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On the complexity of $$k$$-SAT. (English) Zbl 0990.68079
Summary: The $$k$$-SAT problem is to determine if a given $$k$$-CNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve $$k$$-SAT for $$k\geq 3$$. Here exponential time means $$2^{\delta n}$$ for some $$\delta> 0$$. In this paper, assuming that, for $$k\geq 3$$, $$k$$-SAT requires exponential time complexity, we show that the complexity of $$k$$-SAT increases as $$k$$ increases. More precisely, for $$k\geq 3$$, define $$s_k= \inf\{\delta$$: there exists $$2^{\delta n}$$ algorithm for solving $$k$$-SAT}. Define ETH (Exponential-Time Hypothesis) for $$k$$-SAT as follows: for $$k\geq 3$$, $$s_k> 0$$. In this paper, we show that $$s_k$$ is increasing infinitely often assuming ETH for $$k$$-SAT. Let $$s_\infty$$ be the limit of $$s_k$$. We will in fact show that $$s_k\leq (1-d/k)s_\infty$$ for some constant $$d> 0$$. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a $$k$$-CNF to the satisfiability of a disjunction of $$2^{\varepsilon n}$$ $$k'$$-CNFs in fewer variables for some $$k'\geq k$$ and arbitrarily small $$\varepsilon> 0$$. We also show that such a disjunction can be computed in time $$2^{\varepsilon n}$$ for arbitrarily small $$\varepsilon> 0$$.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
##### Keywords:
$$k$$-SAT problem; critical clauses; Sparsification Lemma
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##### References:
 [1] Alon, N.; Spencer, J.; Erdős, P., The probabilistic method, (1992), Wiley New York [2] Beigel, R.; Eppstein, R., 3-coloring in time O(1.3446n) time: a no-MIS algorithm, Proceedings of the 36th annual IEEE symposium on foundations of computer science, (1995), p. 444-453 · Zbl 0938.68940 [3] Crawford, J.M.; Auton, L.D., Experimental results on the crossover point in random 3SAT, Artificial intelligence, 81, (1996) [4] E. A. Hirsch, Two new upper bounds for SAT, in, SIAM Conference on Discrete Algorithms, 1997. [5] R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strongly exponential complexity, in 1998 Annual IEEE Symposium on Foundations of Computer Science, pp. 653-662. · Zbl 1006.68052 [6] Jian, T., An O(20.304n) algorithm for solving maximum independent set problem, IEEE trans. comput., 35, 847-851, (1986) · Zbl 0606.68062 [7] O. Kullmann, and, H. Luckhardt, Deciding propositional tautologies: algorithms and their complexity, submitted. [8] Lawler, E., A note on the complexity of the chromatic number problem, Inform. process. lett., 5, 66-67, (1976) · Zbl 0336.68021 [9] Monien, B.; Speckenmeyer, E., Solving satisfiability in less than 2^n steps, Discrete appl. math., 10, 287-295, (1985) · Zbl 0603.68092 [10] Paturi, R.; Pudlák, P.; Zane, F., Satisfiability coding lemma, Proceedings of the 38th annual IEEE symposium on foundations of computer science, (1997), p. 566-574 [11] Paturi, R.; Pudlák, P.; Saks, M.; Zane, F., An improved exponential-time algorithm for k-SAT, 1998 annual IEEE symposium on foundations of computer science, (1998), p. 628-637 [12] Robson, J., Algorithms for maximum independent sets, J. algorithms, 7, 425-440, (1986) · Zbl 0637.68080 [13] Schiermeyer, I., Solving 3-satisfiability in less than 1.579^n steps, Selected papers from CSL ’92, Lecture notes in computer science, 702, (1993), Springer-Verlag Berlin/New York, p. 379-394 · Zbl 0788.68066 [14] Schiermeyer, I., Pure literal look ahead: an O(1.497n) 3-satisfiability algorithm, Technical report, university of Köln, (1996) [15] Schöning, U., A probabilistic algorithm for k-SAT and constraint satisfaction problems, 1999 annual IEEE symposium on foundations of computer science, (1999), p. 410-414 [16] B. Selman, Personal communication, 1999. [17] Shindo, M.; Tomita, E., A simple algorithm for finding a maximum clique and its worst-case time complexity, Systems comput. Japan, 21, 1-13, (1990) [18] Tarjan, R.; Trojanowski, A., Finding a maximum independent set, SIAM J. comput., 6, 537-546, (1977) · Zbl 0357.68035
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