Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses
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 ≥ 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 . For “non mixed Max 3Sat”, we give a PCP construction which is a variant of one in  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).
KeywordsApproximation Algorithm Constraint Satisfaction Problem Proof System Hardness Result Satisfiability Problem
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- 3.P. Crescenzi, R. Silvestri and L. Trevisan. To weight or not to weight: Where is the question? Proc. of 4th Israel Symposium on Theory of Computing and Systems, pp. 68–77, 1996.Google Scholar
- 4.U. Feige and M. Goemans. Approximating the value of two prover proof systems, with applications to MAX-2SAT and MAX-DICUT. Proc. of the 3rd Israel Symposium on Theory and Computing Systems, Tel Aviv, pp. 182–189, 1995.Google Scholar
- 6.V. Guruswami. The approximability of set splitting problems and satisfiability problems with no mixed clauses. ECCC Technical Report TR-99-043, 1999.Google Scholar
- 7.V. Guruswami. Query-efficient Checking of Proofs and Improved PCP Characterizations of NP. S.M Thesis, MIT, May 1999.Google Scholar
- 8.V. Guruswami, D. Lewin, M. Sudan and L. Trevisan. A tight characterization of NP with 3 query PCPs. ECCC Technical Report TR98-034, 1998. Preliminary Version in Proc. of FOCS’98.Google Scholar
- 9.J. Håstad Some optimal inapproximability results. ECCC Technical Report TR97-37, 1997. Preliminary version in Proc. of STOC’97.Google Scholar
- 11.H. Karlo. and U. Zwick. A (7/8-ε)-approximation algorithm for MAX 3SAT? In Proceedings of the 38th FOCS, 1997.Google Scholar
- 12.S. Khanna, M. Sudan and D. Williamson. A complete classification of the approximability of maximization problems derived from Boolean constraint satisfaction. Proc. of the 29th STOC, 1997.Google Scholar
- 13.L. Lovász. Coverings and colorings of hypergraphs. Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, pp. 3–12, Utilitas Mathematica Publishing, Winnipeg, 1973.Google Scholar
- 16.L. Trevisan, G. Sorkin, M. Sudan and D. Williamson. Gadgets, approximation and linear programming. Proceedings of the 37th FOCS, pp. 517–626, 1996.Google Scholar
- 17.U. Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, 1998.Google Scholar
- 18.U. Zwick. Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In Proceedings of STOC’99.Google Scholar