### Abstract

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.

Original language | English (US) |
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Pages | 810-813 |

Number of pages | 4 |

DOIs | |

State | Published - Feb 28 2006 |

Event | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms - Miami, FL, United States Duration: Jan 22 2006 → Jan 24 2006 |

### Other

Other | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | Miami, FL |

Period | 1/22/06 → 1/24/06 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

*Certifying large branch-width*. 810-813. Paper presented at Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL, United States. https://doi.org/10.1145/1109557.1109646