Volume
LIPIcs, Volume 14
29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
Event
STACS 2012, February 29 to March 3, 2012, Paris, FranceEditors
Publication Details
- published at: 2012-02-24
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-939897-35-4
- DBLP: db/conf/stacs/stacs2012
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Complete Volume
LIPIcs, Volume 14, STACS'12, Complete Volume
Abstract
LIPIcs, Volume 14, STACS'12, Complete Volume
Cite as
29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)
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@Proceedings{durr_et_al:LIPIcs.STACS.2012,
title = {{LIPIcs, Volume 14, STACS'12, Complete Volume}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2013},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012},
URN = {urn:nbn:de:0030-drops-41081},
doi = {10.4230/LIPIcs.STACS.2012},
annote = {Keywords: Models of Computation, Nonnumerical Algorithms and Problems, Mathematical Logic, Formal Languages, Combinatorics, Graph Theory}
}
Document
Front Matter
Frontmatter, Foreword, Conference Organization, External Reviewers, Table of Contents
Abstract
Frontmatter, Foreword, Conference Organization, External Reviewers, Table of Contents
Cite as
29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. i-xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{durr_et_al:LIPIcs.STACS.2012.i,
author = {D\"{u}rr, Christoph and Wilke, Thomas},
title = {{Frontmatter, Foreword, Conference Organization, External Reviewers, Table of Contents}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {i--xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.i},
URN = {urn:nbn:de:0030-drops-33850},
doi = {10.4230/LIPIcs.STACS.2012.i},
annote = {Keywords: Frontmatter, Foreword, Conference Organization, External Reviewers, Table of Contents}
}
Document
Invited Talk
Forms of Determinism for Automata (Invited Talk)
Abstract
We survey in this paper some variants of the notion of determinism, refining the spectrum between non-determinism and determinism. We present unambiguous automata, strongly unambiguous automata, prophetic automata, guidable automata, and history-deterministic automata. We instantiate these various notions for finite words, infinite words, finite trees, infinite trees, data languages, and cost functions. The main results are underlined and some open problems proposed.
Cite as
Thomas Colcombet. Forms of Determinism for Automata (Invited Talk). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 1-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{colcombet:LIPIcs.STACS.2012.1,
author = {Colcombet, Thomas},
title = {{Forms of Determinism for Automata}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {1--23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.1},
URN = {urn:nbn:de:0030-drops-33862},
doi = {10.4230/LIPIcs.STACS.2012.1},
annote = {Keywords: Automata, determinism, unambiguity, words, infinite trees}
}
Document
Invited Talk
Iterative Methods in Combinatorial Optimization (Invited Talk)
Abstract
In these lectures, I will describe a simple iterative method that supplies new proofs of integrality of linear characterizations of various basic problems in combinatorial optimization, and also allows adaptations to design approximation algorithms for NP-hard variants of these problems involving extra "degree-like" budget constraints. It is inspired by Jain's iterative rounding method for designing approximation algorithms for survivable network design problems, and augmented with a relaxation idea in the work of Lau, Naor, Salvatipour and Singh in their work on designing the approximation algorithm for its degree bounded version. Its application was further refined in recent work of Bansal, Khandekar and Nagarajan on degree-bounded directed network design.
I will begin by reviewing the background material on LP relaxations and their solvability and properties of extreme point or vertex solutions to such problems. I will then introduce the basic framework of the method using the assignment problem, and show its application by re-deriving the approximation results of Shmoys and Tardos for the generalized assignment problem.
I will then discuss linear characterizations for the spanning tree polyhedron in undirected graphs and give a new proof of integrality using an iterative method. I will then illustrate an application to approximating the degree-bounded version of the undirected problem, by proving the results of Goemans and Lau & Singh.
I will continue with showing how these methods for spanning trees simplify and generalize to showing linear descriptions of maximum weight matroid bases and also maximum weight sets that are independent in two different matroids. This also leads to good additive approximation algorithms for a bounded degree version of the matroid basis problem.
I will close with applications of the iterative method by revisiting Jain's original proof for the SNDP and giving a new proof that unifies its treatment with that for the Symmetric TSP polyhedron (describing joint work with Nagarajan and Singh). I will also outline the versatility of the method by pointing out the other problems for which the method has been applied, summarizing the discussion in a recent monograph I have co-authored on this topic with Lap Chi Lau and Mohit Singh (published by Cambridge University Press, 2011).
Cite as
R. Ravi. Iterative Methods in Combinatorial Optimization (Invited Talk). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, p. 24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{ravi:LIPIcs.STACS.2012.24,
author = {Ravi, R.},
title = {{Iterative Methods in Combinatorial Optimization}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {24--24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.24},
URN = {urn:nbn:de:0030-drops-33876},
doi = {10.4230/LIPIcs.STACS.2012.24},
annote = {Keywords: combinatorial optimization, linear programming, matroid}
}
Document
Invited Talk
On Randomness in Hash Functions (Invited Talk)
Abstract
In the talk, we shall discuss quality measures for hash functions used
in data structures and algorithms, and survey positive and negative
results. (This talk is not about cryptographic hash functions.)
For the analysis of algorithms involving hash functions, it is often
convenient to assume the hash functions used behave fully randomly; in
some cases there is no analysis known that avoids this assumption. In
practice, one needs to get by with weaker hash functions that can be
generated by randomized algorithms. A well-studied range of applications concern realizations of dynamic dictionaries (linear
probing, chained hashing, dynamic perfect hashing, cuckoo hashing and
its generalizations) or Bloom filters and their variants.
A particularly successful and useful means of classification are
Carter and Wegman's universal or k-wise independent classes,
introduced in 1977. A natural and widely used approach to analyzing
an algorithm involving hash functions is to show that it works if a
sufficiently strong universal class of hash functions is used, and to
substitute one of the known constructions of such classes. This
invites research into the question of just how much independence in
the hash functions is necessary for an algorithm to work. Some recent
analyses that gave impossibility results constructed rather artificial
classes that would not work; other results pointed out natural, widely
used hash classes that would not work in a particular application.
Only recently it was shown that under certain assumptions on some
entropy present in the set of keys even 2-wise independent hash
classes will lead to strong randomness properties in the hash values.
The negative results show that these results may not be taken as
justification for using weak hash classes indiscriminately, in
particular for key sets with structure.
When stronger independence properties are needed for a theoretical
analysis, one may resort to classic constructions. Only in 2003 it
was found out how full randomness can be simulated using only linear
space overhead (which is optimal). The "split-and-share" approach
can be used to justify the full randomness assumption in some
situations in which full randomness is needed for the analysis to go
through, like in many applications involving multiple hash functions
(e.g., generalized versions of cuckoo hashing with multiple hash
functions or larger bucket sizes, load balancing, Bloom filters and
variants, or minimal perfect hash function constructions).
For practice, efficiency considerations beyond constant factors are
important. It is not hard to construct very efficient 2-wise
independent classes. Using k-wise independent classes for constant k
bigger than 3 has become feasible in practice only by new
constructions involving tabulation. This goes together well with the
quite new result that linear probing works with 5-independent hash
functions.
Recent developments suggest that the classification of hash function
constructions by their degree of independence alone may not be
adequate in some cases. Thus, one may want to analyze the behavior of
specific hash classes in specific applications, circumventing the
concept of k-wise independence. Several such results were recently
achieved concerning hash functions that utilize tabulation. In
particular if the analysis of the application involves using
randomness properties in graphs and hypergraphs (generalized cuckoo
hashing, also in the version with a "stash", or load balancing), a
hash class combining k-wise independence with tabulation has turned
out to be very powerful.
Cite as
Martin Dietzfelbinger. On Randomness in Hash Functions (Invited Talk). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 25-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{dietzfelbinger:LIPIcs.STACS.2012.25,
author = {Dietzfelbinger, Martin},
title = {{On Randomness in Hash Functions}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {25--28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.25},
URN = {urn:nbn:de:0030-drops-33884},
doi = {10.4230/LIPIcs.STACS.2012.25},
annote = {Keywords: Algorithms, hash functions, randomized algorithms, data structures, graphs, hypergraphs}
}
Document
Invited Talk
Pseudo-deterministic Algorithms (Invited Talk)
Abstract
In this talk we describe a new type of probabilistic algorithm which we call "Bellagio" Algorithms: a randomized algorithm which is guaranteed to run in expected polynomial time, and to produce a correct and unique solution with high probability. These algorithms are pseudo-deterministic: they can not be distinguished from deterministic algorithms in polynomial time by a probabilistic polynomial time observer with black box access to the algorithm.
We show a necessary and sufficient condition for the existence of a
Bellagio Algorithm for an NP relation R: R has a Bellagio
algorithm if and only if it is deterministically reducible to some
decision problem in BPP. Several examples of Bellagio algorithms, for
well known problems in algebra and graph theory which improve on
deterministic solutions, follow.
The notion of pseudo-deterministic algorithms (or more generally
computations) is interesting beyond just sequential algorithms. In
particular, it has long been known that it is impossible to solve
deterministically tasks such as "consensus" in a faulty
distributed systems, whereas randomized protocols can achieve
consensus in expected constant time. We thus explore the notion of
pseudo-deterministic fault tolerant distributed protocols: randomized
protocols which are polynomial time indistinguishable from
deterministic protocols in presence of faults.
