Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
Constructive Linear Time Algorithms for Branchwidth
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Graphs of bounded rank-width
Computing representations of matroids of bounded branch-width
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
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Branch-width is defined for graphs, matroids, and, more generally, arbitrary symmetric submodular functions. For a finite set V, a function f on the set of subsets 2V of V is submodular if f(X) + f(Y) ≥ f(X ∩ Y) + f(X ∪ Y), and symmetric if f(X) = f(V \ X). We discuss the computational complexity of recognizing that symmetric submodular functions have branch-width at most k for fixed k. An integer-valued symmetric submodular function f on 2V is a connectivity function if f(θ) = 0 and f({v}) ≤ 1 for all v ∈ V. We show that for each constant k, if a connectivity function f on 2V is presented by an oracle and the branch-width of f is larger than k, then there is a certificate of polynomial size (in |V|) such that a polynomial-time algorithm can verify the claim that branch-width of f is larger than k. In particular it is in coNP to recognize matroids represented over a fixed field with branch-width at most k for fixed k.