Fig. 2

Dependency graph and tree decompositions of the running example (feature network \(\mathcal {N} _{\text {col}}\)). A The dependency graph contains one (binary) edge per dependency due to a constraint \({\text {\textsf{NotEquals}}}\in \mathcal {C} _{\text {col}}\). The dependency hyperedges due to the three network functions \({\text {\textsf{Card}}}\in F_{\text {card}}\) are colored. B Two possible tree decompositions of this dependency graph (and therefore \(\mathcal {N} _{\text {col}}\)). The difference set is underlined in each bag. Solving of the network could be based on either one, but with different run time, which is dominated by the largest bag (bold). Due to their largest bags of size 5 and 6, the two tree decompositions have respective width 4 and 5. The bags handling the 4-ary functions are highlighted, where colors correspond to the hyperedges of A. C Tree decomposition of the network without 4-ary functions \({\text {\textsf{Card}}}\). The functions don’t allow any tree decomposition with width 3; thus they make the problem more complex