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Approximation of Natural W[P]-Complete Minimisation Problems Is Hard

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3 Author(s)
Kord Eickmeyer ; Inst. fur Inf., Humboldt Univ. Berlin, Berlin ; Martin Grohe ; Magdalena Grüber

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, 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] ne 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.

Published in:

Computational Complexity, 2008. CCC '08. 23rd Annual IEEE Conference on

Date of Conference:

23-26 June 2008