Semi-nice tree-decompositions: The best of branchwidth, treewidth and pathwidth with one algorithm

Branchwidth and treewidth are connectivity parameters of graphs of high importance in algorithm design. By dynamic programming along the associated branch- or tree-decomposition one can solve most graph optimization problems in time linear in the graph size and exponential in the parameter. If one of these parameters is bounded on a class of graphs, then so is the other, but they can differ by a small constant factor and this difference can be crucial for the resulting runtime. In this paper we introduce semi-nice tree-decompositions and show that they combine the best of both branchwidth and treewidth. We first give simple algorithms to transform a given tree-decomposition or branch-decomposition into a semi-nice tree-decomposition. We then give two templates for dynamic programming along a semi-nice tree-decomposition, one for optimization problems over vertex subsets and another for optimization problems over edge subsets. We show that the resulting runtime will match or beat the runtimes achieved by doing dynamic programming directly on either a branch- or tree-decomposition. For example, given a graph G on n vertices with path-, tree- and branch-decompositions of width pw,tw and bw respectively, the Minimum Dominating Set problem on G is solved in time O(n2^m^i^n^{^1^.^5^8^p^w^,^2^t^w^,^2^.^3^8^b^w^}) by a single dynamic programming algorithm along a semi-nice tree-decomposition. On the applied side the immediate gain is that for each optimization problem one can achieve the benefits of both treewidth, branchwidth and pathwidth while developing and implementing only one dynamic programming algorithm. On the theoretical side the combination of the best properties of both branchwidth and treewidth in a single decomposition is a step towards a more general framework giving the fastest possible algorithms on tree-like graphs.