On Approximation Hardness of the Minimum 2SAT-DELETION Problem

The Minimum 2SAT-Deletion problem is to delete the minimum number of clauses in a 2SAT instance to make it satisfiable. It is one of the prototypes in the approximability hierarchy of minimization problems [8], and its approximability is largely open. We prove a lower approximation bound of \(8\sqrt 5-15\approx 2.88854\), improving the previous bound of \(10\sqrt5-21\approx 1.36067\) by Dinur and Safra [5]. For highly restricted instances with exactly 4 occurrences of every variable we provide a lower bound of \(\frac32\). Both inapproximability results apply to instances with no mixed clauses (the literals in every clause are both either negated, or unnegated).

[1]  Uri Zwick,et al.  Finding almost-satisfying assignments , 1998, STOC '98.

[2]  Philip N. Klein,et al.  Approximation Algorithms for Steiner and Directed Multicuts , 1997, J. Algorithms.

[3]  D. West.  Introduction to Graph Theory , 1995 .

[4]  Marek Karpinski,et al.  Improved Approximation Lower Bounds on Small Occurrence Optimization , 2003, Electron. Colloquium Comput. Complex..

[5]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[6]  Uri Zwick,et al.  Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems , 2002, IPCO.

[7]  Luca Trevisan,et al.  Constraint satisfaction: the approximability of minimization problems , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[8]  Miroslav Chlebík,et al.  Inapproximability Results for Bounded Variants of Optimization Problems , 2003, FCT.

[9]  Charles H. C. Little,et al.  On defect-d matchings in graphs , 1975, Discret. Math..

[10]  Irit Dinur,et al.  The importance of being biased , 2002, STOC '02.

[11]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[12]  Jaikumar Radhakrishnan,et al.  Greed is good: Approximating independent sets in sparse and bounded-degree graphs , 1997, Algorithmica.

[13]  Jianer Chen,et al.  On Approximating Minimum Vertex Cover for Graphs with Perfect Matching , 2000, ISAAC.

[14]  Ewald Speckenmeyer,et al.  Ramsey numbers and an approximation algorithm for the vertex cover problem , 1985, Acta Informatica.

[15]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.