Abstract
The degree of a variable \(x_i\) in a MaxSAT instance is the number of times \(x_i\) and \(\bar{x}_i\) appearing in the given formula. The degree of a MaxSAT instance is equal to the largest variable degree in the instance. In this paper, we study techniques for solving the MaxSAT problem on instances of degree 3 (briefly, (n, 3)-MaxSAT), which is NP-hard. Two new non-trivial reduction rules are introduced based on the resolution principle. As applications, we present two algorithms for the (n, 3)-MaxSAT problem: a parameterized algorithm of time \(O^*(1.194^k)\), and an exact algorithm of time \(O^*(1.237^n)\), improving the previous best upper bounds \(O^*(1.2721^k)\) and \(O^*(1.2600^n)\), respectively.
This work is supported by the National Natural Science Foundation of China, under grants 61173051, 61232001, 61472449, and 61420106009.
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Notes
- 1.
Following the current convention in the research in exact and parameterized algorithms, we will use the notation \(O^*(f)\) to denote the bound \(f \cdot m^{O(1)}\), where f is an arbitrary function and m is the instance size.
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Xu, C., Chen, J., Wang, J. (2015). Improved MaxSAT Algorithms for Instances of Degree 3. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_2
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