Cite as
Shafi Goldwasser. Pseudo-deterministic Algorithms (Invited Talk). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, p. 29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{goldwasser:LIPIcs.STACS.2012.29,
author = {Goldwasser, Shafi},
title = {{Pseudo-deterministic Algorithms}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {29--29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.29},
URN = {urn:nbn:de:0030-drops-34435},
doi = {10.4230/LIPIcs.STACS.2012.29},
annote = {Keywords: randomized algorithms, distributed computing, Monte Carlo, Las Vegas}
}
Document
13/9-approximation for Graphic TSP
Abstract
The Travelling Salesman Problem is one of the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3/2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4/3.
Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (2011), and then by Momke and Svensson (2011). In this paper, we provide an improved analysis of the approach used by the latter, yielding a bound of 13/9 on the approximation factor, as well as a bound of 19/12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.
Cite as
Marcin Mucha. 13/9-approximation for Graphic TSP. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 30-41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{mucha:LIPIcs.STACS.2012.30,
author = {Mucha, Marcin},
title = {{13/9-approximation for Graphic TSP}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {30--41},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.30},
URN = {urn:nbn:de:0030-drops-34025},
doi = {10.4230/LIPIcs.STACS.2012.30},
annote = {Keywords: approximation algorithms, travelling salesman problem}
}
Document
A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems
Abstract
We consider the monotone submodular k-set packing problem in the context of the more general problem of maximizing a monotone submodular function in a k-exchange system. These systems, introduced by Feldman et al. [Feldman,2011], generalize the matroid k-parity problem in a wide class of matroids and capture many other combinatorial optimization problems. We give a deterministic, non-oblivious local search algorithm that attains an approximation ratio of (k + 3)/2 + epsilon for the problem of maximizing a monotone submodular function in a k-exchange system, improving on the best known result of k+epsilon, and answering an open question posed by Feldman et al.
Cite as
Justin Ward. A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 42-53, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{ward:LIPIcs.STACS.2012.42,
author = {Ward, Justin},
title = {{A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {42--53},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.42},
URN = {urn:nbn:de:0030-drops-34315},
doi = {10.4230/LIPIcs.STACS.2012.42},
annote = {Keywords: k-set packing, k-exchange systems, submodular maximization, local search, approximation algorithms}
}
Document
A Pumping Lemma for Pushdown Graphs of Any Level
Abstract
We present a pumping lemma for the class of epsilon-contractions of pushdown graphs of level n, for each n. A pumping lemma was proposed by Blumensath, but there is an irrecoverable error in his proof; we present a new proof. Our pumping lemma also improves the bounds given in the invalid paper of Blumensath.
Cite as
Pawel Parys. A Pumping Lemma for Pushdown Graphs of Any Level. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 54-65, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{parys:LIPIcs.STACS.2012.54,
author = {Parys, Pawel},
title = {{A Pumping Lemma for Pushdown Graphs of Any Level}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {54--65},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.54},
URN = {urn:nbn:de:0030-drops-33940},
doi = {10.4230/LIPIcs.STACS.2012.54},
annote = {Keywords: pushdown graph, epsilon-contraction, pumping lemma}
}
Document
Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth
Abstract
An algorithmic meta theorem for a logic and a class C of structures
states that all problems expressible in this logic can be solved
efficiently for inputs from $C$. The prime example is Courcelle's
Theorem, which states that monadic second-order (MSO) definable
problems are linear-time solvable on graphs of bounded tree width. We
contribute new algorithmic meta theorems, which state that
MSO-definable problems are (a) solvable by uniform constant-depth
circuit families (AC0 for decision problems and TC0 for counting
problems) when restricted to input structures of bounded tree depth
and (b) solvable by uniform logarithmic-depth circuit families (NC1
for decision problems and #NC1 for counting problems) when a tree
decomposition of bounded width in term representation is part of the
input. Applications of our theorems include a TC0-completeness proof
for the unary version of integer linear programming with a fixed
number of equations and extensions of a recent result that counting
the number of accepting paths of a visible pushdown automaton lies in
#NC1. Our main technical contributions are a new tree automata model
for unordered, unranked, labeled trees; a method for representing the
tree automata's computations algebraically using convolution circuits;
and a lemma on computing balanced width-3 tree decompositions of trees
in TC0, which encapsulates most of the technical difficulties
surrounding earlier results connecting tree automata and NC1.
Cite as
Michael Elberfeld, Andreas Jakoby, and Till Tantau. Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 66-77, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{elberfeld_et_al:LIPIcs.STACS.2012.66,
author = {Elberfeld, Michael and Jakoby, Andreas and Tantau, Till},
title = {{Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {66--77},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.66},
URN = {urn:nbn:de:0030-drops-34059},
doi = {10.4230/LIPIcs.STACS.2012.66},
annote = {Keywords: algorithmic meta theorem, monadic second-order logic, circuit complexity, tree width, tree depth}
}
Document
An Approximation Algorithm for #k-SAT
Abstract
We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k>=3 within a running time that is not only non-trivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For 4-CNF input the bound increases to O(1.6155^n).
We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way. Relaxing the requirements on the quality of the approximation, on k-CNF input we obtain significantly reduced running times in comparison to the above bounds.
Cite as
Marc Thurley. An Approximation Algorithm for #k-SAT. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 78-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{thurley:LIPIcs.STACS.2012.78,
author = {Thurley, Marc},
title = {{An Approximation Algorithm for #k-SAT}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {78--87},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.78},
URN = {urn:nbn:de:0030-drops-34400},
doi = {10.4230/LIPIcs.STACS.2012.78},
annote = {Keywords: #k-SAT, approximate counting, exponential algorithms}
}
Document
Asymptotic enumeration of Minimal Automata
Abstract
We determine the asymptotic proportion of minimal automata, within
n-state accessible deterministic complete automata over a k-letter
alphabet, with the uniform distribution over the possible transition
structures, and a binomial distribution over terminal states, with
arbitrary parameter b. It turns out that a fraction ~ 1-C(k,b)
n^{-k+2} of automata is minimal, with C(k,b) a function, explicitly
determined, involving the solution of a transcendental equation.
Cite as
Frédérique Bassino, Julien David, and Andrea Sportiello. Asymptotic enumeration of Minimal Automata. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 88-99, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bassino_et_al:LIPIcs.STACS.2012.88,
author = {Bassino, Fr\'{e}d\'{e}rique and David, Julien and Sportiello, Andrea},
title = {{Asymptotic enumeration of Minimal Automata}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {88--99},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.88},
URN = {urn:nbn:de:0030-drops-34328},
doi = {10.4230/LIPIcs.STACS.2012.88},
annote = {Keywords: minimal automata, regular languages, enumeration of random structures}
}
Document
Balanced Partitions of Trees and Applications
Abstract
We study the k-BALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ceil(n/k) while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it cannot be approximated within any factor in polynomial time. However, little is known about restricted graph classes. We show that for trees k-BALANCED PARTITIONING remains surprisingly hard. In particular, approximating the cut size is APX-hard even if the maximum degree of the tree is constant. If instead the diameter of the tree is bounded by a constant, we show
that it is NP-hard to approximate the cut size within n^c, for any constant c<1.
In the face of the hardness results, we show that allowing near-balanced solutions, in which there are at most (1+eps)ceil(n/k)
vertices in any of the k sets, admits a PTAS for trees. Remarkably, the computed cut size is no larger than that of an optimal balanced solution. In the final section of our paper, we harness results on embedding graph metrics into tree metrics to extend our PTAS for trees to general graphs. In addition to being conceptually simpler and easier to analyse, our scheme improves the best factor known on the cut size of near-balanced solutions from O(log^{1.5}(n)/eps^2) [Andreev and Räcke TCS 2006] to 0(log n), for weighted graphs. This also settles a question posed by Andreev and Räcke of whether an algorithm with approximation guarantees on the cut size independent from eps exists.
Cite as
Andreas Emil Feldmann and Luca Foschini. Balanced Partitions of Trees and Applications. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 100-111, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{feldmann_et_al:LIPIcs.STACS.2012.100,
author = {Feldmann, Andreas Emil and Foschini, Luca},
title = {{Balanced Partitions of Trees and Applications}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {100--111},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.100},
URN = {urn:nbn:de:0030-drops-34081},
doi = {10.4230/LIPIcs.STACS.2012.100},
annote = {Keywords: balanced partitioning, bicriteria approximation, hardness of approximation, tree embeddings}
}
Document
Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property
Abstract
In this paper we present an implicit dynamic dictionary with the working-set property, supporting insert(e) and delete(e) in O(log n) time, predecessor(e) in O(log l_{p(e)}) time, successor(e) in O(log l_{s(e)}) time and search(e) in O(log min(l_{p(e)},l_{e}, l_{s(e)})) time, where n is the number of elements stored in the dictionary, l_{e} is the number of distinct elements searched for since element e was last searched for and p(e) and s(e) are the predecessor and successor of e, respectively. The time-bounds are all worst-case. The dictionary stores the elements in an array of size n using *no* additional space. In the cache-oblivious model the log is base B and the cache-obliviousness is due to our black box use of an existing cache-oblivious implicit dictionary. This is the first implicit dictionary supporting predecessor and successor searches in the working-set bound. Previous implicit structures required O(log n) time.
