Decoding Tree Decompositions from Permutations

Most algorithmic strategies for solving problems considering treewidth parameterization require that a tree decomposition is given. Given a graph \(G=(V,E)\) and denoting by \(\mathcal {C_G}\) the family of chordal graphs (triangulations) \(G'\) such that \(V(G)=V(G')\) and \( E(G)\subseteq E(G')\) , the treewidth of a graph G can be defined alternatively as the size of the smallest maximum clique of a graph in \(\mathcal {C_G}\) , minus one. In addition, any tree decomposition \(\mathcal {T}\) of a graph \(G'\in \mathcal {C_G}\) is also a tree decomposition of G. In this paper, we are interested in the main subproblem to be solved by the most popular heuristics for treewidth computation, called Tree Decomposition Decoding. In such a problem, we are given a graph \(G=(V, E)\) and a permutation \(\rho \) of V(G) and asked to determine the width of the tree decomposition \(\mathcal {T}\) of G that is an optimum tree decomposition of the minimal triangulation \(G'\in \mathcal {C_G}\) having \(\rho \) as perfect elimination ordering. From \((G,\rho )\) , it is easy to find the solution to the problem by first constructing the triangulation \(G'\) arising from \(\rho \) . However, in the worst case, such constructions of \(G'\) require \(\varTheta (|V(G)|^2)\) space. In this work, we propose two algorithms for solving the problem; both avoid the construction of triangulations \(G'\) . The first performers in \(\mathcal {O}(|V(G)|\cdot \ell )\) space and \(\mathcal {O}(|V(G)|^2\cdot \ell )\) time, where \(\ell \) is the number of leaves of the tree decomposition encoded by \(\rho \) . The second is faster in practice and achieves a different trade-off, solving the problem within \(\mathcal {O}(|E(G)| + |V(G)|)\) space and \(\mathcal {O}(|E(G)|\cdot \log |V(G)|)\) time.