Füchsle, Hendrik Molter, Rolf Niedermeier, and Malte Renken. Delay-Robust Routes in Temporal Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fuchsle_et_al:LIPIcs.STACS.2022.30,
author = {F\"{u}chsle, Eugen and Molter, Hendrik and Niedermeier, Rolf and Renken, Malte},
title = {{Delay-Robust Routes in Temporal Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {30:1--30:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.30},
URN = {urn:nbn:de:0030-drops-158403},
doi = {10.4230/LIPIcs.STACS.2022.30},
annote = {Keywords: algorithms and complexity, parameterized complexity, time-varying networks, temporal paths, journeys}
}
Document
Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties
Abstract
Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings.
In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.
Cite as
Hiromichi Goko, Kazuhisa Makino, Shuichi Miyazaki, and Yu Yokoi. Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goko_et_al:LIPIcs.STACS.2022.31,
author = {Goko, Hiromichi and Makino, Kazuhisa and Miyazaki, Shuichi and Yokoi, Yu},
title = {{Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {31:1--31:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.31},
URN = {urn:nbn:de:0030-drops-158414},
doi = {10.4230/LIPIcs.STACS.2022.31},
annote = {Keywords: Stable matching, Hospitals/Residents problem, Lower quota, NP-hardness, Approximation algorithm, Strategy-proofness}
}
Document
Online Scheduling on Identical Machines with a Metric State Space
Abstract
This paper introduces an online scheduling problem on m identical machines with a metric state space, which generalizes the classical online scheduling problem on identical machines, the online traveling salesman problem, and the online dial-a-ride problem. Each job is associated with a source state, a destination state, a processing time, and a release time. Each machine can process a job on and after its release time. Before processing a job, a machine needs to change its state to the source state (in a time corresponding to the distance), and after the process of the job, the machine’s state becomes the destination state. While related research deals with a model in which only release times are unknown to the algorithm, this paper focuses on a general model in which destination states and processing times are also unknown. The main result of this paper is to propose a O(log m/log log m)-competitive online algorithm for the problem, which is best possible. A key approach is to divide the difficulty of the problem. To cope with unknown release times, we provide frameworks to produce a min{2ρ+1/2, ρ+2}-competitive algorithm using a ρ-competitive algorithm for a basic case where all jobs are released at time 0. Then, focusing on unknown destination states and processing times, we construct an O(log m/log log m)-competitive algorithm for the basic case. We also provide improved algorithms for some special cases.
Cite as
Hiromichi Goko, Akitoshi Kawamura, Yasushi Kawase, Kazuhisa Makino, and Hanna Sumita. Online Scheduling on Identical Machines with a Metric State Space. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goko_et_al:LIPIcs.STACS.2022.32,
author = {Goko, Hiromichi and Kawamura, Akitoshi and Kawase, Yasushi and Makino, Kazuhisa and Sumita, Hanna},
title = {{Online Scheduling on Identical Machines with a Metric State Space}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {32:1--32:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.32},
URN = {urn:nbn:de:0030-drops-158421},
doi = {10.4230/LIPIcs.STACS.2022.32},
annote = {Keywords: Online scheduling, Competitive analysis, Online dial-a-ride}
}
Document
A Simplicial Model for KB4_n: Epistemic Logic with Agents That May Die
Abstract
The standard semantics of multi-agent epistemic logic S5_n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work in Information and Computation 2021 [Éric Goubault et al., 2021]. Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4_n. Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash.
Cite as
Éric Goubault, Jérémy Ledent, and Sergio Rajsbaum. A Simplicial Model for KB4_n: Epistemic Logic with Agents That May Die. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goubault_et_al:LIPIcs.STACS.2022.33,
author = {Goubault, \'{E}ric and Ledent, J\'{e}r\'{e}my and Rajsbaum, Sergio},
title = {{A Simplicial Model for KB4\underlinen: Epistemic Logic with Agents That May Die}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {33:1--33:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.33},
URN = {urn:nbn:de:0030-drops-158434},
doi = {10.4230/LIPIcs.STACS.2022.33},
annote = {Keywords: Epistemic logic, Simplicial complexes, Distributed computing}
}
Document
Star Transposition Gray Codes for Multiset Permutations
Abstract
Given integers k ≥ 2 and a_1,…,a_k ≥ 1, let a: = (a_1,…,a_k) and n: = a_1+⋯+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i = 1,…,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a_1 = ⋯ = a_k = 1) can be generated by star transpositions, while combinations (k = 2) can be generated by these operations if and only if they are balanced (a_1 = a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Δ(a): = n-2max{a_1,…,a_k} that allows us to distinguish three different regimes for this problem. We show that if Δ(a) < 0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Δ(a) > 0, assuming that they exist for the case Δ(a) = 0. For the case Δ(a) = 0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.