Cite as
Gerth Stølting Brodal and Casper Kejlberg-Rasmussen. Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 112-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{stltingbrodal_et_al:LIPIcs.STACS.2012.112,
author = {St{\o}lting Brodal, Gerth and Kejlberg-Rasmussen, Casper},
title = {{Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {112--123},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.112},
URN = {urn:nbn:de:0030-drops-34101},
doi = {10.4230/LIPIcs.STACS.2012.112},
annote = {Keywords: working-set property, dictionary, implicit, cache-oblivious, worst-case, external memory, I/O efficient}
}
Document
Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified
Abstract
We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating from its mean by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is at most exp(-Omega( delta^2 mu t / T )) for 0 <= delta <= 1, and exp(-Omega( delta mu t / T )) for delta > 1. In fact, the bounds hold even if the weight functions f_i's for i in [t] are distinct, provided that all of them have the same mean mu.
We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on the spectral expansion lambda of M, which is the square root of the second largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t| >= delta mu t ] is at most exp(-Omega( delta^2 (1-lambda) mu t )) for 0 <= delta <= 1, and exp(-Omega( delta (1-lambda) mu t )) for delta > 1.
Both of our results extend to continuous-time Markov chains, and to the case where the walk starts from an arbitrary distribution x, at a price of a multiplicative factor depending on the distribution x in the concentration bounds.
Cite as
Kai-Min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher. Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 124-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{chung_et_al:LIPIcs.STACS.2012.124,
author = {Chung, Kai-Min and Lam, Henry and Liu, Zhenming and Mitzenmacher, Michael},
title = {{Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {124--135},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.124},
URN = {urn:nbn:de:0030-drops-34374},
doi = {10.4230/LIPIcs.STACS.2012.124},
annote = {Keywords: probabilistic analysis, tail bounds, Markov chains}
}
Document
Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)
Abstract
In this paper, a compressed membership problem for finite automata, both deterministic (DFAs) and non-deterministic (NFAs), with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, in order to reflect the recompression in the NFA, we need to modify it only a little, in particular its size stays polynomial in the input size.
Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter [Plandowski Rytter 1999] and extending the partial result of Lohrey and Mathissen [Lohrey and Mathissen 2011];
as this problem is known to be NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P, and this problem is known to be P-hard.
Cite as
Artur Jez. Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 136-147, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{jez:LIPIcs.STACS.2012.136,
author = {Jez, Artur},
title = {{Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {136--147},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.136},
URN = {urn:nbn:de:0030-drops-34068},
doi = {10.4230/LIPIcs.STACS.2012.136},
annote = {Keywords: Compressed membership problem, SLP, Finite Automata, Algorithms for compressed data}
}
Document
Concurrency Makes Simple Theories Hard
Abstract
A standard way of building concurrent systems is by composing several
individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like Hennessy-Milner (HM) logic and its extension with the reachability operator (EF-logic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EF-logic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by Feferman-Vaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems.
Cite as
Stefan Göller and Anthony Widjaja Lin. Concurrency Makes Simple Theories Hard. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 148-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{goller_et_al:LIPIcs.STACS.2012.148,
author = {G\"{o}ller, Stefan and Widjaja Lin, Anthony},
title = {{Concurrency Makes Simple Theories Hard}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {148--159},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.148},
URN = {urn:nbn:de:0030-drops-34214},
doi = {10.4230/LIPIcs.STACS.2012.148},
annote = {Keywords: Modal Logic, Model Checking, Asynchronous Product, Pushdown Systems, Feferman-Vaught, Nonelementary lower bounds}
}
Document
Conflict-free Chromatic Art Gallery Coverage
Abstract
We consider a chromatic variant of the art gallery problem, where each
guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen
by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that
ensure a conflict-free covering of all n-vertex polygons? We call this
the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor
networks where colors indicate the wireless frequencies assigned to a
set of covering "landmarks" in the environment so that a mobile robot
can always communicate with at least one landmark in its line-of-sight
range without interference.
Our main result shows that k(n) is O(log n) for orthogonal and for
monotone polygons, and O(log^2 n) for arbitrary simple polygons. By
contrast, if all guards visible from each point must have distinct
colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011].
Cite as
Andreas Bärtschi and Subhash Suri. Conflict-free Chromatic Art Gallery Coverage. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 160-171, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bartschi_et_al:LIPIcs.STACS.2012.160,
author = {B\"{a}rtschi, Andreas and Suri, Subhash},
title = {{Conflict-free Chromatic Art Gallery Coverage}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {160--171},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.160},
URN = {urn:nbn:de:0030-drops-33952},
doi = {10.4230/LIPIcs.STACS.2012.160},
annote = {Keywords: art gallery problem, conflict-free coloring, visibility}
}
Document
Constant compression and random weights
Abstract
Omega numbers, as considered in algorithmic randomness, are by definition real numbers that are equal to the halting probability of a universal prefix-free Turing machine. Omega numbers are obviously left-r.e., i.e., are effectively approximable from below. Furthermore, among all left-r.e. real numbers in the appropriate range between 0 and 1, the Omega numbers admit well-known characterizations as the ones that are Martin-Löf random, as well as the ones such that any of their effective approximation from below is slower than any other effective approximation from below to any other real, up to a constant factor. In what follows, we obtain a further characterization of Omega numbers in terms of Theta numbers.
Tadaki considered for a given prefix-free Turing machine and some natural number a the set of all strings that are compressed by this machine by at least a bits relative to their length, and he introduced Theta numbers as the weight of sets of this form.
He showed that in the case of a universal prefix-free Turing machine any Theta number is an Omega number and he asked whether this implication can be reversed. We answer his question in the affirmative and thus obtain a new characterization of Omega numbers.
In addition to the one-sided case of the set of all strings compressible by at least a certain number a of bits, we consider sets that comprise all strings that are compressible by at least a but no more than b bits, and we call the weight of such a set a two-sided Theta number. We demonstrate that in the case of a universal prefix-free Turing machine, for given a and all sufficiently large b the corresponding two-sided Theta number is again an Omega number. Conversely, any Omega number can be realized as two-sided Theta number for any pair of natural numbers a and b>a.
Cite as
Wolfgang Merkle and Jason Teutsch. Constant compression and random weights. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 172-181, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{merkle_et_al:LIPIcs.STACS.2012.172,
author = {Merkle, Wolfgang and Teutsch, Jason},
title = {{Constant compression and random weights}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {172--181},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.172},
URN = {urn:nbn:de:0030-drops-34351},
doi = {10.4230/LIPIcs.STACS.2012.172},
annote = {Keywords: computational complexity, Kolmogorov complexity, algorithmic randomness, Omega number}
}
Document
Contraction checking in graphs on surfaces
Abstract
The Contraction Checking problem asks, given two graphs H and G as input, whether H can be obtained from G by a sequence of edge contractions. Contraction Checking remains NP-complete, even when H is fixed. We show that this is not the case when G is embeddable in a surface of fixed Euler genus. In particular, we give an algorithm that solves Contraction Checking in f(h,g)*|V(G)|^3 steps, where h is the size of H and g is the Euler genus of the input graph G.
Cite as
Marcin Kaminski and Dimitrios M. Thilikos. Contraction checking in graphs on surfaces. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 182-193, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{kaminski_et_al:LIPIcs.STACS.2012.182,
author = {Kaminski, Marcin and Thilikos, Dimitrios M.},
title = {{Contraction checking in graphs on surfaces}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {182--193},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.182},
URN = {urn:nbn:de:0030-drops-34032},
doi = {10.4230/LIPIcs.STACS.2012.182},
annote = {Keywords: Surfaces, Topological Minors, Contractions, Parameterized algorithms, Linkages}
}
Document
Distribution of the number of accessible states in a random deterministic automaton
Abstract
We study the distribution of the number of accessible states in deterministic and complete automata with n states over a k-letters alphabet. We show that as n tends to infinity and for a fixed alphabet size, the distribution converges in law toward a Gaussian centered around vk n and of standard deviation equivalent to sk n^(1/2), for some explicit constants vk and sk. Using this characterization, we give a simple algorithm for random uniform generation of accessible deterministic and complete automata of size n of expected complexity O(n^(3/2)), which matches the best methods known so far. Moreover, if we allow a variation around n in the size of the output automaton, our algorithm is the first solution of linear expected complexity. Finally we show how this work can be used to study accessible automata (which are difficult to apprehend from a combinatorial point of view) through the prism of the simpler deterministic and complete automata. As an example, we show how the average complexity in O(n log log n) for Moore's minimization algorithm obtained by David for deterministic and complete automata can be extended to accessible automata.
