On the complexity of computing treelength
Discrete Applied Mathematics
A space-time tradeoff for permutation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Feedback vertex sets in tournaments
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Subexponential parameterized algorithm for minimum fill-in
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Computing hypergraph width measures exactly
Information Processing Letters
Feedback Vertex Sets in Tournaments
Journal of Graph Theory
Efficient algorithms for the max k-vertex cover problem
TCS'12 Proceedings of the 7th IFIP TC 1/WG 202 international conference on Theoretical Computer Science
Decomposing combinatorial auctions and set packing problems
Journal of the ACM (JACM)
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For a given graph G and integers b,f≥ 0, let S be a subset of vertices of G ofsize b + 1 such that the subgraph of G induced byS is connected and S can be separated from othervertices of G by removing f vertices. We provethat every graph on n vertices contains at most$n\binom{b+f}{b}$ such vertex subsets. This result from extremalcombinatorics appears to be very useful in the design of severalenumeration and exact algorithms. In particular, we use it toprovide algorithms that for a given n-vertex graphGcompute the treewidth of G in time$\mathcal{O}(1.7549^n)$ by making use of exponential space and intime $\mathcal{O}(2.6151^n)$ and polynomial space;decide in time $\mathcal{O}(({\frac{2n+k+1}{3})^{k+1}\cdotkn^6})$ if the treewidth of G is at most k;list all minimal separators of G in time$\mathcal{O}(1.6181^n)$ and all potential maximal cliques ofG in time $\mathcal{O}(1.7549^n)$.This significantly improves previous algorithms for theseproblems.