Abstract
Parameterized algorithms are a way to solve hard problems more efficiently, given that a specific parameter of the input is small. In this paper, we apply this idea to the field of answer set programming (ASP). To this end, we propose two kinds of graph representations of programs to exploit their treewidth as a parameter. Treewidth roughly measures to which extent the internal structure of a program resembles a tree. Our main contribution is the design of parameterized dynamic programming algorithms, which run in linear time if the treewidth and weights of the given program are bounded. Compared to previous work, our algorithms handle the full syntax of ASP. Finally, we report on an empirical evaluation that shows good runtime behaviour for benchmark instances of low treewidth, especially for counting answer sets.
For additional details and proofs, we refer to an extended self-archived version [8]. A preliminary version of the paper was presented on the workshop TAASP’16. Research was supported by the Austrian Science Fund (FWF), Grant Y698. The first and second author are also affiliated with the University of Potsdam, Germany.
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Notes
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- 2.
\(\sigma \)
\(\rho \)
\(\,\mathrel {\mathop :}=\{(x,{\underset{(x,c_1)\in \sigma }{\varSigma }}c_1+ {\underset{(x,c_2)\in \rho }{\varSigma }c_2}) \mid (x,\cdot )\in \sigma \cup \rho \}\); \(\sigma ^+_{r}\,\mathrel {\mathop :}=\sigma \cup \{(r,0)\}\); \(\sigma ^-_{S}\,\mathrel {\mathop :}=\{(x,y) \in \sigma \mid {x\not \in S}\}\). - 3.
We require to add \(\{\leftarrow B_r \mid r\in {{\mathrm{CH}}}(\mathcal{P}), H_r \subsetneq \text {at}_{\le t}\}\) in order to decide satisfiability for corner cases of choice rules involving counterwitnesses of Line 3 in Algorithm 3.
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Fichte, J.K., Hecher, M., Morak, M., Woltran, S. (2017). Answer Set Solving with Bounded Treewidth Revisited. In: Balduccini, M., Janhunen, T. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2017. Lecture Notes in Computer Science(), vol 10377. Springer, Cham. https://doi.org/10.1007/978-3-319-61660-5_13
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