Cite as
Arnaud Carayol and Cyril Nicaud. Distribution of the number of accessible states in a random deterministic automaton. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 194-205, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{carayol_et_al:LIPIcs.STACS.2012.194,
author = {Carayol, Arnaud and Nicaud, Cyril},
title = {{Distribution of the number of accessible states in a random deterministic automaton}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {194--205},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.194},
URN = {urn:nbn:de:0030-drops-34422},
doi = {10.4230/LIPIcs.STACS.2012.194},
annote = {Keywords: finite automata, random sampling, average complexity}
}
Document
Edge-disjoint Odd Cycles in 4-edge-connected Graphs
Abstract
Finding edge-disjoint odd cycles is one of the most important problems
in graph theory, graph algorithm and combinatorial optimization. In
fact, it is closely related to the well-known max-cut problem. One of
the difficulties of this problem is that the Erdös-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following: For any 4-edge-connected graph G=(V,E),
either G has edge-disjoint k odd cycles or there exists an edge set F
subseteq E with |F| <= f(k) such that G-F is bipartite. We note that
the 4-edge-connectivity is best possible in this statement.
Similar approach can be applied to an algorithmic question. Suppose
that the input graph G is a 4-edge-connected graph with n vertices.
We show that, for any epsilon > 0, if k = O ((log log log n)^{1/2-epsilon}), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n.
Cite as
Ken-ichi Kawarabayashi and Yusuke Kobayashi. Edge-disjoint Odd Cycles in 4-edge-connected Graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 206-217, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{kawarabayashi_et_al:LIPIcs.STACS.2012.206,
author = {Kawarabayashi, Ken-ichi and Kobayashi, Yusuke},
title = {{Edge-disjoint Odd Cycles in 4-edge-connected Graphs}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {206--217},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.206},
URN = {urn:nbn:de:0030-drops-34173},
doi = {10.4230/LIPIcs.STACS.2012.206},
annote = {Keywords: odd-cycles, disjoint paths problem, Erd\"{o}s-Posa property, packing algorithm, 4-edge-connectivity}
}
Document
Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P
Abstract
Power circuits are data structures which support efficient algorithms
for highly compressed integers. Using this new data structure it has
been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. (The algorithm has been implemented and is available in the CRAG library.) 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of Power Circuits.
Cite as
Volker Diekert, Jürn Laun, and Alexander Ushakov. Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 218-229, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{diekert_et_al:LIPIcs.STACS.2012.218,
author = {Diekert, Volker and Laun, J\"{u}rn and Ushakov, Alexander},
title = {{Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {218--229},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.218},
URN = {urn:nbn:de:0030-drops-34204},
doi = {10.4230/LIPIcs.STACS.2012.218},
annote = {Keywords: Algorithmic group theory, Data structures, Compression, Word Problem}
}
Document
Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion
Abstract
We present two recursive techniques to construct compressed sensing schemes that can be "decoded" in sub-linear time. The first technique is based on the well studied code composition method called code concatenation where the "outer" code has strong list recoverability properties. This technique uses only one level of recursion and critically uses the power of list recovery. The second recursive technique is conceptually similar, and has multiple recursion levels. The following compressed sensing results are obtained using these techniques:
- Strongly explicit efficiently decodable l_1/l_1 compressed sensing matrices: We present a strongly explicit ("for all") compressed sensing measurement matrix with O(d^2log^2 n) measurements that can output near-optimal d-sparse approximations in time poly(d log n).
- Near-optimal efficiently decodable l_1/l_1 compressed sensing matrices for non-negative signals: We present two randomized constructions of ("for all") compressed sensing matrices with near optimal number of measurements: O(d log n loglog_d n) and O_{m,s}(d^{1+1/s} log n (log^(m) n)^s), respectively, for any integer parameters s,m>=1. Both of these constructions can output near optimal d-sparse approximations for non-negative signals in time poly(d log n).
To the best of our knowledge, none of the results are dominated by existing results in the literature.
Cite as
Hung Q. Ngo, Ely Porat, and Atri Rudra. Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 230-241, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{ngo_et_al:LIPIcs.STACS.2012.230,
author = {Ngo, Hung Q. and Porat, Ely and Rudra, Atri},
title = {{Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {230--241},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.230},
URN = {urn:nbn:de:0030-drops-34011},
doi = {10.4230/LIPIcs.STACS.2012.230},
annote = {Keywords: Compressed Sensing, Sub-Linear Time Decoding, List-Recoverable Codes}
}
Document
Ehrenfeucht-Fraïssé goes elementarily automatic for structures of bounded degree
Abstract
Many relational structures are automatically presentable, i.e. elements of the domain can be seen as words over a finite alphabet and equality and other atomic relations are represented with finite automata. The first-order theories over such structures are known to be primitive recursive, which is shown by the inductive construction of an automaton representing any relation definable in the first-order logic. We propose a general method based on Ehrenfeucht-Fraïssé games to give upper bounds on the size of these automata and on the time required to build them. We apply this method for two different automatic structures which have elementary decision procedures, Presburger Arithmetic and automatic structures of bounded
degree. For the latter no upper bound on the size of the automata was known. We conclude that the very general and simple automata-based algorithm works well to decide the first-order theories over these structures.
Cite as
Antoine Durand-Gasselin and Peter Habermehl. Ehrenfeucht-Fraïssé goes elementarily automatic for structures of bounded degree. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 242-253, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{durandgasselin_et_al:LIPIcs.STACS.2012.242,
author = {Durand-Gasselin, Antoine and Habermehl, Peter},
title = {{Ehrenfeucht-Fra\"{i}ss\'{e} goes elementarily automatic for structures of bounded degree}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {242--253},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.242},
URN = {urn:nbn:de:0030-drops-34198},
doi = {10.4230/LIPIcs.STACS.2012.242},
annote = {Keywords: Automata-based decision procedures for logical theories, Automatic Structures, Ehrenfeucht-Fra\"{i}ss\'{e} Games, Logics, Complexity}
}
Document
Improved Bounds for Bipartite Matching on Surfaces
Abstract
We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus:
(1) BPM in planar graphs is in UL (improves upon the SPL bound from Datta, Kulkarni, and Roy;
(2) BPM in constant genus graphs is in NL (orthogonal to the SPL bound from Datta, Kulkarni, Tewari, and Vinodchandran.;
(3) BPM in poly-logarithmic genus graphs is in NC; (extends the NC bound for O(log n) genus graphs from Mahajan and Varadarajan, and Kulkarni, Mahajan, and Varadarajan.
For Part (1) we combine the flow technique of Miller and Naor with the double counting technique of Reinhardt and Allender . For Part (2) and (3) we extend Miller and Naor's result to higher genus surfaces in the spirit of Chambers, Erickson and Nayyeri.
Cite as
Samir Datta, Arjun Gopalan, Raghav Kulkarni, and Raghunath Tewari. Improved Bounds for Bipartite Matching on Surfaces. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 254-265, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{datta_et_al:LIPIcs.STACS.2012.254,
author = {Datta, Samir and Gopalan, Arjun and Kulkarni, Raghav and Tewari, Raghunath},
title = {{Improved Bounds for Bipartite Matching on Surfaces}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {254--265},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.254},
URN = {urn:nbn:de:0030-drops-34141},
doi = {10.4230/LIPIcs.STACS.2012.254},
annote = {Keywords: Perfect Matching, Graphs on Surfaces, Space Complexity, NC, UL}
}
Document
Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices
Abstract
We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.
Cite as
Ioannis Koutis, Alex Levin, and Richard Peng. Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 266-277, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{koutis_et_al:LIPIcs.STACS.2012.266,
author = {Koutis, Ioannis and Levin, Alex and Peng, Richard},
title = {{Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {266--277},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.266},
URN = {urn:nbn:de:0030-drops-34348},
doi = {10.4230/LIPIcs.STACS.2012.266},
annote = {Keywords: Spectral sparsification, linear system solving}
}
Document
Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid
Abstract
A key theorem in algorithmic graph-minor theory is a min-max relation
between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson
and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties.
In 2008, Demaine and Hajiaghayi proved a remarkable linear min-max
relation for graphs excluding any fixed minor H: every H-minor-free
graph of treewidth at least c_H r has an r times r-grid minor for some
constant c_H. However, as they pointed out, there is still a major
problem left in this theorem. The problem is that their proof heavily
depends on Graph Minor Theory, most of which lacks explicit bounds and
is believed to have very large bounds. Hence c_H is not explicitly
given in the paper and therefore this result is usually not strong
enough to derive efficient algorithms.
Motivated by this problem, we give another (relatively short and
simple) proof of this result without using big machinery of Graph
Minor Theory. Hence we can give an explicit bound for c_H (an exponential function of a polynomial of |H|). Furthermore, our result
gives a constant w=2^O(r^2 log r) such that every graph of treewidth
at least w has an r times r-grid minor, which improves the previously
known best bound 2^Theta(r^5)$ given by Robertson, Seymour, and Thomas
in 1994.
