Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In

@inproceedings{Villanger2006ImprovedEA,
  title={Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In},
  author={Yngve Villanger},
  booktitle={LATIN},
  year={2006}
}
Exact exponential-time algorithms for NP-hard problems is an emerging field, and an increasing number of new results are being added continuously. Two important NP-hard problems that have been studied for decades are the treewidth and the minimum fill problems. Recently, an exact algorithm was presented by Fomin, Kratsch, and Todinca to solve both of these problems in time ${\mathcal O}^{*}$(1.9601n). Their algorithm uses the notion of potential maximal cliques, and is able to list these in… 
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24 Citations
Background Citations
8
Methods Citations
5
Results Citations
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24 Citations

Faster Parameterized Algorithms for Minimum Fill-in
TLDR
A new lemma is presented describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k, which improves the base of the exponential part of the best known parameterized algorithm time for this problem so far.
Exact Algorithms for Treewidth and Minimum Fill-In
TLDR
It is shown that the treewidth and the minimum fill-in of an $n$-vertex graph can be computed in time $\mathcal{O}(1.8899^n)$ and the running time of the algorithms can be reduced to 1.4142 minutes.
On exact algorithms for treewidth
TLDR
Experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential-time algorithms using exponential space or using only polynomial space are given and it is shown that with a more complicated algorithm using balanced separators, Treewidth can be computed in O*(2.9512n) time and polynometric space.
Treewidth computation and extremal combinatorics
TLDR
It is proved that every graph on nvertices contains at most $n\binom{b+f}{b}$ such vertex subsets, and this result from extremalcombinatorics appears to be very useful in the design of severalumeration and exact algorithms.
Exact Algorithms for Edge Domination
TLDR
It is shown that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226n) time and polynomial space using modifications of the algorithm for edge dominate set.
Combinatorial Optimization on Graphs of Bounded Treewidth
TLDR
The concepts of treewidth and tree decompositions are introduced, and the technique with the Weighted Independent Set problem is illustrated, to survey some of the latest developments.
A measure & conquer approach for the analysis of exact algorithms
TLDR
The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis, in order to step beyond limitations of current algorithms design.
A Note on Exact Algorithms for Vertex Ordering Problems on Graphs
In this note, we give a proof that several vertex ordering problems can be solved in O∗(2n) time and O∗(2n) space, or in O∗(4n) time and polynomial space. The algorithms generalize algorithms for the…
Computing branchwidth via efficient triangulations and blocks
TLDR
It is shown how blocks can be used to construct an algorithm computing the branchwidth of a graph on n vertices in time, based on the recent results of Mazoit.
Exact Algorithms for Graph Homomorphisms
TLDR
It is shown that for an odd integer $k\ge 5$, whether an input graph G with n vertices is homomorphic to the cycle of length k, can be decided in time, which is the first NP-hard case different from graph coloring.
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References

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