Certifying large branch-width

Branch-width is defined for graphs, matroids, and, more generally, arbitrary symmetric submodular functions. For a finite set <i>V</i>, a function <i>f</i> on the set of subsets 2^<i>V</i> of <i>V</i> is <i>submodular</i> if <i>f</i>(<i>X</i>) + <i>f</i>(<i>Y</i>) ≥ <i>f</i>(<i>X</i> ∩ <i>Y</i>) + <i>f</i>(<i>X</i> ∪ <i>Y</i>), and <i>symmetric</i> if <i>f</i>(<i>X</i>) = <i>f</i>(<i>V \ X</i>). We discuss the computational complexity of recognizing that symmetric submodular functions have branch-width at most <i>k</i> for fixed <i>k</i>. An integer-valued symmetric submodular function <i>f</i> on 2^<i>V</i> is a <i>connectivity function</i> if <i>f</i>(θ) = 0 and <i>f</i>({<i>v</i>}) ≤ 1 for all <i>v</i> ∈ <i>V</i>. We show that for each constant <i>k</i>, if a connectivity function <i>f</i> on 2^<i>V</i> is presented by an oracle and the branch-width of <i>f</i> is larger than <i>k</i>, then there is a certificate of polynomial size (in |<i>V</i>|) such that a polynomial-time algorithm can verify the claim that branch-width of <i>f</i> is larger than <i>k.</i> In particular it is in coNP to recognize matroids represented over a fixed field with branch-width at most <i>k</i> for fixed <i>k.</i>