Cite as
Ken-ichi Kawarabayashi and Yusuke Kobayashi. Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 278-289, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{kawarabayashi_et_al:LIPIcs.STACS.2012.278,
author = {Kawarabayashi, Ken-ichi and Kobayashi, Yusuke},
title = {{Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {278--289},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.278},
URN = {urn:nbn:de:0030-drops-34165},
doi = {10.4230/LIPIcs.STACS.2012.278},
annote = {Keywords: grid minor, treewidth, graph minor}
}
Document
Linear-Space Data Structures for Range Mode Query in Arrays
Abstract
A mode of a multiset S is an element a in S of maximum multiplicity;
that is, a occurs at least as frequently as any other element in S.
Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O(sqrt(n) loglog n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O(sqrt(n / log n)) worst-case time. Furthermore, we present strong evidence that a query time significantly below sqrt(n) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two sqrt(n) by sqrt(n) matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogonal range mode in higher dimensions (queries in near O(n^(1-1/2d)) time) and for halfspace range mode in higher dimensions (queries in O(n^(1-1/d^2)) time).
Cite as
Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T. Wilkinson. Linear-Space Data Structures for Range Mode Query in Arrays. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 290-301, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{chan_et_al:LIPIcs.STACS.2012.290,
author = {Chan, Timothy M. and Durocher, Stephane and Larsen, Kasper Green and Morrison, Jason and Wilkinson, Bryan T.},
title = {{Linear-Space Data Structures for Range Mode Query in Arrays}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {290--301},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.290},
URN = {urn:nbn:de:0030-drops-34254},
doi = {10.4230/LIPIcs.STACS.2012.290},
annote = {Keywords: mode, range query, data structure, linear space, array}
}
Document
Log-supermodular functions, functional clones and counting CSPs
Abstract
Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that non-negative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard as any problem in #P. Furthermore, any non-trivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all non-trivial counting CSPs (with non-negative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.
Cite as
Andrei A. Bulatov, Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. Log-supermodular functions, functional clones and counting CSPs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 302-313, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bulatov_et_al:LIPIcs.STACS.2012.302,
author = {Bulatov, Andrei A. and Dyer, Martin and Goldberg, Leslie Ann and Jerrum, Mark},
title = {{Log-supermodular functions, functional clones and counting CSPs}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {302--313},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.302},
URN = {urn:nbn:de:0030-drops-34078},
doi = {10.4230/LIPIcs.STACS.2012.302},
annote = {Keywords: counting constraint satisfaction problems, approximation, complexity}
}
Document
Low Randomness Rumor Spreading via Hashing
Abstract
We consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple push-based protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudo-random generator, respectively. For several well-studied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability.
Previous explicit algorithms require Omega(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashing-based algorithm is within an O(log n)-factor of the theoretical minimum determined by [Giakkoupis and Woelfel, 2011].
Cite as
George Giakkoupis, Thomas Sauerwald, He Sun, and Philipp Woelfel. Low Randomness Rumor Spreading via Hashing. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 314-325, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{giakkoupis_et_al:LIPIcs.STACS.2012.314,
author = {Giakkoupis, George and Sauerwald, Thomas and Sun, He and Woelfel, Philipp},
title = {{Low Randomness Rumor Spreading via Hashing}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {314--325},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.314},
URN = {urn:nbn:de:0030-drops-34417},
doi = {10.4230/LIPIcs.STACS.2012.314},
annote = {Keywords: Parallel and Distributed Computing, Randomness, Rumor Spreading}
}
Document
Lower Bounds on the Complexity of MSO_1 Model-Checking
Abstract
One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.
Cite as
Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326,
author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath},
title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {326--337},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326},
URN = {urn:nbn:de:0030-drops-34185},
doi = {10.4230/LIPIcs.STACS.2012.326},
annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity}
}
Document
LP can be a cure for Parameterized Problems
Abstract
We investigate the parameterized complexity of Vertex Cover parameterized above the optimum value of the linear programming (LP) relaxation of the integer linear programming formulation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that even the most straightforward branching algorithm (after some preprocessing) results in an O^*(2.6181^r) algorithm for the problem where r is the excess of the
vertex cover size over the LP optimum. We write O^*(f(k)) for a time complexity of the form O(f(k)n^{O(1)}), where f(k) grows exponentially with k.
Then, using known and new reductions, we give O^*(2.6181^k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an O^*(1.6181^k) algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The notable improvement is the bound for Odd Cycle Transversal for which this is the first major improvement after the first algorithm that showed it fixed-parameter tractable in 2003. We also observe that using our algorithm, one can obtain a simple kernel for the classical vertex cover problem with at most 2k-O(log k) vertices.
Cite as
N.S. Narayanaswamy, Venkatesh Raman, M.S. Ramanujan, and Saket Saurabh. LP can be a cure for Parameterized Problems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 338-349, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{narayanaswamy_et_al:LIPIcs.STACS.2012.338,
author = {Narayanaswamy, N.S. and Raman, Venkatesh and Ramanujan, M.S. and Saurabh, Saket},
title = {{LP can be a cure for Parameterized Problems}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {338--349},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.338},
URN = {urn:nbn:de:0030-drops-34291},
doi = {10.4230/LIPIcs.STACS.2012.338},
annote = {Keywords: Algorithms and data structures. Graph Algorithms, Parameterized Algorithms.}
}
Document
Mind Change Speed-up for Learning Languages from Positive Data
Abstract
Within the frameworks of learning in the limit of indexed classes of
recursive languages from positive data and automatic learning in the
limit of indexed classes of regular languages (with automatically
computable sets of indices), we study the problem of minimizing the
maximum number of mind changes F_M(n) by a learner M on all languages
with indices not exceeding n. For inductive inference of recursive
languages, we establish two conditions under which F_M(n) can be made
smaller than any recursive unbounded non-decreasing function. We also
establish how F_M(n) is affected if at least one of these two
conditions does not hold. In the case of automatic learning, some
partial results addressing speeding up the function F_M(n) are obtained.
Cite as
Sanjay Jain and Efim Kinber. Mind Change Speed-up for Learning Languages from Positive Data. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 350-361, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{jain_et_al:LIPIcs.STACS.2012.350,
author = {Jain, Sanjay and Kinber, Efim},
title = {{Mind Change Speed-up for Learning Languages from Positive Data}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {350--361},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.350},
URN = {urn:nbn:de:0030-drops-33936},
doi = {10.4230/LIPIcs.STACS.2012.350},
annote = {Keywords: Algorithmic and automatic learning, mind changes, speedup}
}
Document
Monomials in arithmetic circuits: Complete problems in the counting hierarchy
Abstract
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before. We also study these questions for circuits computing multilinear polynomials.
Cite as
Hervé Fournier, Guillaume Malod, and Stefan Mengel. Monomials in arithmetic circuits: Complete problems in the counting hierarchy. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 362-373, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{fournier_et_al:LIPIcs.STACS.2012.362,
author = {Fournier, Herv\'{e} and Malod, Guillaume and Mengel, Stefan},
title = {{Monomials in arithmetic circuits: Complete problems in the counting hierarchy}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {362--373},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.362},
URN = {urn:nbn:de:0030-drops-34240},
doi = {10.4230/LIPIcs.STACS.2012.362},
annote = {Keywords: arithmetic circuits, counting problems, polynomials}
}
Document
Motion planning with pulley, rope, and baskets
Abstract
We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported into the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given
threshold.
We prove that it is Pi-2-p-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions.
Cite as
Christian E.J. Eggermont and Gerhard J. Woeginger. Motion planning with pulley, rope, and baskets. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 374-383, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{eggermont_et_al:LIPIcs.STACS.2012.374,
author = {Eggermont, Christian E.J. and Woeginger, Gerhard J.},
title = {{Motion planning with pulley, rope, and baskets}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {374--383},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.374},
URN = {urn:nbn:de:0030-drops-33900},
doi = {10.4230/LIPIcs.STACS.2012.374},
annote = {Keywords: planning and scheduling; computational complexity}
}
Document
On Computing Pareto Stable Assignments
Abstract
Assignment between two parties in a two-sided matching market has been one of the central questions studied in economics, due to its extensive applications, focusing on different solution concepts with different objectives. One of the most important and well-studied ones is that of stability, proposed by Gale and Shapley, which captures fairness condition in a model where every individual in the market has a preference of the other side. When the preferences have indifferences (i.e., ties), a stable outcome need not be Pareto efficient, causing a loss in efficiency. The solution concept Pareto stability, which requires both stability and Pareto efficiency, offers a refinement of the solution concept stability in the sense that it captures both fairness and efficiency.
We study the algorithmic question of computing a Pareto stable assignment in a many-to-many matching market model, where both sides of the market can have multiunit capacities (i.e., demands) and can be matched with multiple partners given the capacity constraints. We provide an algorithm to efficiently construct an assignment that is simultaneously stable and Pareto efficient; our result immediately implies the existence of a Pareto stable assignment for this model.
Cite as
Ning Chen. On Computing Pareto Stable Assignments. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 384-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{chen:LIPIcs.STACS.2012.384,
author = {Chen, Ning},
title = {{On Computing Pareto Stable Assignments}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {384--395},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.384},
URN = {urn:nbn:de:0030-drops-34042},
doi = {10.4230/LIPIcs.STACS.2012.384},
annote = {Keywords: Algorithm, stable matching, Pareto efficiency}
}
Document
On the separation question for tree languages
Abstract
We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game.
