Abstract
Counting models for two Conjunctive Normal Form formulae (2-CFs), known as the #2SAT problem, is a classic #P complete problem. It is known that if the constraint graph of a 2-CF F is acyclic or contains loops and parallel edges, \(\#2SAT(F)\) can be computed efficiently. In this paper we address the cyclic case different from loops and parallel edges.
If the constraint graph G of a 2-CF F is cyclic, T a spanning tree plus loops and parallel edges of G and \(\overline{T}=G\setminus T\), what we called its cotree, we show that by building a set partition \(\cup T_i\) of \(\overline{T}\), where each \(T_i\) of the partition is formed by the frond edges of the cycles that are chained via other intersected cycles, then a parametric polynomial deterministic procedure for computing #2SAT with time complexity for the worst case of \(O(2^{k} \cdot poly(|E(T)|))\) can be obtained, where poly is a polynomial function, and k is the cardinality of the largest set in the partition.
This method shows that #2SAT is in the class of fixed-parameter tratable (FPT) problems, where the fixed-parameter k in our proposal, depends on the number of edges of a subcotree of a decomposition of the constraint graph (tree+loops+parallel:cotree) associated to the formula.
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Marcial-Romero, J.R., De Ita Luna, G., Hernández, J.A., Valdovinos, R.M. (2015). A Parametric Polynomial Deterministic Algorithm for #2SAT. In: Sidorov, G., Galicia-Haro, S. (eds) Advances in Artificial Intelligence and Soft Computing. MICAI 2015. Lecture Notes in Computer Science(), vol 9413. Springer, Cham. https://doi.org/10.1007/978-3-319-27060-9_16
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DOI: https://doi.org/10.1007/978-3-319-27060-9_16
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