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
Dominating sets in planar graphs: branch-width and exponential speed-up
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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 the (k, r)-center in planar graphs and map graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Efficient exact algorithms on planar graphs: exploiting sphere cut branch decompositions
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Dynamic programming and fast matrix multiplication
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Dynamic programming and planarity: Improved tree-decomposition based algorithms
Discrete Applied Mathematics
How to use planarity efficiently: new tree-decomposition based algorithms
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
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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 $O(n2^{min\{1.58 {\it pw},2{\it tw},2.38{\it bw}\}}$) by a single dynamic programming algorithm along a semi-nice tree-decomposition.