Cite as
Petr Gregor, Torsten Mütze, and Arturo Merino. Star Transposition Gray Codes for Multiset Permutations. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gregor_et_al:LIPIcs.STACS.2022.34,
author = {Gregor, Petr and M\"{u}tze, Torsten and Merino, Arturo},
title = {{Star Transposition Gray Codes for Multiset Permutations}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {34:1--34:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.34},
URN = {urn:nbn:de:0030-drops-158448},
doi = {10.4230/LIPIcs.STACS.2022.34},
annote = {Keywords: Gray code, permutation, combination, transposition, Hamilton cycle}
}
Document
Improved Quantum Lower and Upper Bounds for Matrix Scaling
Abstract
Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting.
We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n × n matrices with at most m non-zero entries and with 𝓁₂-error ε = Θ~(1/m) must make Ω(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for ε ∈ [1/n,1/2], any quantum algorithm capable of producing ε/100-𝓁₁-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε) queries, and that there exists a constant ε₀ > 0 for which this problem takes Ω(n^{1.5}) queries.
To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε-𝓁₁-scaling in time O~(n^{1.5}/ε²), whereas the best previously known bounds were O~(n²polylog(1/ε)) (classical) and O~(n^{1.5}/ε³) (quantum).
Cite as
Sander Gribling and Harold Nieuwboer. Improved Quantum Lower and Upper Bounds for Matrix Scaling. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 35:1-35:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gribling_et_al:LIPIcs.STACS.2022.35,
author = {Gribling, Sander and Nieuwboer, Harold},
title = {{Improved Quantum Lower and Upper Bounds for Matrix Scaling}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {35:1--35:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.35},
URN = {urn:nbn:de:0030-drops-158458},
doi = {10.4230/LIPIcs.STACS.2022.35},
annote = {Keywords: Matrix scaling, quantum algorithms, lower bounds}
}
Document
Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth
Abstract
We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators.
Let p,q ∈ ℕ such that p is a prime and q ≥ 3. We show:
- If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q-1-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH).
- If p does not divide q-1, there is no ε > 0 for which there exists a (q-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.'18].
Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ε)^{ctw} n^{O(1)}, assuming SETH. We also give an algorithm with matching running time for this problem.
Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms'14] only excluded 2^{o(tw)}n^{O(1)} time algorithms for this problem.
Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain "compatibility matrix" in a non-trivial way.
Cite as
Carla Groenland, Isja Mannens, Jesper Nederlof, and Krisztina Szilágyi. Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{groenland_et_al:LIPIcs.STACS.2022.36,
author = {Groenland, Carla and Mannens, Isja and Nederlof, Jesper and Szil\'{a}gyi, Krisztina},
title = {{Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {36:1--36:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.36},
URN = {urn:nbn:de:0030-drops-158464},
doi = {10.4230/LIPIcs.STACS.2022.36},
annote = {Keywords: connected edge sets, cutwidth, parameterized algorithms, colorings, counting modulo p}
}
Document
Satisfiability of Circuits and Equations over Finite Malcev Algebras
Abstract
We show that the satisfiability of circuits over finite Malcev algebra A is NP-complete or A is nilpotent. This strengthens the result from our earlier paper [Idziak and Krzaczkowski, 2018] where nilpotency has been enforced, however with the use of a stronger assumption that no homomorphic image of A has NP-complete circuits satisfiability. Our methods are moreover strong enough to extend our result of [Idziak et al., 2021] from groups to Malcev algebras. Namely we show that tractability of checking if an equation over such an algebra A has a solution enforces its nice structure: A must have a nilpotent congruence ν such that also the quotient algebra A/ν is nilpotent. Otherwise, if A has no such congruence ν then the Exponential Time Hypothesis yields a quasipolynomial lower bound. Both our results contain important steps towards a full characterization of finite algebras with tractable circuit satisfiability as well as equation satisfiability.
Cite as
Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski. Satisfiability of Circuits and Equations over Finite Malcev Algebras. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{idziak_et_al:LIPIcs.STACS.2022.37,
author = {Idziak, Pawe{\l} M. and Kawa{\l}ek, Piotr and Krzaczkowski, Jacek},
title = {{Satisfiability of Circuits and Equations over Finite Malcev Algebras}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {37:1--37:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.37},
URN = {urn:nbn:de:0030-drops-158474},
doi = {10.4230/LIPIcs.STACS.2022.37},
annote = {Keywords: Circuit satisfiability, solving equations, Exponential Time Hypothesis}
}
Document
Classes of Intersection Digraphs with Good Algorithmic Properties
Abstract
While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs such as (Independent) Dominating Set, Kernel, and H-Homomorphism. Third, we give a new width measure of digraphs, bi-mim-width, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results.
