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**1 - 4**of**4**### D.: Independent set, induced matching, and pricing: connections and tight (subexponential time) approximation hardnesses

- CoRR abs/1308.2617
, 2013

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### The Constant Inapproximability of the Parameterized Dominating Set Problem

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### Fixed Parameter hardness in opt

"... For a minimization (resp., maximization) problem P, an algorithm is called an (r(k), t(k))-FPT-approximation algorithm in the optimum value of the instance if for the (unknown) optimum value opt for I, the algorithm either computes a feasible solution for I, with value at most k·r(k) (resp., at leas ..."

Abstract
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For a minimization (resp., maximization) problem P, an algorithm is called an (r(k), t(k))-FPT-approximation algorithm in the optimum value of the instance if for the (unknown) optimum value opt for I, the algorithm either computes a feasible solution for I, with value at most k·r(k) (resp., at least k/r(k)) or it computes a certificate that k < opt (resp., k> opt) in time t(k) · poly(n). A problem P is (r, t)-FPT-inapproximable (or, (r, t)-FPT-hard) when the problem does not admit an (r, t)-FPT-approximation algorithm. Inapproximability results in k, many times start by drastically reducing k, rendering it unrelated to any optimum of any instance. From inapproximability theory perspective this means that the inapproximability is in k but not in opt. We define inapproximability in opt because we want to define the exact contrary statement to approximation in opt. A hardness (in the usual sense) is (also) a hardness in opt if k equals opt(I) some instance I. In such case a (r(k), t(k))-hardness is also (r(opt(I)), t(opt(I)))-hardness, hence the name hardness in opt. Note that we did not change the definition of FPT-hardness but only the way it is implemented. Before this paper the gap between instance that is in-