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Approximation of Natural W[P]complete Minimisation Problems is Hard
"... We prove that the weighted monotone circuit satisfiability problem has no fixedparameter 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 cir ..."
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We prove that the weighted monotone circuit satisfiability problem has no fixedparameter 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 fixedparameter tractable 2approximation 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.
The Complexity of the MatroidGreedoid Partition Problem
, 2008
"... We show that the maximum matroidgreedoid partition problem is NPhard to approximate to within 1/2 + ε for any ε> 0, which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and ..."
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We show that the maximum matroidgreedoid partition problem is NPhard to approximate to within 1/2 + ε for any ε> 0, which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and listsize, together with an efficient algorithm for listdecoding. We show that the recent extractor construction of Guruswami, Umans and Vadhan [5] can be used to obtain codes with these properties. We also show that the parameterized matroidgreedoid partition problem is fixedparameter tractable.