Cite as
Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Classes of Intersection Digraphs with Good Algorithmic Properties. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{jaffke_et_al:LIPIcs.STACS.2022.38,
author = {Jaffke, Lars and Kwon, O-joung and Telle, Jan Arne},
title = {{Classes of Intersection Digraphs with Good Algorithmic Properties}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {38:1--38:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.38},
URN = {urn:nbn:de:0030-drops-158480},
doi = {10.4230/LIPIcs.STACS.2022.38},
annote = {Keywords: intersection digraphs, H-digraphs, reflexive digraphs, directed domination, directed H-homomorphism}
}
Document
Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems
Abstract
For a positive integer c, a graph G is said to be c-closed if every pair of non-adjacent vertices in G have at most c-1 neighbours in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by Fox et al. [ICALP `18 and SICOMP `20]. Koana et al. [ESA `20] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set and (Threshold) Dominating Set, and showed that each of these problems admits a polynomial kernel, either w.r.t. the parameter k+c or w.r.t. the parameter k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable w.r.t. the parameter k+cl(G), whereas Partial Dominating Set, parameterized by k is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs, unless NP ⊆ co-NP/poly.
Cite as
Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, and Saket Saurabh. Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kanesh_et_al:LIPIcs.STACS.2022.39,
author = {Kanesh, Lawqueen and Madathil, Jayakrishnan and Roy, Sanjukta and Sahu, Abhishek and Saurabh, Saket},
title = {{Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {39:1--39:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.39},
URN = {urn:nbn:de:0030-drops-158494},
doi = {10.4230/LIPIcs.STACS.2022.39},
annote = {Keywords: c-closed graphs, domination problems, perfect code, connected dominating set, fixed-parameter tractable, polynomial kernel}
}
Document
Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k
Abstract
Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general.
We consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2^{2^{O(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
Cite as
Mamadou Moustapha Kanté, Eun Jung Kim, O-joung Kwon, and Sang-il Oum. Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kante_et_al:LIPIcs.STACS.2022.40,
author = {Kant\'{e}, Mamadou Moustapha and Kim, Eun Jung and Kwon, O-joung and Oum, Sang-il},
title = {{Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {40:1--40:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.40},
URN = {urn:nbn:de:0030-drops-158507},
doi = {10.4230/LIPIcs.STACS.2022.40},
annote = {Keywords: path-width, matroid, linear rank-width, graph, forbidden minor, vertex-minor, pivot-minor}
}
Document
Fairly Popular Matchings and Optimality
Abstract
We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity.
Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation.
We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K.
Cite as
Telikepalli Kavitha. Fairly Popular Matchings and Optimality. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kavitha:LIPIcs.STACS.2022.41,
author = {Kavitha, Telikepalli},
title = {{Fairly Popular Matchings and Optimality}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {41:1--41:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.41},
URN = {urn:nbn:de:0030-drops-158516},
doi = {10.4230/LIPIcs.STACS.2022.41},
annote = {Keywords: Bipartite graphs, Stable matchings, Mixed matchings, Polytopes}
}
Document
Covering Many (Or Few) Edges with k Vertices in Sparse Graphs
Abstract
We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed α between zero and one we are given a graph and two numbers k ∈ ℕ and t ∈ ℚ. The task is to find a vertex subset S of exactly k vertices that has value at least (resp. at most for minimization) t. Here, the value of a vertex set computes as α times the number of edges with exactly one endpoint in S plus 1-α times the number of edges with both endpoints in S. These two problems generalize many prominent graph problems, such as Densest k-Subgraph, Sparsest k-Subgraph, Partial Vertex Cover, and Max (k,n-k)-Cut.
In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max (k,n-k)-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.
Cite as
Tomohiro Koana, Christian Komusiewicz, André Nichterlein, and Frank Sommer. Covering Many (Or Few) Edges with k Vertices in Sparse Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{koana_et_al:LIPIcs.STACS.2022.42,
author = {Koana, Tomohiro and Komusiewicz, Christian and Nichterlein, Andr\'{e} and Sommer, Frank},
title = {{Covering Many (Or Few) Edges with k Vertices in Sparse Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {42:1--42:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.42},
URN = {urn:nbn:de:0030-drops-158525},
doi = {10.4230/LIPIcs.STACS.2022.42},
annote = {Keywords: Parameterized Complexity, Kernelization, Partial Vertex Cover, Densest k-Subgraph, Max (k,n-k)-Cut, Degeneracy}
}
Document
One-To-Two-Player Lifting for Mildly Growing Memory
Abstract
We investigate a phenomenon of "one-to-two-player lifting" in infinite-duration two-player games on graphs with zero-sum objectives. More specifically, let 𝒞 be a class of strategies. It turns out that in many cases, to show that all two-player games on graphs with a given payoff function are determined in 𝒞, it is sufficient to do so for one-player games. That is, in many cases the determinacy in 𝒞 can be "lifted" from one-player games to two-player games. Namely, Gimbert and Zielonka (CONCUR 2005) have shown this for the class of positional strategies. Recently, Bouyer et al. (CONCUR 2020) have extended this to the classes of arena-independent finite-memory strategies. Informally, these are finite-memory strategies that use the same way of storing memory in all game graphs.
