Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Algorithms for Vertex Partitioning Problems on Partial k-Trees
SIAM Journal on Discrete Mathematics
Treewidth: Algorithmoc Techniques and Results
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Improved Tree Decomposition Based Algorithms for Domination-like Problems
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Tour Merging via Branch-Decomposition
INFORMS Journal on Computing
Algorithm Design
Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs
ACM Transactions on Algorithms (TALG)
Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up
SIAM Journal on Computing
Efficient exact algorithms on planar graphs: exploiting sphere cut branch decompositions
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Subexponential parameterized algorithms for degree-constrained subgraph problems on planar graphs
Journal of Discrete Algorithms
A local search algorithm for branchwidth
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
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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.