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Improved exact algorithms for mildly sparse instances of MAX SAT. (English) Zbl 1379.68178
Summary: We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form $$O(2^{(1 - \mu(c)) n})$$ for instances with $$n$$ variables and $$m = c n$$ clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with $$\mu(c) = \frac{1}{O(c \log c)}$$ due to E. Dantsin and A. Wolpert [Lect. Notes Comput. Sci. 4121, 266–276 (2006; Zbl 1187.68258)], a randomized polynomial space algorithm with $$\mu(c) = \frac{1}{O(c \log^3 c)}$$ and a deterministic polynomial space algorithm with $$\mu(c) = \frac{1}{O(c^2 \log^2 c)}$$ due to T. Sakai et al. [Theory Comput. Syst. 57, No. 2, 426–443 (2015; Zbl 1347.68312)]. Our first result is a deterministic polynomial space algorithm with $$\mu(c) = \frac{1}{O(c \log c)}$$ that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than $$2^n$$ only if $$m = O(n^2)$$. Our second results are deterministic exponential space algorithms for Max SAT with $$\mu(c) = \frac{1}{O((c \log c)^{2 / 3})}$$ and for Max 3-SAT with $$\mu(c) = \frac{1}{O(c^{1 / 2})}$$ that run super-polynomially faster than $$2^n$$ when $$m = o(n^{5 / 2} / \log^{5 / 2} n)$$ and $$m = o(n^3 / \log^2 n)$$ respectively. Our third result is a randomized exponential space algorithm for Max SAT with $$\mu(c) = \frac{1}{O(c^{1 / 2} \log^{3 / 2} c)}$$ that run super-polynomially faster than $$2^n$$ when $$m = o(n^3 / \log^5 n)$$.
##### MSC:
 68Q25 Analysis of algorithms and problem complexity 68W05 Nonnumerical algorithms 68W20 Randomized algorithms 68W40 Analysis of algorithms
MAX-2-SAT
Full Text:
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