In this paper, we put the lifting technique into the context of memory complexity. The memory complexity of a payoff function measures, how many states of memory we need to play optimally in game graphs with up to n nodes, depending on n. We address the following question. Assume that we know the memory complexity of our payoff function in one-player games. Then what can be said about its memory complexity in two-player games? In particular, when is it finite?
In this paper, we answer this questions for strategies with "chromatic" memory. These are strategies that only accumulate sequences of colors of edges in their memory. We obtain the following results.
- Assume that the chromatic memory complexity in one-player games is sublinear in n on some infinite subsequence. Then the chromatic memory complexity in two-player games is finite.
- We provide an example in which (a) the chromatic memory complexity in one-player games is linear in n; (b) the memory complexity in two-player games is infinite. Thus, we obtain the exact barrier for the one-to-two-player lifting theorems in the setting of chromatic finite-memory strategies. Previous results only cover payoff functions with constant chromatic memory complexity.
Cite as
Alexander Kozachinskiy. One-To-Two-Player Lifting for Mildly Growing Memory. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 43:1-43:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kozachinskiy:LIPIcs.STACS.2022.43,
author = {Kozachinskiy, Alexander},
title = {{One-To-Two-Player Lifting for Mildly Growing Memory}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {43:1--43:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.43},
URN = {urn:nbn:de:0030-drops-158535},
doi = {10.4230/LIPIcs.STACS.2022.43},
annote = {Keywords: Games on graphs, one-to-two-player lifting, strategy complexity, positional determinacy, finite-memory determinacy}
}
Document
If VNP Is Hard, Then so Are Equations for It
Abstract
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size.
In a recent work of Chatterjee, Kumar, Ramya, Saptharishi and Tengse (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.
Cite as
Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse. If VNP Is Hard, Then so Are Equations for It. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kumar_et_al:LIPIcs.STACS.2022.44,
author = {Kumar, Mrinal and Ramya, C. and Saptharishi, Ramprasad and Tengse, Anamay},
title = {{If VNP Is Hard, Then so Are Equations for It}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {44:1--44:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.44},
URN = {urn:nbn:de:0030-drops-158547},
doi = {10.4230/LIPIcs.STACS.2022.44},
annote = {Keywords: Computational Complexity, Algebraic Circuits, Algebraic Natural Proofs}
}
Document
Determining a Slater Winner Is Complete for Parallel Access to NP
Abstract
We consider the complexity of deciding the winner of an election under the Slater rule. In this setting we are given a tournament T = (V,A), where the vertices of V represent candidates and the direction of each arc indicates which of the two endpoints is preferable for the majority of voters. The Slater score of a vertex v ∈ V is defined as the minimum number of arcs that need to be reversed so that T becomes acyclic and v becomes the winner. We say that v is a Slater winner in T if v has minimum Slater score in T.
Deciding if a vertex is a Slater winner in a tournament has long been known to be NP-hard. However, the best known complexity upper bound for this problem is the class Θ₂^p, which corresponds to polynomial-time Turing machines with parallel access to an NP oracle. In this paper we close this gap by showing that the problem is Θ₂^p-complete, and that this hardness applies to instances constructible by aggregating the preferences of 7 voters.
Cite as
Michael Lampis. Determining a Slater Winner Is Complete for Parallel Access to NP. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lampis:LIPIcs.STACS.2022.45,
author = {Lampis, Michael},
title = {{Determining a Slater Winner Is Complete for Parallel Access to NP}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {45:1--45:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.45},
URN = {urn:nbn:de:0030-drops-158555},
doi = {10.4230/LIPIcs.STACS.2022.45},
annote = {Keywords: Slater winner, Feedback Arc Set, Tournaments}
}
Document
Improved Ackermannian Lower Bound for the Petri Nets Reachability Problem
Abstract
Petri nets, equivalently presentable as vector addition systems with states, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there exists a sequence of valid execution steps reaching a given final configuration, is the central algorithmic problem for this model. The complexity of the problem has remained, until recently, one of the hardest open questions in verification of concurrent systems. A first upper bound has been provided only in 2015 by Leroux and Schmitz, then refined by the same authors to non-primitive recursive Ackermannian upper bound in 2019. The exponential space lower bound, shown by Lipton already in 1976, remained the only known for over 40 years until a breakthrough non-elementary lower bound by Czerwiński, Lasota, Lazic, Leroux and Mazowiecki in 2019. Finally, a matching Ackermannian lower bound announced this year by Czerwiński and Orlikowski, and independently by Leroux, established the complexity of the problem.
