## Bounded Nondeterminism and Alternation in Parameterized Complexity Theory (2003)

### Cached

### Download Links

- [basics.sjtu.edu.cn]
- [www.informatik.hu-berlin.de]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of the 18th IEEE Conference on Computational Complexity |

Citations: | 13 - 5 self |

### BibTeX

@INPROCEEDINGS{Chen03boundednondeterminism,

author = {Yijia Chen and Jörg Flum and Martin Grohe},

title = {Bounded Nondeterminism and Alternation in Parameterized Complexity Theory},

booktitle = {In Proceedings of the 18th IEEE Conference on Computational Complexity},

year = {2003},

pages = {13--29}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give machine characterisations and logical descriptions of a number of parameterized complexity classes. The focus of our attention is the class W[P], which we characterise as the class of all parameterized problems decidable by a nondeterministic fixed-parameter tractable algorithm whose use of nondeterminism is bounded in terms of the parameter. We give similar characterisations for AW[P], the “alternating version of W[P]”, and various other parameterized complexity classes. We also give logical characterisations of the classes W[P] and AW[P] in terms of fragments of least fixed-point logic, thereby putting these two classes into a uniform framework that we have developed in earlier work. Furthermore, we investigate the relation between alternation and space in parameterized complexity theory. In this context, we prove that the COMPACT TURING MACHINE COMPUTATION problem, shown to be hard for the class AW[SAT] in [1], is complete for the class uniform-XNL. 1.

### Citations

2419 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...em A-hierarchy ¡ FPT + parameter-bounded alternating nondeterminism of bounded alternation depth (Theorem 17(3))� Our machine model is based on the standard random access machines (RAMs) described in =-=[20]-=-. The arithmetic operations are addition, subtraction, and division by two (rounded off), and we use a uniform cost measure. For details, we refer the reader to Section 2.6 of [20]. Our model is non-s... |

916 | Parametrized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...ome parameter of the problem instance that can be expected to be small in the typical applications. A core structural parameterized complexity theory has been developed over the last 10–15 years (see =-=[7]-=-). Unfortunately, it has led to a bewildering variety of parameterized complexity classes, the most important of which are displayed in Figure 1. As the reader will have guessed, none of the inclusion... |

539 |
The complexity of relational query languages
- Vardi
- 1982
(Show Context)
Citation Context ...ea of this area is to characterise the computational complexity of problems in terms of logical definability. Most standard complexity classes have natural descriptive characterisations (for example, =-=[12, 17, 18, 23]-=-). In two earlier papers [13, 14], two of us gave descriptive characterisations of the classes FPT, W¥ � ¨ , and A¥ � ¨ for ��� ¦ , and AW¥�� ¨ . Here, we give such characterisations of the classes W¥... |

510 |
Personality Theory
- Hjelle, Ziegler
- 1992
(Show Context)
Citation Context ...putation problem, which was only known to be hard for the class AW¥ SAT¨ before (cf. [1]), is complete for the class uniform-XNL under parameterized logspace reductions. Descriptive complexity theory =-=[11, 19]-=- provides a machine independent way of understanding complexity classes. The idea of this area is to characterise the computational complexity of problems in terms of logical definability. Most standa... |

287 |
Descriptive Complexity
- Immerman
- 1999
(Show Context)
Citation Context ...putation problem, which was only known to be hard for the class AW¥ SAT¨ before (cf. [1]), is complete for the class uniform-XNL under parameterized logspace reductions. Descriptive complexity theory =-=[11, 19]-=- provides a machine independent way of understanding complexity classes. The idea of this area is to characterise the computational complexity of problems in terms of logical definability. Most standa... |

283 | Generalized First-Order Spectra and Polynomial-Time Recognizable Sets - Fagin - 1974 |

275 | Relational queries computable in polynomial time
- Immerman
- 1986
(Show Context)
Citation Context ...ea of this area is to characterise the computational complexity of problems in terms of logical definability. Most standard complexity classes have natural descriptive characterisations (for example, =-=[12, 17, 18, 23]-=-). In two earlier papers [13, 14], two of us gave descriptive characterisations of the classes FPT, W¥ � ¨ , and A¥ � ¨ for ��� ¦ , and AW¥�� ¨ . Here, we give such characterisations of the classes W¥... |

