Dynamic programming on graphs with bounded treewidth
- Hans L. BodlaenderAffiliated withDept. of Computer Science, University of Utrecht
In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth ≤k, (or equivalently, the class of partial k-trees), for fixed k. We introduce two classes of graph decision problems, LCC and ECC, and subclasses C-LCC, and C-ECC. We show that each problem in LCC (or C-LCC) is solvable in polynomial (O(n C )) time, when restricted to graphs with fixed upper bounds on the treewidth and degree; and that each problem in ECC (or C-ECC) is solvable in polynomial (O(n C )) time, when restricted to graphs with a fixed upper bound on the treewidth (with given corresponding tree-decomposition). Also, problems in C-LCC and C-ECC are solvable in polynomial time for graphs with a logarithmic treewidth, and in the case of C-LCC-problems, a fixed upper bound on the degree of the graph.
Also, we show for a large number of graph decision problems, their membership in LCC, ECC, C-LCC and/or C-ECC, thus showing the existence of O(n C ) or polynomial algorithms for these problems, restricted to the graphs with bounded treewidth (and bounded degree). In several cases, C=1, hence our method gives in these cases linear algorithms.
For several NP-complete problems, and subclasses of the graphs with bounded treewidth, polynomial algorithms have been obtained. In a certain sense, the results in this paper unify these results.
KeywordsTreewidth partial k-trees graph decision problems restrictions of NP-complete problems polynomial time algorithms dynamic programming local condition compositions
- Dynamic programming on graphs with bounded treewidth
- Book Title
- Automata, Languages and Programming
- Book Subtitle
- 15th International Colloquium Tampere, Finland, July 11–15, 1988 Proceedings
- pp 105-118
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
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- Series ISSN
- Springer Berlin Heidelberg
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- partial k-trees
- graph decision problems
- restrictions of NP-complete problems
- polynomial time algorithms
- dynamic programming
- local condition compositions
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