Cite as
André Arnold, Henryk Michalewski, and Damian Niwinski. On the separation question for tree languages. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 396-407, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{arnold_et_al:LIPIcs.STACS.2012.396,
author = {Arnold, Andr\'{e} and Michalewski, Henryk and Niwinski, Damian},
title = {{On the separation question for tree languages}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {396--407},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.396},
URN = {urn:nbn:de:0030-drops-34156},
doi = {10.4230/LIPIcs.STACS.2012.396},
annote = {Keywords: Alternating automata on infinite trees, Index hierarchy, Separation property}
}
Document
On the treewidth and related parameters of random geometric graphs
Abstract
We give asymptotically exact values for the treewidth tw(G) of a random geometric graph G(n,r) in [0,sqrt(n)]^2. More precisely, we show that there exists some c_1 > 0, such that for any constant 0 < r < c_1, tw(G)=Theta(log(n)/loglog(n)), and also, there exists some
c_2 > c_1, such that for any r=r(n)> c_2, tw(G)=Theta(r sqrt(n)). Our proofs show that for the corresponding values of r the same asymptotic bounds also hold for the pathwidth and treedepth of a random geometric graph.
Cite as
Dieter Mitsche and Guillem Perarnau. On the treewidth and related parameters of random geometric graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 408-419, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{mitsche_et_al:LIPIcs.STACS.2012.408,
author = {Mitsche, Dieter and Perarnau, Guillem},
title = {{On the treewidth and related parameters of random geometric graphs}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {408--419},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.408},
URN = {urn:nbn:de:0030-drops-34280},
doi = {10.4230/LIPIcs.STACS.2012.408},
annote = {Keywords: Random geometric graphs, treewidth, treedepth}
}
Document
Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation
Abstract
The analysis of randomized search heuristics on classes of functions
is fundamental for the understanding of the underlying stochastic
process and the development of suitable proof techniques. Recently,
remarkable progress has been made in bounding the expected
optimization time of the simple (1+1) EA on the class of linear
functions. We improve the best known bound in this setting from
(1.39+o(1))(en ln n) to (en ln n)+O(n) in expectation and with high
probability, which is tight up to lower-order terms. Moreover, upper
and lower bounds for arbitrary mutations probabilities p are derived,
which imply expected polynomial optimization time as long as
p=O((ln n)/n) and which are tight if p=c/n for a constant c. As a
consequence, the standard mutation probability p=1/n is optimal for
all linear functions, and the (1+1) EA is found to be an optimal
mutation-based algorithm. Furthermore, the algorithm turns out to be
surprisingly robust since large neighborhood explored by the mutation
operator does not disrupt the search.
Cite as
Carsten Witt. Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 420-431, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{witt:LIPIcs.STACS.2012.420,
author = {Witt, Carsten},
title = {{Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {420--431},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.420},
URN = {urn:nbn:de:0030-drops-33920},
doi = {10.4230/LIPIcs.STACS.2012.420},
annote = {Keywords: Randomized Search Heuristics, Evolutionary Algorithms, Linear Functions, Running Time Analysis}
}
Document
Parameterized Complexity of Connected Even/Odd Subgraph Problems
Abstract
Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about
- a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and
- a connected k- vertex induced subgraph with all vertices of even degrees, the problem known as k-Vertex Eulerian Subgraph.
We resolve both open problems and thus complete the characterization of even/odd subgraph problems from parameterized complexity perspective. We show that k-Edge Connected Odd Subgraph is FPT and that k-Vertex Eulerian Subgraph is W[1]-hard.
Our FPT algorithm is based on a novel combinatorial result on the treewidth of minimal connected odd graphs with even amount of edges.
Cite as
Fedor V. Fomin and Petr A. Golovach. Parameterized Complexity of Connected Even/Odd Subgraph Problems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 432-440, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{fomin_et_al:LIPIcs.STACS.2012.432,
author = {Fomin, Fedor V. and Golovach, Petr A.},
title = {{Parameterized Complexity of Connected Even/Odd Subgraph Problems}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {432--440},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.432},
URN = {urn:nbn:de:0030-drops-33986},
doi = {10.4230/LIPIcs.STACS.2012.432},
annote = {Keywords: Parameterized complexity, Euler graph, even graph, odd graph, treewidth}
}
Document
Playing Mastermind With Constant-Size Memory
Abstract
We analyze the classic board game of Mastermind with n holes and a
constant number of colors. The classic result of Chvatal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with Theta(n / log n) questions. We show that this bound remains valid if the codebreaker may only store a constant number of guesses and answers. In addition to an intrinsic interest in this question, our result also disproves a conjecture of Droste, Jansen, and Wegener (Theory of Computing Systems 39 (2006), 525-544) on the memory-restricted black-box complexity of the OneMax function class.
Cite as
Benjamin Doerr and Carola Winzen. Playing Mastermind With Constant-Size Memory. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 441-452, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{doerr_et_al:LIPIcs.STACS.2012.441,
author = {Doerr, Benjamin and Winzen, Carola},
title = {{Playing Mastermind With Constant-Size Memory}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {441--452},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.441},
URN = {urn:nbn:de:0030-drops-34112},
doi = {10.4230/LIPIcs.STACS.2012.441},
annote = {Keywords: Algorithms, Mastermind, black-box complexity, memory-restricted algorithms, query complexity}
}
Document
Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers
Abstract
We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn's theorem on semisimple algebras, and M.E. Harris's 1980
work on p'-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest.
Cite as
László Babai and Youming Qiao. Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 453-464, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{babai_et_al:LIPIcs.STACS.2012.453,
author = {Babai, L\'{a}szl\'{o} and Qiao, Youming},
title = {{Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {453--464},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.453},
URN = {urn:nbn:de:0030-drops-34008},
doi = {10.4230/LIPIcs.STACS.2012.453},
annote = {Keywords: polynomial-time algorithm, group isomorphism, solvable group}
}
Document
Preemptive and Non-Preemptive Generalized Min Sum Set Cover
Abstract
In the (non-preemptive) Generalized Min Sum Set Cover Problem, we
are given n ground elements and a collection of sets S = {S_1,
S_2, ..., S_m} where each set S_i in 2^{[n]} has a positive
requirement k(S_i) that has to be fulfilled. We would like to order all elements to minimize the total (weighted) cover time of all sets. The cover time of a set S_i is defined as the first index j in the ordering such that the first j elements in the ordering contain k(S_i) elements in S_i. This problem was introduced by [Azar, Gamzu and Yin, 2009] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011].
We study the version where preemption is allowed. The difference is
that elements can be fractionally scheduled and a set S is
covered in the moment when k(S) amount of elements in S are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming and analysis are completely different from [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We also show that any preemptive solution can be transformed into a non-preemptive one by losing a factor of 6.2 in the objective function. As a byproduct, we obtain an improved 12.4-approximation for the non-preemptive problem.
Cite as
Sungjin Im, Maxim Sviridenko, and Ruben van der Zwaan. Preemptive and Non-Preemptive Generalized Min Sum Set Cover. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 465-476, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{im_et_al:LIPIcs.STACS.2012.465,
author = {Im, Sungjin and Sviridenko, Maxim and van der Zwaan, Ruben},
title = {{Preemptive and Non-Preemptive Generalized Min Sum Set Cover}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {465--476},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.465},
URN = {urn:nbn:de:0030-drops-33991},
doi = {10.4230/LIPIcs.STACS.2012.465},
annote = {Keywords: Set Cover, Approximation, Preemption, Latency, Average cover time}
}
Document
Randomized Communication Complexity for Linear Algebra Problems over Finite Fields
Abstract
Finding the singularity of a matrix is a basic problem in linear algebra. Chu and Schnitger [SC95] first considered this problem in the communication complexity model, in which Alice holds the first half of the matrix and Bob holds the other half. They proved that the deterministic communication complexity is Omega(n^2 log p) for an n by n matrix over the finite field F_p. Then, Clarkson and Woodruff [CW09] introduced the singularity problem to the streaming model. They proposed a randomized one pass streaming algorithm that uses O(k^2 log n) space to decide if the rank of a matrix is k, and proved an Omega(k^2) lower bound for randomized one-way protocols in the communication complexity model.
We prove that the randomized/quantum communication complexity of the singularity problem over F_p is Omega(n^2 log p), which implies the same space lower bound for randomized streaming algorithms, even for a constant number of passes. The proof uses the framework by Lee and Shraibman [LS09], but we choose Fourier coefficients as the witness for the dual approximate norm of the communication matrix. Moreover, we use Fourier analysis to show the same randomized/quantum lower bound when deciding if the determinant of a non-singular matrix is a or b for non-zero a and b.
Cite as
Xiaoming Sun and Chengu Wang. Randomized Communication Complexity for Linear Algebra Problems over Finite Fields. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 477-488, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{sun_et_al:LIPIcs.STACS.2012.477,
author = {Sun, Xiaoming and Wang, Chengu},
title = {{Randomized Communication Complexity for Linear Algebra Problems over Finite Fields}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {477--488},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.477},
URN = {urn:nbn:de:0030-drops-34385},
doi = {10.4230/LIPIcs.STACS.2012.477},
annote = {Keywords: communication complexity, streaming, matrix, singularity, determinant}
}
Document
Regular tree languages, cardinality predicates, and addition-invariant FO
Abstract
This paper considers the logic FOcard, i.e., first-order logic with cardinality predicates that can specify the size of a structure modulo some number. We study the expressive power of FOcard on the class of languages of ranked, finite, labelled trees with successor relations.