231 | Languages that capture complexity classes
- Immerman
- 1987
(Show Context)
Citation Context ...ea of this area is to characterise the computational complexity of problems in terms of logical definability. Most standard complexity classes have natural descriptive characterisations (for example, =-=[12, 17, 18, 23]-=-). In two earlier papers [13, 14], two of us gave descriptive characterisations of the classes FPT, W¥ � ¨ , and A¥ � ¨ for ��� ¦ , and AW¥�� ¨ . Here, we give such characterisations of the classes W¥... |

77 |
On the complexity of bounded-variable queries
- Vardi
- 1995
(Show Context)
Citation Context ...the form ¡ § � �����s� § � � , where ¤ � ¦ ,s���©�������©�s� �¦� ¢ ©s¤� The proof of the following theorem parallels that of Theorem 7.1 in [13]. We use the following two facts, the first implicit in =-=[24]-=- and the second in the proof of the Immerman-Vardi Theorem [17, 23] (also see [11]). Fix � � ¦ : ¡ � , and� � LFP ¤s¦¥ . ) There is an algorithm that for every vocabularys, every formula � ��� (� ¢ le... |

54 |
On the complexity of database queries
- Papadimitriou, Yannakakis
- 1999
(Show Context)
Citation Context ...k for a fine-grain complexity analysis of algorithmic problems that are intractable in general. It has been used to analyse problems in various areas of computer science, for example, database theory =-=[16, 21]-=-, artificial intelligence [15], and computational biology [3, 22]. The theory is built on a weakened notion of tractability called fixed-parameter tractability, which relaxes the classical notion of t... |

48 | Resolution is not automatizable unless W[P] is tractable
- Alekhnovich, Razborov
(Show Context)
Citation Context ...ircuits. Thus in some sense, W¥ P¨ is one of the most natural parameterized complexity classes. W¥ P¨ has received some recent attention because of an important result due to Alekhnovich and Razborov =-=[2]-=- showing that resolution is not automatizable unless FPT ¡ W¥ P¨ . Our first theorem is a simple machine characterisation of the class W¥ P¨ . Intuitively, it states that a problem is in W ¥ P¨ if, an... |

44 | On the parameterized complexity of short computation and factorization
- Cai, Chen, et al.
- 1997
(Show Context)
Citation Context ...§ ����� � ©������ BNTMC© which gives the desired reduction. � We now turn to a machine characterisation of the class W¥ ¦ ¨ . Our proof uses the following result due to Cai, Chen, Downey, and Fellows =-=[4]-=-: Theorem 14 (Cai et al. [4]). The following parameterized problem SHORT TURING MACHINE ACCEPTANCE (STMA) is W ¥§¦©¨ -complete under FPT-reductions: ¦ ��� ¥ ¦ � STMA Input: A nondeterministic Turing m... |

41 |
Fixed-parameter tractability and completeness IV: On completeness for W[P] and PSPACE analogs
- Abrahamson, Downey, et al.
- 1995
(Show Context)
Citation Context ...he relation between alternation and space in parameterized complexity theory. In this context, we prove that the COMPACT TURING MACHINE COMPUTATION problem, shown to be hard for the class AW£ SAT¤ in =-=[1]-=-, is complete for the class uniform-XNL. 1. Introduction Parameterized complexity theory provides a framework for a fine-grain complexity analysis of algorithmic problems that are intractable in gener... |

34 |
Parameterized complexity analysis
- Bodlaender, Downey, et al.
- 1995
(Show Context)
Citation Context ...t are intractable in general. It has been used to analyse problems in various areas of computer science, for example, database theory [16, 21], artificial intelligence [15], and computational biology =-=[3, 22]-=-. The theory is built on a weakened notion of tractability called fixed-parameter tractability, which relaxes the classical notion of tractability, polynomial time computability, by admitting algorith... |

33 | Fixed-parameter tractability, definability, and model checking
- Flum, Grohe
(Show Context)
Citation Context ...mputational complexity of problems in terms of logical definability. Most standard complexity classes have natural descriptive characterisations (for example, [12, 17, 18, 23]). In two earlier papers =-=[13, 14]-=-, two of us gave descriptive characterisations of the classes FPT, W¥ � ¨ , and A¥ � ¨ for ��� ¦ , and AW¥�� ¨ . Here, we give such characterisations of the classes W¥ P¨ and AW¥ P¨ . All these charac... |

