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## On parameterized approximability (2006)

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Venue: | Proc. of IWPEC, Lecture Notes in Computer Science 4169 |

Citations: | 21 - 5 self |

### Citations

1212 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...s, and R≥1 the real numbers greater than or equal to 1. We recall a few basic definitions on optimisation problems and parameterized complexity. For further background, we refer the reader to [2] and =-=[11, 15]-=-. 2s2.1 Optimisation Problems In this paper we consider NP-optimisation problems O over a finite alphabet Σ consisting of triples (solO,costO,goal O ) where 1. solO is a function that associates to an... |

639 |
Parameterized Complexity Theory
- Flum, Grohe
- 2006
(Show Context)
Citation Context ...s, and R≥1 the real numbers greater than or equal to 1. We recall a few basic definitions on optimisation problems and parameterized complexity. For further background, we refer the reader to [2] and =-=[11, 15]-=-. 2s2.1 Optimisation Problems In this paper we consider NP-optimisation problems O over a finite alphabet Σ consisting of triples (solO,costO,goal O ) where 1. solO is a function that associates to an... |

414 | Probabilistic checking of proofs : A new characterization of NP.
- Arora, Safra
- 1998
(Show Context)
Citation Context ...he maximum clique problem is known to be hard to approximate — unless ZPP = NP not approximable with ratio n 1−ε for any ε > 0 [16] — and, most likely, not fixed-parameter tractable — the problem is W=-=[1]-=--complete [10], and unless the exponential time hypothesis fails, it is not even solvable in time n o(k) [6, 7]. Here, and in the following, k denotes the size of the desired clique and n the number o... |

194 | Complexity and approximation,
- Ausiello, Crescenzi, et al.
- 1999
(Show Context)
Citation Context ...iven parameter k ∈ N whether M accepts the empty string in at most k steps, and it was shown in [3], respectively [5] that the single-tape version is complete for W[1] and the other one complete for W=-=[2]-=-. Theorem 20. MIN-SHORT-NTM-HALT is not fpt cost approximable unless W[2] = FPT. Proof. Without loss of generality the following proof only considers Turing machines that are either accepting or not h... |

166 | The PCP theorem by gap amplification,” - Dinur - 2007 |

120 | Fixed-parameter tractability and completeness II: On completeness for W[1].
- Downey, Fellows
- 1995
(Show Context)
Citation Context ...ique problem is known to be hard to approximate — unless ZPP = NP not approximable with ratio n 1−ε for any ε > 0 [16] — and, most likely, not fixed-parameter tractable — the problem is W[1]-complete =-=[10]-=-, and unless the exponential time hypothesis fails, it is not even solvable in time n o(k) [6, 7]. Here, and in the following, k denotes the size of the desired clique and n the number of vertices of ... |

114 | Approximating clique-width and branch-width.
- Oum, Seymour
- 2006
(Show Context)
Citation Context ...ead of computing a solution of cost approximately k, an algorithm only has to decide if such a solution exists. We observe that a few known results yield fpt approximation algorithms: Oum and Seymour =-=[18]-=- showed that the problem of finding a clique decomposition of minimum width is fpt approximable. Based on results due to Seymour [20], Even et al. [13] showed that the directed feedback vertex set pro... |

106 | Approximating minimum feedback sets and multicuts in directed graphs,”
- Even, Naor, et al.
- 1998
(Show Context)
Citation Context ...ld fpt approximation algorithms: Oum and Seymour [18] showed that the problem of finding a clique decomposition of minimum width is fpt approximable. Based on results due to Seymour [20], Even et al. =-=[13]-=- showed that the directed feedback vertex set problem is fpt approximable. Then it follows from a result due to Reed et al. [19] that the linear programming dual of the feedback vertex set problem, th... |