Our first main result characterises the class of FOcard-definable tree
languages in terms of algebraic closure properties of the tree languages. As it can be effectively checked whether the language of a given tree automaton satisfies these closure properties, we obtain a
decidable characterisation of the class of regular tree languages definable in FOcard.
Our second main result considers first-order logic with unary relations, successor relations, and two additional designated symbols < and + that must be interpreted as a linear order and its associated addition. Such a formula is called addition-invariant if, for each fixed interpretation of the unary relations and successor relations, its result is independent of the particular interpretation of < and +.
We show that the FOcard-definable tree languages are exactly the regular tree languages definable in addition-invariant first-order logic.
Our proof techniques involve tools from algebraic automata theory,
reasoning with locality arguments, and the use of logical interpretations. We combine and extend methods developed by Benedikt and Segoufin (ACM ToCL, 2009) and Schweikardt and Segoufin (LICS, 2010).
Cite as
Frederik Harwath and Nicole Schweikardt. Regular tree languages, cardinality predicates, and addition-invariant FO. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 489-500, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{harwath_et_al:LIPIcs.STACS.2012.489,
author = {Harwath, Frederik and Schweikardt, Nicole},
title = {{Regular tree languages, cardinality predicates, and addition-invariant FO}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {489--500},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.489},
URN = {urn:nbn:de:0030-drops-34394},
doi = {10.4230/LIPIcs.STACS.2012.489},
annote = {Keywords: regular tree languages, algebraic closure properties, decidable characterisations, addition-invariant first-order logic, logical interpretations}
}
Document
Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem
Abstract
We give a very simple approximation algorithm for the maximum asymmetric traveling salesman problem. The approximation guarantee of our algorithm is 2/3, which matches the best known approximation guarantee by Kaplan, Lewenstein, Shafrir and Sviridenko. Our algorithm is simple to analyze, and contrary to previous approaches, which need an optimal solution to a linear program, our algorithm is combinatorial and only uses maximum weight perfect matching algorithm.
Cite as
Katarzyna Paluch, Khaled Elbassioni, and Anke van Zuylen. Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 501-506, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{paluch_et_al:LIPIcs.STACS.2012.501,
author = {Paluch, Katarzyna and Elbassioni, Khaled and van Zuylen, Anke},
title = {{Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {501--506},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.501},
URN = {urn:nbn:de:0030-drops-34129},
doi = {10.4230/LIPIcs.STACS.2012.501},
annote = {Keywords: approximation algorithm, maximum asymmetric traveling salesman problem}
}
Document
Stabilization of Branching Queueing Networks
Abstract
Queueing networks are gaining attraction for the performance analysis of parallel computer systems. A Jackson network is a set of interconnected servers, where the completion of a job at server i may result in the creation of a new job for server j. We propose to extend Jackson networks by "branching" and by "control" features. Both extensions are new and substantially expand the modelling power of Jackson networks. On the other hand, the extensions raise computational questions, particularly concerning the stability of the networks, i.e, the ergodicity of the underlying Markov chain. We show for our extended model that it is decidable in polynomial time if there exists a controller that achieves stability. Moreover, if such a controller exists, one can efficiently compute a static randomized controller which stabilizes the network in a very strong sense; in particular, all moments of the queue sizes are finite.
Cite as
Tomáš Brázdil and Stefan Kiefer. Stabilization of Branching Queueing Networks. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 507-518, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{brazdil_et_al:LIPIcs.STACS.2012.507,
author = {Br\'{a}zdil, Tom\'{a}\v{s} and Kiefer, Stefan},
title = {{Stabilization of Branching Queueing Networks}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {507--518},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.507},
URN = {urn:nbn:de:0030-drops-34133},
doi = {10.4230/LIPIcs.STACS.2012.507},
annote = {Keywords: continuous-time Markov decision processes, infinite-state systems, performance analysis}
}
Document
Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes
Abstract
We associate to each Boolean language complexity class C the algebraic class a.C consisting of families of polynomials {f_n} for which the evaluation problem over the integers is in C. We prove the following lower bound and randomness-to-hardness results:
1. If polynomial identity testing (PIT) is in NSUBEXP then a.NEXP does not have poly size constant-free arithmetic circuits.
2. a.NEXP^RP does not have poly size constant-free arithmetic circuits.
3. For every fixed k, a.MA does not have arithmetic circuits of size n^k.
Items 1 and 2 strengthen two results due to (Kabanets and Impagliazzo, 2004). The third item improves a lower bound due to (Santhanam, 2009).
We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case.
Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT is in i.o-NTIME[2^{n^{o(1)}}]/n^{o(1)} if and only if there exists a family of multilinear polynomials in a.NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.
Cite as
Maurice Jansen and Rahul Santhanam. Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 519-530, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{jansen_et_al:LIPIcs.STACS.2012.519,
author = {Jansen, Maurice and Santhanam, Rahul},
title = {{Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {519--530},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.519},
URN = {urn:nbn:de:0030-drops-34307},
doi = {10.4230/LIPIcs.STACS.2012.519},
annote = {Keywords: Computational Complexity, Circuit Lower Bounds, Polynomial Identity Testing, Derandomization}
}
Document
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
Abstract
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity 2^{O(k)} O(n).
We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. In addition, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (JGAA '99).
Cite as
Paul Bonsma. Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 531-542, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bonsma:LIPIcs.STACS.2012.531,
author = {Bonsma, Paul},
title = {{Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {531--542},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.531},
URN = {urn:nbn:de:0030-drops-34224},
doi = {10.4230/LIPIcs.STACS.2012.531},
annote = {Keywords: Analysis of algorithms, parameterized algorithms, graphs on surfaces, subgraph isomorphism, dynamic programming, branch decompositions, counting probl}
}
Document
The Denjoy alternative for computable functions
Abstract
The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary function f:R->R, the Denjoy alternative holds outside a null set, i.e., for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the limit superior of the slopes around x equals +infinity, and the limit inferior -infinity. Algorithmic randomness allows us to define randomness notions giving rise to different concepts of almost everywhere. It is then natural to wonder which of these concepts corresponds to the almost everywhere notion appearing in the Denjoy-Young-Saks theorem. To answer this question Demuth investigated effective versions of the theorem and proved that Demuth randomness is strong enough to ensure the Denjoy alternative for Markov computable functions. In this paper, we show that the set of these points is indeed strictly bigger than the set of Demuth random reals - showing that Demuth's sufficient condition was too strong - and moreover is incomparable with Martin-Löf randomness; meaning in particular that it does not correspond to any known set of random reals. To prove these two theorems, we study density-type theorems, such as the Lebesgue density theorem and obtain results of independent interest. We show for example that the classical notion of Lebesgue density can be characterized by the only very recently defined notion of difference randomness. This is to our knowledge the first analytical characterization of difference randomness. We also consider the concept of porous points, a special type of Lebesgue nondensity points that are well-behaved in the sense that the "density holes" around the point are continuous intervals whose length follows a certain systematic rule. An essential part of our proof will be to argue that porous points of effectively closed classes can never be difference random.
Cite as
Laurent Bienvenu, Rupert Hölzl, Joseph S. Miller, and André Nies. The Denjoy alternative for computable functions. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 543-554, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bienvenu_et_al:LIPIcs.STACS.2012.543,
author = {Bienvenu, Laurent and H\"{o}lzl, Rupert and Miller, Joseph S. and Nies, Andr\'{e}},
title = {{The Denjoy alternative for computable functions}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {543--554},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.543},
URN = {urn:nbn:de:0030-drops-34095},
doi = {10.4230/LIPIcs.STACS.2012.543},
annote = {Keywords: Differentiability, Denjoy alternative, density, porosity, randomness}
}
Document
The Determinacy of Context-Free Games
Abstract
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).
Cite as
Olivier Finkel. The Determinacy of Context-Free Games. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 555-566, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{finkel:LIPIcs.STACS.2012.555,
author = {Finkel, Olivier},
title = {{The Determinacy of Context-Free Games}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {555--566},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.555},
URN = {urn:nbn:de:0030-drops-33897},
doi = {10.4230/LIPIcs.STACS.2012.555},
annote = {Keywords: Automata and formal languages, logic in computer science, Gale-Stewart games, Wadge games, determinacy, context-free games}
}
Document
The dimension of ergodic random sequences
Abstract
Let m be a computable ergodic shift-invariant measure over the set of infinite binary sequences. Providing a constructive proof of Shannon-McMillan-Breiman theorem, V'yugin proved that if x is a Martin-Löf random binary sequence w.r.t. m then its strong effective dimension Dim(x) equals the entropy of m. Whether its effective dimension dim(x) also equals the entropy was left as an open problem. In this paper we settle this problem, providing a positive answer. A key step in the proof consists in extending recent results on Birkhoff's ergodic theorem for Martin-Löf random sequences. At the same time, we present extensions of some previous results.