Our primary contribution is an improvement of the former construction, making it conceptually simpler and more direct. On the way we improve the lower bound for vector addition systems with states in fixed dimension (or, equivalently, Petri nets with fixed number of places): while Czerwiński and Orlikowski prove F_k-hardness (hardness for kth level in Grzegorczyk Hierarchy) in dimension 6k, our simplified construction yields F_k-hardness already in dimension 3k+2.
Cite as
Sławomir Lasota. Improved Ackermannian Lower Bound for the Petri Nets Reachability Problem. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lasota:LIPIcs.STACS.2022.46,
author = {Lasota, S{\l}awomir},
title = {{Improved Ackermannian Lower Bound for the Petri Nets Reachability Problem}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {46:1--46:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.46},
URN = {urn:nbn:de:0030-drops-158561},
doi = {10.4230/LIPIcs.STACS.2022.46},
annote = {Keywords: Petri nets, reachability problem, vector addition systems}
}
Document
Scheduling with Communication Delay in Near-Linear Time
Abstract
We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. Such precedence-constrained jobs can be modeled as a directed acyclic graph, G = (V, E). In this setting, if two precedence-constrained jobs u and v, with v dependent on u (u ≺ v), are scheduled on different machines, then v must start at least ρ time units after u completes. The scheduling objective is to minimize makespan, i.e. the total time from when the first job starts to when the last job finishes. The focus of this paper is to provide an efficient approximation algorithm with near-linear running time. We build on the algorithm of Lepere and Rapine [STACS 2002] for this problem to give an O((ln ρ)/(ln ln ρ))-approximation algorithm that runs in Õ(|V|+|E|) time.
Cite as
Quanquan C. Liu, Manish Purohit, Zoya Svitkina, Erik Vee, and Joshua R. Wang. Scheduling with Communication Delay in Near-Linear Time. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{liu_et_al:LIPIcs.STACS.2022.47,
author = {Liu, Quanquan C. and Purohit, Manish and Svitkina, Zoya and Vee, Erik and Wang, Joshua R.},
title = {{Scheduling with Communication Delay in Near-Linear Time}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {47:1--47:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.47},
URN = {urn:nbn:de:0030-drops-158570},
doi = {10.4230/LIPIcs.STACS.2022.47},
annote = {Keywords: near-linear time scheduling, scheduling with duplication, precedence-constrained jobs, graph algorithms}
}
Document
Extending the Reach of the Point-To-Set Principle
Abstract
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces ℝⁿ. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.
In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two fractal dimensions - computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x) and Dim(x) to individual points x ∈ X - to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.
We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages E₀, E₁, E₂, … used to construct a self-similar fractal E in the plane are elements of the hyperspace of the plane, and they converge to E in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets E that are analytic, i.e., Σ¹₁, a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.
Cite as
Jack H. Lutz, Neil Lutz, and Elvira Mayordomo. Extending the Reach of the Point-To-Set Principle. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lutz_et_al:LIPIcs.STACS.2022.48,
author = {Lutz, Jack H. and Lutz, Neil and Mayordomo, Elvira},
title = {{Extending the Reach of the Point-To-Set Principle}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {48:1--48:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.48},
URN = {urn:nbn:de:0030-drops-158585},
doi = {10.4230/LIPIcs.STACS.2022.48},
annote = {Keywords: algorithmic dimensions, geometric measure theory, hyperspace, point-to-set principle}
}
Document
One-Way Communication Complexity and Non-Adaptive Decision Trees
Abstract
We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2b input bits for all b ≥ 2, denoted IP.
- If f is a total Boolean function that depends on all of its n input bits, then the bounded-error one-way quantum communication complexity of f∘IP equals Ω(n(b-1)).
- If f is a partial Boolean function, then the deterministic one-way communication complexity of f∘IP is at least Ω(b ⋅ 𝖣_{dt}^ → (f)), where 𝖣_{dt}^ → (f) denotes non-adaptive decision tree complexity of f. To prove our quantum lower bound, we first show a lower bound on the VC-dimension of f∘IP. We then appeal to a result of Klauck [STOC'00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.'98], and Frankl and Tokushige [Comb.'99].
It is known due to a result of Montanaro and Osborne [arXiv'09] that the deterministic one-way communication complexity of f∘XOR equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following when the inner gadget is the AND function on 2 input bits.
- There exists a function for which even the quantum non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f∘AND.
- However, for symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f∘AND. In view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f∘AND. The proof of the first bullet above uses the well-studied Odd-Max-Bit function. For the second bullet, we first observe a connection between the one-way communication complexity of f and the Möbius sparsity of f, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result.
