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Approximation of Natural W[P]-complete Minimisation Problems is Hard
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BibTeX
@MISC{Eickmeyer_approximationof,
author = {Kord Eickmeyer and Martin Grohe and Magdalena Grüber},
title = {Approximation of Natural W[P]-complete Minimisation Problems is Hard},
year = {}
}
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Abstract
We prove that the weighted monotone circuit satisfiability problem has no fixed-parameter tractable approximation algorithm with constant or polylogarithmic approximation ratio unless FPT = W[P]. Our result answers a question of Alekhnovich and Razborov [2], who proved that the weighted monotone circuit satisfiability problem has no fixed-parameter tractable 2-approximation algorithm unless every problem in W[P] can be solved by a randomized fpt algorithm and asked whether their result can be derandomized. Alekhnovich and Razborov used their inapproximability result as a lemma for proving that resolution is not automatizable unless W[P] is contained in randomized FPT. It is an immediate consequence of our result that the complexity theoretic assumption can be weakened to W[P] ̸ = FPT. The decision version of the monotone circuit satisfiability problem is known to be complete for the class W[P]. By reducing them to the monotone circuit satisfiability problem with suitable approximation preserving reductions, we prove similar inapproximability results for all other natural minimisation problems known to be W[P]-complete.
Keyphrases
natural complete minimisation problem monotone circuit satisfiability problem weighted monotone circuit satisfiability problem fixed-parameter tractable approximation algorithm inapproximability result similar inapproximability result fixed-parameter tractable 2-approximation algorithm decision version complexity theoretic assumption natural minimisation problem immediate consequence randomized fpt suitable approximation randomized fpt algorithm polylogarithmic approximation ratio