As pointed out by a referee the main result can also be derived from results by Hochman [Upcrossing inequalities for stationary sequences and applications. The Annals of Probability, 37(6):2135--2149, 2009], using rather different considerations.
Cite as
Mathieu Hoyrup. The dimension of ergodic random sequences. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 567-576, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{hoyrup:LIPIcs.STACS.2012.567,
author = {Hoyrup, Mathieu},
title = {{The dimension of ergodic random sequences}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {567--576},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.567},
URN = {urn:nbn:de:0030-drops-33917},
doi = {10.4230/LIPIcs.STACS.2012.567},
annote = {Keywords: Shannon-McMillan-Breiman theorem, Martin-L\"{o}f random sequence, effective Hausdorff dimension, compression rate, entropy}
}
Document
The Field of Reals is not omega-Automatic
Abstract
We investigate structural properties of omega-automatic presentations of infinite structures in order to sharpen our methods to determine whether a given structure is omega-automatic. We apply these methods to show that no field of characteristic 0 admits an injective omega-automatic presentation, and that uncountable fields with a definable linear order cannot be omega-automatic.
Cite as
Faried Abu Zaid, Erich Grädel, and Lukasz Kaiser. The Field of Reals is not omega-Automatic. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 577-588, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{abuzaid_et_al:LIPIcs.STACS.2012.577,
author = {Abu Zaid, Faried and Gr\"{a}del, Erich and Kaiser, Lukasz},
title = {{The Field of Reals is not omega-Automatic}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {577--588},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.577},
URN = {urn:nbn:de:0030-drops-34234},
doi = {10.4230/LIPIcs.STACS.2012.577},
annote = {Keywords: Logic, Algorithmic Model Theory, Automatic Structures}
}
Document
The Limits of Decidability for First Order Logic on CPDA Graphs
Abstract
Higher-order pushdown automata (n-PDA) are abstract machines equipped with a nested 'stack of stacks of stacks'. Collapsible pushdown automata (n-CPDA) extend these devices by adding `links' to the stack and are equi-expressive for tree generation with simply typed lambda-Y terms. Whilst the configuration graphs of HOPDA are well understood, relatively little is known about the CPDA graphs. The order-2 CPDA graphs already have undecidable MSO theories but it was only recently shown by Kartzow [Kartzow 2010] that first-order logic is decidable at the second level. In this paper we show the surprising result that first-order logic ceases to be decidable at order-3 and above. We delimit the fragments of the decision problem to which our undecidability result applies in terms of quantifer alternation and the orders of CPDA links used. Additionally we exhibit a natural sub-hierarchy enjoying limited decidability.
Cite as
Christopher H. Broadbent. The Limits of Decidability for First Order Logic on CPDA Graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 589-600, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{broadbent:LIPIcs.STACS.2012.589,
author = {Broadbent, Christopher H.},
title = {{The Limits of Decidability for First Order Logic on CPDA Graphs}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {589--600},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.589},
URN = {urn:nbn:de:0030-drops-34334},
doi = {10.4230/LIPIcs.STACS.2012.589},
annote = {Keywords: Collapsible Pushdown Automata, First Order Logic, Logical Reflection}
}
Document
The Power of Local Search: Maximum Coverage over a Matroid
Abstract
We present an optimal, combinatorial 1-1/e approximation algorithm for Maximum Coverage over a matroid constraint, using non-oblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1-1/e approximation algorithm for the more general problem of monotone submodular maximization over a matroid constraint. The advantage of our algorithm is that it is entirely combinatorial, and in many circumstances also faster, as well as conceptually simpler.
Following previous work on satisfiability problems by Alimonti, as well as by Khanna, Motwani, Sudan and Vazirani, our local search algorithm is *non-oblivious*. That is, our algorithm uses an auxiliary linear objective function to evaluate solutions. This function gives more weight to elements covered multiple times. We show that the locality ratio of the resulting local search procedure is at least 1-1/e. Our local search procedure only considers improvements of size 1. In contrast, we show that oblivious local search, guided only by the problem's objective function, achieves an approximation ratio of only (n-1)/(2n-1-k) when improvements of size k are considered.
In general, our local search algorithm could take an exponential amount of time to converge to an *exact* local optimum. We address this situation by using a combination of *approximate* local search and the same partial enumeration techniques as Calinescu et al., resulting in a clean 1 - 1/e-approximation algorithm running in polynomial time.
Cite as
Yuval Filmus and Justin Ward. The Power of Local Search: Maximum Coverage over a Matroid. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 601-612, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{filmus_et_al:LIPIcs.STACS.2012.601,
author = {Filmus, Yuval and Ward, Justin},
title = {{The Power of Local Search: Maximum Coverage over a Matroid}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {601--612},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.601},
URN = {urn:nbn:de:0030-drops-33968},
doi = {10.4230/LIPIcs.STACS.2012.601},
annote = {Keywords: approximation algorithms; maximum coverage; matroids; local search}
}
Document
Trichotomy for Integer Linear Systems Based on Their Sign Patterns
Abstract
In this paper, we consider solving the integer linear systems, i.e.,
given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b,
where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science.
For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy:
- ILS_=(r) is linearly solvable, if r < 1,
- ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and
- ILS_=(r) is strongly NP-hard, if r > 1.
This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.
Cite as
Kei Kimura and Kazuhisa Makino. Trichotomy for Integer Linear Systems Based on Their Sign Patterns. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 613-623, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{kimura_et_al:LIPIcs.STACS.2012.613,
author = {Kimura, Kei and Makino, Kazuhisa},
title = {{Trichotomy for Integer Linear Systems Based on Their Sign Patterns}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {613--623},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.613},
URN = {urn:nbn:de:0030-drops-34367},
doi = {10.4230/LIPIcs.STACS.2012.613},
annote = {Keywords: Integer linear system, Sign pattern, Complexity index, TVPI system, Horn system}
}
Document
Tying up the loose ends in fully LZW-compressed pattern matching
Abstract
We consider a natural generalization of the classical pattern matching problem: given compressed representations of a pattern p[1..M] and a text t[1..N] of sizes m and n, respectively, does p occur in t? We develop an optimal linear time solution for the case when p and t are compressed using the LZW method. This improves the previously known O((n+m)log(n+m)) time solution of Gasieniec and Rytter, and essentially closes the line of research devoted to tudying LZW-compressed exact pattern matching.
Cite as
Pawel Gawrychowski. Tying up the loose ends in fully LZW-compressed pattern matching. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 624-635, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{gawrychowski:LIPIcs.STACS.2012.624,
author = {Gawrychowski, Pawel},
title = {{Tying up the loose ends in fully LZW-compressed pattern matching}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {624--635},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.624},
URN = {urn:nbn:de:0030-drops-33975},
doi = {10.4230/LIPIcs.STACS.2012.624},
annote = {Keywords: pattern matching, compression, Lempel-Ziv-Welch}
}
Document
Variable time amplitude amplification and quantum algorithms for linear algebra problems
Abstract
Quantum amplitude amplification is a method of increasing a success probability of an algorithm from a small epsilon>0 to Theta(1) with less repetitions than classically. In this paper, we generalize quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times.
We then apply the new variable time amplitude amplification to give two new quantum algorithms for linear algebra problems. Our first algorithm is an improvement of Harrow et al. algorithm for solving systems of linear equations. We improve the running time of the algorithm from O(k^2 log N) to O(k log^3 k log N) where k is the condition number of the system of equations. Our second algorithm tests whether a matrix A is singular or far from singular, faster then the previously known algorithms.
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Andris Ambainis. Variable time amplitude amplification and quantum algorithms for linear algebra problems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 636-647, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{ambainis:LIPIcs.STACS.2012.636,
author = {Ambainis, Andris},
title = {{Variable time amplitude amplification and quantum algorithms for linear algebra problems}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {636--647},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.636},
URN = {urn:nbn:de:0030-drops-34261},
doi = {10.4230/LIPIcs.STACS.2012.636},
annote = {Keywords: quantum computing, quantum algorithms, amplitude amplification, linear equations}
}
Document
Weak MSO+U over infinite trees
Abstract
We prove that, over infinite trees, satisfiability is decidable for Weak Monadic Second-Order Logic extended by the unbounding quantifier U. We develop an automaton model, prove that it is effectively equivalent to the logic, and that the automaton model has decidable emptiness.
Cite as
Mikolaj Bojanczyk and Szymon Torunczyk. Weak MSO+U over infinite trees. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 648-660, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
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@InProceedings{bojanczyk_et_al:LIPIcs.STACS.2012.648,
author = {Bojanczyk, Mikolaj and Torunczyk, Szymon},
title = {{Weak MSO+U over infinite trees}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {648--660},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {D\"{u}rr, Christoph and Wilke, Thomas},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.648},
URN = {urn:nbn:de:0030-drops-34279},
doi = {10.4230/LIPIcs.STACS.2012.648},
annote = {Keywords: Infinite trees, distance automata, MSO+U, profinite words}
}