On Parameterized Approximability

Chen, Yijia; Grohe, Martin; Grüber, Magdalena

doi:10.1007/11847250_10

Cited by 59 publications

(81 citation statements)

References 14 publications

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“…Clearly, DTIME 2 o eff (n) ⊆ DTIME 2 o(n) ⊆ ε>0 DTIME (2 ε·n ) (9) To the best of our knowledge it is open whether the first inclusion is strict. Here we show the strictness of the second inclusion in (9). We remark that in [8, Proposition 5] we proved that the first class, that is, the effective version of the second one, coincides with an effective version of the third class.…”

Section: An Examplesupporting

confidence: 50%

“…As in [9], we say that an algorithm A is an fpt uni parameterized approximation algorithm for p-Clique with approximation ratio ρ if…”

Section: Part (1) Was Shown As Theorem 27 Inmentioning

confidence: 99%

“…The basic idea of parameterized approximability is explained in [9] as follows: "Suppose we have a problem that is hard to approximate. Can we at least approximate it efficiently for instances for which the optimum is small?…”

Section: Introductionmentioning

confidence: 99%

“…The classical theory of inapproximability does not seem to help answering this question, because usually the hardness proofs require fairly large solutions." In [9] and [10] the framework of parameterized approximability was introduced. In particular, the concept of an fpt approximation algorithm with a given approximation ratio was coined and some (in)approximability results were proven.…”

Section: Introductionmentioning

confidence: 99%

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The Exponential Time Hypothesis and the Parameterized Clique Problem

Chen

Eickmeyer

Flum

2012

Parameterized and Exact Computation

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Abstract. In parameterized complexity there are three natural definitions of fixed-parameter tractability called strongly uniform, weakly uniform and nonuniform fpt. Similarly, there are three notions of subexponential time, yielding three flavours of the exponential time hypothesis (ETH) stating that 3Sat is not solvable in subexponential time. It is known that ETH implies that p-Clique is not fixed-parameter tractable if both are taken to be strongly uniform or both are taken to be uniform, and we extend this to the nonuniform case. We also show that even the containment of weakly uniform subexponential time in nonuniform subexponential time is strict. Furthermore, we deduce from nonuniform ETH that no single exponent d allows for arbitrarily good fpt-approximations of clique.

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Section: An Examplesupporting

confidence: 50%

“…As in [9], we say that an algorithm A is an fpt uni parameterized approximation algorithm for p-Clique with approximation ratio ρ if…”

Section: Part (1) Was Shown As Theorem 27 Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Exponential Time Hypothesis and the Parameterized Clique Problem

Chen

Eickmeyer

Flum

2012

Parameterized and Exact Computation

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“…The concrete question above concerning parameterized approximation of the Dominating Set problem has been open for many years, and is still unresolved. What is well-known is that determining whether a graph G has a dominating set of size k (the exact formulation of the problem) is complete for the parameterized complexity class W [2].…”

Section: Dominating Set G(k)-approximationmentioning

confidence: 99%

Parameterized approximation of dominating set problems

Downey

Fellows

McCartin

et al. 2008

Information Processing Letters

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Abstract.A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g(k) of the form k + c (where c is a fixed constant, termed an additive FPT approximation), unless F P T = W [2]. We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g(k), unless F P T = W [2].

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Fixed-Parameter Approximation: Conceptual Framework and Approximability Results

Cai

Huang

2006

Parameterized and Exact Computation

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The notion of fixed-parameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixed-parameter computation. This work partially aims at enhancing the world of fixed-parameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixed-parameter approximability is closely related to the approximation of small-cost solutions in polynomial time. It is also demonstrated that many fixed-parameter intractable problems are not fixed-parameter approximable. On the other hand, fixed-parameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixed-parameter approximation schemes in time O(2 O((1−ǫ/O(1))k) p(n)) for any small ǫ > 0.

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On Parameterized Approximability

Cited by 59 publications

References 14 publications

The Exponential Time Hypothesis and the Parameterized Clique Problem

The Exponential Time Hypothesis and the Parameterized Clique Problem

Parameterized approximation of dominating set problems

Fixed-Parameter Approximation: Conceptual Framework and Approximability Results

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