Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses

Abstract

We prove hardness results for approximating set splitting problems and also instances of satisfiability problems which have no “mixed” clauses, i.e., every clause has either all its literals unnegated or all of them negated. Results of Håstad [9] imply tight hardness results for set splitting when all sets have size exactly k ≥ 4 elements and also for non-mixed satisfiability problems with exactly k literals in each clause for k ≥ 4. We consider the case k = 3. For the Max E3-Set Splitting problem in which all sets have size exactly 3, we prove an NP-hardness result for approximating within any factor better than 19/20. This result holds even for satisfiable instances of Max E3-Set Splitting, and is based on a PCP construction due to Håstad [9]. For “non mixed Max 3Sat”, we give a PCP construction which is a variant of one in [8] and use it to prove the NP-hardness of approximating within a factor better than 11/12, and also a hardness factor of 15/16 + ε (for any ε > 0) for the version where each clause has exactly 3 literals (as opposed to up to 3 literals).