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

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 p...

• Type: Conference
• Year: 2008
• Where: COCO
• Authors: Kord Eickmeyer, Martin Grohe, Magdalena Grüber
• Keywords: Algorithms, Circuit Satisfiability Problem, COCO 2008, Monotone Circuit Satisfiability, Weighted Monotone Circuit