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Minimum Size Tree-Decompositions

Bi Li 1, 2 Fatima Zahra Moataz 1 Nicolas Nisse 1, 3
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Tree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree- decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic program- ming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree- decompositions. It is then interesting to try to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k >= 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k >= 4 and polynomial for k <= 2. On going work also suggests it is polynomial for k = 3.
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https://hal.inria.fr/hal-01023904
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Submitted on : Tuesday, July 15, 2014 - 2:09:01 PM
Last modification on : Monday, October 12, 2020 - 10:30:39 AM
Long-term archiving on: : Friday, November 21, 2014 - 6:05:59 PM

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Bi Li, Fatima Zahra Moataz, Nicolas Nisse. Minimum Size Tree-Decompositions. 9th International colloquium on graph theory and combinatorics (ICGT), 2014, Grenoble, France. ⟨hal-01023904⟩

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