26 | The parameterized complexity of relational database queries and an improved characterization ofW [1
- Downey, Fellows, et al.
- 1996
(Show Context)
Citation Context ...A structure ¡ and an L-sentence� . Parameter: � � � . Problem: Decide if ¡ �¡�� . The parameterized model-checking problem for first-order logic is complete for the class AW¥�� ¨ under FPT-reductions =-=[10]-=-. For each of the classes of the W-hierarchy and the A-hierarchy there is a fragment of first-order logic whose parameterized model checking problem is complete for the class [8, 13]. Using Theorems 3... |

17 | Fixed-parameter complexity in AI and nonmonotonic reasoning
- Gottlob, Leone, et al.
- 1999
(Show Context)
Citation Context ...ysis of algorithmic problems that are intractable in general. It has been used to analyse problems in various areas of computer science, for example, database theory [16, 21], artificial intelligence =-=[15]-=-, and computational biology [3, 22]. The theory is built on a weakened notion of tractability called fixed-parameter tractability, which relaxes the classical notion of tractability, polynomial time c... |

16 | Describing parameterized complexity classes - Flum, Grohe - 2002 |

11 |
Descriptive complexity and the W - hierarchy
- Downey, Fellows, et al.
- 1998
(Show Context)
Citation Context ...under FPT-reductions [10]. For each of the classes of the W-hierarchy and the A-hierarchy there is a fragment of first-order logic whose parameterized model checking problem is complete for the class =-=[8, 13]-=-. Using Theorems 31 and 32 we get the following: 18sTheorem 33. (1) For all � � ¦ the parameterized model-checking problem for ¡ � LFP ¤s¦¥ is complete for W[P] under FPT-reductions. (2) For all � � ¦... |

8 |
The parameterized complexity of database queries
- Grohe
- 2001
(Show Context)
Citation Context ...k for a fine-grain complexity analysis of algorithmic problems that are intractable in general. It has been used to analyse problems in various areas of computer science, for example, database theory =-=[16, 21]-=-, artificial intelligence [15], and computational biology [3, 22]. The theory is built on a weakened notion of tractability called fixed-parameter tractability, which relaxes the classical notion of t... |

5 |
Resolving Conflicts in Problems from Computational Biology
- Stege
(Show Context)
Citation Context ...t are intractable in general. It has been used to analyse problems in various areas of computer science, for example, database theory [16, 21], artificial intelligence [15], and computational biology =-=[3, 22]-=-. The theory is built on a weakened notion of tractability called fixed-parameter tractability, which relaxes the classical notion of tractability, polynomial time computability, by admitting algorith... |

4 | The Turing way to parameterized intractability
- Cesati
- 2001
(Show Context)
Citation Context ...ndard definition of nondeterministic RAMs is that it also gives us a nice machine description of the class W¥§¦©¨ (see Theorem 15). As a further corollary we get a slight strengthening of a result of =-=[5]-=-: Corollary 13. BOUNDED NONDETERMINISTIC MACHINE COMPUTATION (BNTMC) is W[P]-complete under PL-reductions. Here, (BNTMC) denotes the following problem: ¦ �¦� ¥ ��� ¥ ¦ � � BNTMC Input: A nondeterminis... |

4 |
Parameterized circuit complexity and
- Downey, Fellows, et al.
- 1998
(Show Context)
Citation Context ...RCUIT� ¨ FPT � Remark 5. Of course there are also classes AW¥ � ¨�¡ ¥ � AWSAT��� ��� � � � � � ¦�� ¨ FPT corresponding to the W¥ � ¨ . It turns out, however, that AW¥ � ¨�¡ AW¥ ¦ ¨ for all ��� ¦ (cf. =-=[9]-=-). For that reason, the class AW¥§¦©¨ is usually denoted by AW¥ ��¨ . There is another hierarchy, the so called A-hierarchy, whose classes are obtained by fixing the number of alternations (i.e. the ¤... |