It is well known that the rank of the communication matrix of a function F is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some F. However, in our final result we show that this is not the case when F = f∘AND. More precisely we show that for all f, the deterministic one-way communication complexity of F = f∘AND is at most (rank(M_{F}))(1 - Ω(1)), where M_{F} denotes the communication matrix of F.
Cite as
Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. One-Way Communication Complexity and Non-Adaptive Decision Trees. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{mande_et_al:LIPIcs.STACS.2022.49,
author = {Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail},
title = {{One-Way Communication Complexity and Non-Adaptive Decision Trees}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {49:1--49:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.49},
URN = {urn:nbn:de:0030-drops-158598},
doi = {10.4230/LIPIcs.STACS.2022.49},
annote = {Keywords: Decision trees, communication complexity, composed Boolean functions}
}
Document
Isolation Schemes for Problems on Decomposable Graphs
Abstract
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time.
In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use
- 𝒪(tlog n + log²{n}) random bits on graphs of treewidth at most t;
- 𝒪(√n) random bits on planar or H-minor free graphs; and
- 𝒪(n)-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed H we obtain an algorithm for detecting a Hamiltonian cycle in an H-minor-free graph that runs in deterministic time 2^{𝒪(√n)} and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use 𝒪(d) random bits and assign polynomially-bounded weights.
We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.
Cite as
Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Isolation Schemes for Problems on Decomposable Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{nederlof_et_al:LIPIcs.STACS.2022.50,
author = {Nederlof, Jesper and Pilipczuk, Micha{\l} and Swennenhuis, C\'{e}line M. F. and W\k{e}grzycki, Karol},
title = {{Isolation Schemes for Problems on Decomposable Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {50:1--50:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.50},
URN = {urn:nbn:de:0030-drops-158601},
doi = {10.4230/LIPIcs.STACS.2022.50},
annote = {Keywords: Isolation Lemma, Derandomization, Hamiltonian Cycle, Exact Algorithms}
}
Document
Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)
Abstract
Different models have been proposed to understand natural phenomena at the molecular scale from a computational point of view. Oritatami systems are a model of molecular co-transcriptional folding: the transcript (the "molecule") folds as it is synthesized according to a local energy optimisation process, in a similar way to how actual biomolecules such as RNA fold into complex shapes and functions. We introduce a new model, called turedo, which is a self-avoiding Turing machine on the plane that evolves by marking visited positions and that can only move to unmarked positions. Any oritatami can be seen as a particular turedo. We show that any turedo with lookup radius 1 can conversely be simulated by an oritatami, using a universal bead type set. Our notion of simulation is strong enough to preserve the geometrical and dynamical features of these models up to a constant spatio-temporal rescaling (as in intrinsic simulation). As a consequence, turedo can be used as a readable oritatami "higher-level" programming language to build readily oritatami "smart robots", using our explicit simulation result as a compiler.
As an application of our simulation result, we prove two new complexity results on the (infinite) limit configurations of oritatami systems (and radius-1 turedos), assembled from a finite seed configuration. First, we show that such limit configurations can embed any recursively enumerable set, and are thus exactly as complex as aTAM limit configurations. Second, we characterize the possible densities of occupied positions in such limit configurations: they are exactly the Π₂-computable numbers between 0 and 1. We also show that all such limit densities can be produced by one single oritatami system, just by changing the finite seed configuration.
None of these results is implied by previous constructions of oritatami embedding tag systems or 1D cellular automata, which produce only computable limit configurations with constrained density.
Cite as
Daria Pchelina, Nicolas Schabanel, Shinnosuke Seki, and Guillaume Theyssier. Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 51:1-51:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{pchelina_et_al:LIPIcs.STACS.2022.51,
author = {Pchelina, Daria and Schabanel, Nicolas and Seki, Shinnosuke and Theyssier, Guillaume},
title = {{Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {51:1--51:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.51},
URN = {urn:nbn:de:0030-drops-158618},
doi = {10.4230/LIPIcs.STACS.2022.51},
annote = {Keywords: Molecular Self-assembly, Co-transcriptional folding, Intrinsic simulation, Arithmetical hierarchy of real numbers, 2D Turing machines, Computability}
}
Document
Compact Representation for Matrices of Bounded Twin-Width
Abstract
For every fixed d ∈ ℕ, we design a data structure that represents a binary n × n matrix that is d-twin-ordered. The data structure occupies 𝒪_d(n) bits, which is the least one could hope for, and can be queried for entries of the matrix in time 𝒪_d(log log n) per query.
