Abstract
In this paper, we address two key aspects of solving methods based on tree-decomposition. First, we propose an algorithm computing decompositions that allows to bound the size of separators, which is a crucial parameter to limit the space complexity, and thus the feasibility of such methods. Moreover, we show how it is possible to dynamically modify the considered decomposition during the search. This dynamic modification can offer more freedom to the variable ordering heuristics. This also allows to better use the information gained during the search while controlling the size of the required memory.
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Notes
- 1.
The 2-section of a hypergraph (X, C) is the graph \((X,C')\) where \(C'=\{ \{x,y\} | \exists c \in C, \{x,y\} \subseteq c\}\) [14].
- 2.
For any \(Y \subseteq X\), the subgraph G[Y] of \(G=(X,C)\) induced by Y is the graph \((Y,C_Y)\) where \(C_Y = \{\{x,y\}\in C | x,y \in Y\}\).
- 3.
A structural good (resp. nogood) of \(E_i\) w.r.t. \(E_j\) (with \(E_j\) a child of \(E_i\)) is a consistent assignment of \(E_i \cap E_j\) which can (resp. cannot) be consistently extended on \(Desc(E_j)\) [19].
- 4.
Given a CSP \(P=(X,D,C)\) and a sequence of decisions \(\varSigma \), \(\varDelta \) is a nogood of P if \(P_{|\varDelta }\) has no solution where \(P_{|\varDelta }\) is the CSP \((X,D',C)\) with \(D'=(d'_{x_1},\ldots ,d'_{x_n})\) and for each positive decision \(x_i = v_i\), \(d'_{x_i} =\{v_i\}\) and for each negative decision \(x_i \ne v_i\), \(d'_{x_i} = d_{x_i} \backslash \{v_i\}\). If \(x_i\) does not appear in \(\varDelta \) then \(d'_{x_i} = d_{x_i}\) [23].
- 5.
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Jégou, P., Kanso, H., Terrioux, C. (2016). Towards a Dynamic Decomposition of CSPs with Separators of Bounded Size. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_20
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