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 imply tight hardness results for set splitting when all sets have size exactly $k \ge 4$ elements and also for non-mixed satisfiability problems with exactly $k$ literals in each clause for $k \ge 4$. 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 ${\frac{19}{20}}$. This result holds even for satisfiable instances of MAX E3-SET SPLITTING, and is based on a PCP construction due to Håstad. For “non-mixed MAX 3SAT,” we give a PCP construction which is a slight variant of the one given by Håstad for MAX 3SAT, and use it to prove the NP-hardness of approximating within a factor better than ${\frac{11}{12}}$.
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Guruswami, V. Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses. Algorithmica 38, 451–469 (2004). https://doi.org/10.1007/s00453-003-1072-z
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DOI: https://doi.org/10.1007/s00453-003-1072-z