Cite as
Michał Pilipczuk, Marek Sokołowski, and Anna Zych-Pawlewicz. Compact Representation for Matrices of Bounded Twin-Width. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{pilipczuk_et_al:LIPIcs.STACS.2022.52,
author = {Pilipczuk, Micha{\l} and Soko{\l}owski, Marek and Zych-Pawlewicz, Anna},
title = {{Compact Representation for Matrices of Bounded Twin-Width}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.52},
URN = {urn:nbn:de:0030-drops-158620},
doi = {10.4230/LIPIcs.STACS.2022.52},
annote = {Keywords: twin-width, compact representation, adjacency oracle}
}
Document
On Finer Separations Between Subclasses of Read-Once Oblivious ABPs
Abstract
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials).
Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank.
Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.
Cite as
C. Ramya and Anamay Tengse. On Finer Separations Between Subclasses of Read-Once Oblivious ABPs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 53:1-53:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ramya_et_al:LIPIcs.STACS.2022.53,
author = {Ramya, C. and Tengse, Anamay},
title = {{On Finer Separations Between Subclasses of Read-Once Oblivious ABPs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {53:1--53:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.53},
URN = {urn:nbn:de:0030-drops-158636},
doi = {10.4230/LIPIcs.STACS.2022.53},
annote = {Keywords: Algebraic Complexity Theory, Algebraic Branching Programs, Commutative Matrices}
}
Document
A Relativization Perspective on Meta-Complexity
Abstract
Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where:
1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof.
2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function.
3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0.
4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets.
5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20].
Cite as
Hanlin Ren and Rahul Santhanam. A Relativization Perspective on Meta-Complexity. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ren_et_al:LIPIcs.STACS.2022.54,
author = {Ren, Hanlin and Santhanam, Rahul},
title = {{A Relativization Perspective on Meta-Complexity}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {54:1--54:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.54},
URN = {urn:nbn:de:0030-drops-158646},
doi = {10.4230/LIPIcs.STACS.2022.54},
annote = {Keywords: meta-complexity, relativization, minimum circuit size problem}
}
Document
Superlinear Lower Bounds Based on ETH
Abstract
We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in essentially-linear time unless the exponential time hypothesis (ETH) is false and k-Clique is decidable in essentially-linear time in terms of the graph’s size for all fixed k. Such conditional lower bounds have previously only been demonstrated relative to the strong exponential time hypothesis (SETH). Our results therefore offer significant progress towards proving unconditional superlinear time complexity lower bounds for natural problems in polynomial time.
Cite as
András Z. Salamon and Michael Wehar. Superlinear Lower Bounds Based on ETH. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 55:1-55:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{salamon_et_al:LIPIcs.STACS.2022.55,
author = {Salamon, Andr\'{a}s Z. and Wehar, Michael},
title = {{Superlinear Lower Bounds Based on ETH}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {55:1--55:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.55},
URN = {urn:nbn:de:0030-drops-158652},
doi = {10.4230/LIPIcs.STACS.2022.55},
annote = {Keywords: Circuit Satisfiability, Conditional Lower Bounds, Exponential Time Hypothesis, Limited Nondeterminism}
}
Document
NP-Completeness of Perfect Matching Index of Cubic Graphs
Abstract
The perfect matching index of a cubic graph G, denoted by π(G), is the smallest number of perfect matchings needed to cover all the edges of G; it is correctly defined for every bridgeless cubic graph. The value of π(G) is always at least 3, and if G has no 3-edge-colouring, then π(G) ≥ 4. On the other hand, a long-standing conjecture of Berge suggests that π(G) never exceeds 5. It was proved by Esperet and Mazzuoccolo [J. Graph Theory 77 (2014), 144-157] that it is NP-complete to decide for a 2-connected cubic graph whether π(G) ≤ 4. A disadvantage of the proof (noted by the authors) is that the constructed graphs have 2-cuts. We show that small cuts can be avoided and that the problem remains NP-complete even for nontrivial snarks - cyclically 4-edge-connected cubic graphs of girth at least 5 with no 3-edge-colouring. Our proof significantly differs from the one due to Esperet and Mazzuoccolo in that it combines nowhere-zero flow methods with elements of projective geometry, without referring to perfect matchings explicitly.
Cite as
Martin Škoviera and Peter Varša. NP-Completeness of Perfect Matching Index of Cubic Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{skoviera_et_al:LIPIcs.STACS.2022.56,
author = {\v{S}koviera, Martin and Var\v{s}a, Peter},
title = {{NP-Completeness of Perfect Matching Index of Cubic Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {56:1--56:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.56},
URN = {urn:nbn:de:0030-drops-158667},
doi = {10.4230/LIPIcs.STACS.2022.56},
annote = {Keywords: cubic graph, edge colouring, snark, perfect matching, covering, NP-completeness}
}
Document
Optimal Oracles for Point-To-Set Principles
Abstract
The point-to-set principle [Lutz and Lutz, 2018] characterizes the Hausdorff dimension of a subset E ⊆ ℝⁿ by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e., without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry.
