Abstract
Nested and co-nested formulas are two classes of CNF instances that can be characterized in terms of outerplanar graphs. For these classes, linear time algorithms are known for SAT and (unweighted) Max-SAT. In this paper we devise linear time algorithms for a general optimization version of SAT. Moreover, we show that a general probabilistic version of SAT reduces to solving a system of linear inequalities where the number of variables and constraints is linear in the size of the formula.
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Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: theory, algorithms, and applications. New York: Prentice Hall.
Andersen, K. A., & Pretolani, D. (2001). Easy cases of probabilistic satisfiability. Annals of Mathematics and Artificial Intelligence, 33, 69–91.
Baioletti, G., Capotorti, A., Tulipani, S., & Vantaggi, B. (2000). Elimination of Boolean variables for probabilistic coherence. Soft Computing, 4(2), 81–88.
Baioletti, G., Capotorti, A., Tulipani, S., & Vantaggi, B. (2002). Simplification rules for the coherent probabilistic assessment problem. Annals of Mathematics and Artificial Intelligence, 35, 11–28.
Boole, G. (1854). An investigation of the laws of thought. London: Walton and Maberlay.
Conforti, M., & Cornuéjols, G. (1995). A class of logic problems solvable by linear programming. Journal of the ACM, 42, 1107–1113.
Crama, Y., Hansen, P., & Jaumard, B. (1990). The basic algorithm for pseudo-Boolean programming revisited. Discrete Applied Mathematics, 29, 171–185.
Georgakopoulos, G., Kavvadias, D., & Papadimitriou, C. H. (1988). Parabilistic satisfiability. Journal of Complexity, 4, 1–11.
Hailperin, T. (1986). Boole’s logic and probability (2nd ed.) Studies in logic and the foundations of mathematics (Vol. 85). Amsterdam: North-Holland.
Knuth, D. E. (1990). Nested satisfiability. Acta Informatica, 28, 1–6.
Kratochvíl, J., & Křivánek, M. (1993). Satisfiability of co-nested formulas. Acta Informatica, 30, 397–403.
Nilsson, N. J. (1986). Probabilistic logic. Artificial Intelligence, 28, 71–87.
Pretolani, D. (2004). Hypergraphs reduction and satisfiability problems. In Lecture notes comp. sci.: Vol. 2919. Theory and application of satisfiability testing, 6th international conference, SAT 2003 (pp. 383–397). Berlin: Springer.
Pretolani, D. (2005). Probability logic and optimization SAT: The PSAT and CPA models. Annals of Mathematics and Artificial Intelligence, 43, 211–221.
Yen, J. Y. (1971). Finding the K shortest loopless paths in a network. Management Science, 17(11), 712–716.
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Pretolani, D. Optimization and probabilistic satisfiability on nested and co-nested formulas. Ann Oper Res 188, 371–387 (2011). https://doi.org/10.1007/s10479-008-0502-3
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DOI: https://doi.org/10.1007/s10479-008-0502-3