Certifying large branch-width

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 2 ^V 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 2 ^V is a connectivity function if f(ø) = 0 and f({v}) ≤ 1 for all v ∈ V. We show that for each constant k, if a connectivity function f on 2 ^V 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.