97 |
Packing directed circuits fractionally
- Seymour
- 1995
(Show Context)
Citation Context ... known results yield fpt approximation algorithms: Oum and Seymour [18] showed that the problem of finding a clique decomposition of minimum width is fpt approximable. Based on results due to Seymour =-=[20]-=-, Even et al. [13] showed that the directed feedback vertex set problem is fpt approximable. Then it follows from a result due to Reed et al. [19] that the linear programming dual of the feedback vert... |

74 |
Handle-rewriting hypergraph grammars.
- Courcelle, Engelfriet, et al.
- 1993
(Show Context)
Citation Context ...on is proved similarly to Proposition 12. 9s3.2 Examples Example 15. We first look at the problem MIN-CLIQUE-WIDTH of computing a decomposition of minimum clique-width for a given graph. Clique-width =-=[8]-=- is a graph parameter that is defined by a composition mechanism for vertex-labelled graphs and measures the complexity of a graph according to the difficulty of decomposing the graph into a kind of t... |

72 |
Clique is hard to approximate within n 1−ε
- H˚astad
- 1999
(Show Context)
Citation Context ...require fairly large solutions. Let us illustrate this with an example: The maximum clique problem is known to be hard to approximate — unless ZPP = NP not approximable with ratio n 1−ε for any ε > 0 =-=[16]-=- — and, most likely, not fixed-parameter tractable — the problem is W[1]-complete [10], and unless the exponential time hypothesis fails, it is not even solvable in time n o(k) [6, 7]. Here, and in th... |

62 | Tight lower bounds for certain parameterized NP-hard problems.
- Chen, Chor, et al.
- 2005
(Show Context)
Citation Context ... 1−ε for any ε > 0 [16] — and, most likely, not fixed-parameter tractable — the problem is W[1]-complete [10], and unless the exponential time hypothesis fails, it is not even solvable in time n o(k) =-=[6, 7]-=-. Here, and in the following, k denotes the size of the desired clique and n the number of vertices of the input graph. Now we ask: Is there an fpt algorithm that, given a graph G and a k ∈ N, finds a... |

48 | Clique is hard to approximate within n1−ε,
- Hastad
- 1999
(Show Context)
Citation Context ... require fairly large solutions. Let us illustrate this with an example: The maximum clique problem is known to be hard to approximate — unless ZPP = NP not approximable with ratio n1−ε for any ε > 0 =-=[16]-=- — and, most likely, not fixed-parameter tractable — the problem is W[1]-complete [10], and unless the exponential time hypothesis fails, it is not even solvable in time no(k) [6, 7]. Here, and in the... |

45 | The parameterized complexity of short computations and factorization
- Cai, Chen, et al.
- 1993
(Show Context)
Citation Context ... and p-SHORT-NSTM-HALT are to decide for a given nondeterministic (single-tape) Turing machine M and a given parameter k ∈ N whether M accepts the empty string in at most k steps, and it was shown in =-=[3]-=-, respectively [5] that the single-tape version is complete for W[1] and the other one complete for W[2]. Theorem 20. MIN-SHORT-NTM-HALT is not fpt cost approximable unless W[2] = FPT. Proof. Without ... |

44 | Approximating rank-width and clique-width quickly
- Oum
- 2005
(Show Context)
Citation Context ...ank-width to investigate clique-width and showed that rwd(G ) ≤ cwd(G ) ≤ 2 rwd(G )+1 − 1 for the clique-width cwd(G ) and the rank-width rwd(G ) of a given simple, undirected and finite graph G . In =-=[17]-=-, Oum presents two algorithms to compute rank-decompositions approximately: For a graph G = (V,E) and k ∈ N the algorithms either output a rank-decomposition of width at most f (k) with f (k) = 3k + 1... |

38 |
Linear FPT reductions and computational lower bounds
- Chen, Huang, et al.
- 2004
(Show Context)
Citation Context ... 1−ε for any ε > 0 [16] — and, most likely, not fixed-parameter tractable — the problem is W[1]-complete [10], and unless the exponential time hypothesis fails, it is not even solvable in time n o(k) =-=[6, 7]-=-. Here, and in the following, k denotes the size of the desired clique and n the number of vertices of the input graph. Now we ask: Is there an fpt algorithm that, given a graph G and a k ∈ N, finds a... |

