Parameterized circuit complexity and the W hierarchy

The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic.We show that unless PTIME = NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f (k) · p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We establish a number of similar lower bounds for the model-checking problem for first-order logic, for example, on the class of all trees.

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“…Theorem 12 (Downey et al [6], Flum and Grohe [9]). If AWSAT[3-CNF] is fixed-parameter tractable then…”

Section: First-order Logicmentioning

confidence: 99%

The complexity of first-order and monadic second-order logic revisited

Frick

Grohe

2004

Annals of Pure and Applied Logic

175

162

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“…This theorem is a weak parameterized analogue of Stockmeyer's theorem. In [4] a weak parameterized analogue of the theorem of Valiant and Vazirani was proved. It was proved that, for any t 1, if there exist suitable amplification algorithms, then there is a fpt many-one randomized reduction of W [t] to Unique W [t].…”

Section: Probabilistic Classes and Probability Amplificationmentioning

confidence: 95%

“…There are suitable fpt probability amplification algorithms? This is an important question which was left open in [4]. We cope, in this paper with a related problem:…”

Section: Probabilistic Classes and Probability Amplificationmentioning

confidence: 99%

The parameterized complexity of probability amplification

Montoya

2008

Information Processing Letters

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“…One of the problems is that all known proofs of Toda's Theorem filter through probabilistic classes. Whilst there are known analogs of Valiant's Theorem (Downey, Fellows and Regan [130], and Müller [230,231]), there is no known method of "small" probability amplification. (See Montoya [229] for a thorough discussion of this problem.)…”

Section: Other Parameterized Classesmentioning

confidence: 99%

“…14 (Downey, Fellows and Regan[130]). For all t, there is a randomized FPT-reduction from W[t] to unique W[t].…”

mentioning

confidence: 99%

Confronting intractability via parameters

Downey

Thilikos

2011

Computer Science Review

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One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows [125], Flum and Grohe [155], Niedermeier [240], and the upcoming undergraduate text Downey and Fellows [127].

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Parameterized circuit complexity and the W hierarchy

Cited by 36 publications

References 13 publications

The complexity of first-order and monadic second-order logic revisited

The complexity of first-order and monadic second-order logic revisited

The parameterized complexity of probability amplification

Confronting intractability via parameters

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