In this paper, we introduce an extension of point-to-set principle - the notion of optimal oracles for subsets E ⊆ ℝⁿ. One of the primary motivations of this definition is that, if E has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for E. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of E agree, then E has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on E implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set E is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold.
Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies' theorem on projections.
Cite as
D. M. Stull. Optimal Oracles for Point-To-Set Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{stull:LIPIcs.STACS.2022.57,
author = {Stull, D. M.},
title = {{Optimal Oracles for Point-To-Set Principles}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {57:1--57:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.57},
URN = {urn:nbn:de:0030-drops-158675},
doi = {10.4230/LIPIcs.STACS.2022.57},
annote = {Keywords: Algorithmic randomness, Kolmogorov complexity, geometric measure theory}
}
Document
High Quality Consistent Digital Curved Rays via Vector Field Rounding
Abstract
We consider the consistent digital rays (CDR) of curved rays, which approximates a set of curved rays emanating from the origin by the set of rooted paths (called digital rays) of a spanning tree of a grid graph. Previously, a construction algorithm of CDR for diffused families of curved rays to attain an O(√{n log n}) bound for the distance between digital ray and the corresponding ray is known [Chun et al., 2019]. In this paper, we give a description of the problem as a rounding problem of the vector field generated from the ray family, and investigate the relation of the quality of CDR and the discrepancy of the range space generated from gradient curves of rays. Consequently, we show the existence of a CDR with an O(log ^{1.5} n) distance bound for any diffused family of curved rays.
Cite as
Takeshi Tokuyama and Ryo Yoshimura. High Quality Consistent Digital Curved Rays via Vector Field Rounding. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{tokuyama_et_al:LIPIcs.STACS.2022.58,
author = {Tokuyama, Takeshi and Yoshimura, Ryo},
title = {{High Quality Consistent Digital Curved Rays via Vector Field Rounding}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {58:1--58:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.58},
URN = {urn:nbn:de:0030-drops-158680},
doi = {10.4230/LIPIcs.STACS.2022.58},
annote = {Keywords: Computational Geometry, Discrepancy Theory, Consistent Digital Rays, Digital Geometry}
}
Document
Sharp Indistinguishability Bounds from Non-Uniform Approximations
Abstract
We study the basic problem of distinguishing between two symmetric probability distributions over n bits by observing k bits of a sample, subject to the constraint that all (k-1)-wise marginal distributions of the two distributions are identical to each other. Previous works of Bogdanov et al. [Bogdanov et al., 2019] and of Huang and Viola [Huang and Viola, 2019] have established approximately tight results on the maximal possible statistical distance between the k-wise marginals of such distributions when k is at most a small constant fraction of n. Naor and Shamir [Naor and Shamir, 1994] gave a tight bound for all k in the special case k = n and when distinguishing with the OR function; they also derived a non-tight result for general k and n. Krause and Simon [Krause and Simon, 2000] gave improved upper and lower bounds for general k and n when distinguishing with the OR function, but these bounds are exponentially far apart when k = Ω(n). In this work we provide sharp upper and lower bounds on the maximal statistical distance that hold for all k and n. Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials. This is the first work to construct suitable non-uniform approximations for this purpose; the sharpness and wider applicability of our result stems from this non-uniformity.
Cite as
Christopher Williamson. Sharp Indistinguishability Bounds from Non-Uniform Approximations. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{williamson:LIPIcs.STACS.2022.59,
author = {Williamson, Christopher},
title = {{Sharp Indistinguishability Bounds from Non-Uniform Approximations}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {59:1--59:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.59},
URN = {urn:nbn:de:0030-drops-158692},
doi = {10.4230/LIPIcs.STACS.2022.59},
annote = {Keywords: bounded indistinguishability, randomness, secret sharing, polynomial approximation}
}
Document
Analyzing XOR-Forrelation Through Stochastic Calculus
Abstract
In this note we present a simplified analysis of the quantum and classical complexity of the k-XOR Forrelation problem (introduced in the paper of Girish, Raz and Zhan [Uma Girish et al., 2020]) by a stochastic interpretation of the Forrelation distribution.
Cite as
Xinyu Wu. Analyzing XOR-Forrelation Through Stochastic Calculus. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 60:1-60:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{wu:LIPIcs.STACS.2022.60,
author = {Wu, Xinyu},
title = {{Analyzing XOR-Forrelation Through Stochastic Calculus}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {60:1--60:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.60},
URN = {urn:nbn:de:0030-drops-158705},
doi = {10.4230/LIPIcs.STACS.2022.60},
annote = {Keywords: quantum complexity, Brownian motion}
}