29 | Fixed-parameter approximation: Conceptual framework and approximability results.
- Cai, Huang
- 2010
(Show Context)
Citation Context ...in other words: every parameterized complexity class above FPT that contains a problem in NP contains problems that are approximable and problems that are inapproximable. Independently, Cai and Huang =-=[4]-=- and Downey, Fellows and McCartin [12] introduced similar frameworks of parameterized approximability. 2 Preliminaries N denotes the natural numbers (positive integers), R the real numbers, and R≥1 th... |

27 |
Clique-width minimization is NP-hard
- Fellows, Rosamund, et al.
(Show Context)
Citation Context ...e. A decomposition of cliquewidth k is also called k-expression and given the k-expression, many hard graph problems are solvable in polynomial time for graphs of bounded clique-width. Fellows et al. =-=[14]-=- recently proved that deciding whether the clique-width of G is at most k is NP-hard and that the minimisation problem MIN-CLIQUE-WIDTH cannot be absolutely approximated in polynomial time unless P = ... |

21 | Parameterized tractability of edge-disjoint paths on directed acyclic graphs
- Slivkins
- 2003
(Show Context)
Citation Context ...tion is the following problem: p-DIRECTED-VERTEX-DISJOINT-CYCLES Input: A directed graph G and k ∈ N. Parameter: k. Problem: Decide whether there are k vertex-disjoint cycles in G . It is implicit in =-=[21]-=- that p-DIRECTED-VERTEX-DISJOINT-CYCLES is W[1]-hard. For the maximum number ν(G ) of vertex-disjoint cycles and for the minimum size τ(G ) (τ ∗ (G )) of a (fractional) feedback vertex set in a given ... |

15 |
Context-free handle-rewriting hypergraph grammars
- Courcelle, Engelfriet, et al.
- 1990
(Show Context)
Citation Context ...on is proved similarly to Proposition 12. 93.2 Examples Example 15. We first look at the problem MIN-CLIQUE-WIDTH of computing a decomposition of minimum clique-width for a given graph. Clique-width =-=[8]-=- is a graph parameter that is defined by a composition mechanism for vertex-labelled graphs and measures the complexity of a graph according to the difficulty of decomposing the graph into a kind of t... |

14 | Packing directed circuits
- Reed, Robertson, et al.
- 1996
(Show Context)
Citation Context ... is fpt approximable. Based on results due to Seymour [20], Even et al. [13] showed that the directed feedback vertex set problem is fpt approximable. Then it follows from a result due to Reed et al. =-=[19]-=- that the linear programming dual of the feedback vertex set problem, the vertex disjoint cycle problem, is fpt cost approximable. This is interesting because the standard parameterization of this max... |

10 |
Computation models for parameterized complexity
- Cesati, Ianni
- 1997
(Show Context)
Citation Context ...HALT are to decide for a given nondeterministic (single-tape) Turing machine M and a given parameter k ∈ N whether M accepts the empty string in at most k steps, and it was shown in [3], respectively =-=[5]-=- that the single-tape version is complete for W[1] and the other one complete for W[2]. Theorem 20. MIN-SHORT-NTM-HALT is not fpt cost approximable unless W[2] = FPT. Proof. Without loss of generality... |

8 |
Parameterized approximation algorithms. In:
- Downey, Fellows, et al.
- 2006
(Show Context)
Citation Context ...omplexity class above FPT that contains a problem in NP contains problems that are approximable and problems that are inapproximable. Independently, Cai and Huang [4] and Downey, Fellows and McCartin =-=[12]-=- introduced similar frameworks of parameterized approximability. 2 Preliminaries N denotes the natural numbers (positive integers), R the real numbers, and R≥1 the real numbers greater than or equal t... |