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Incoming links: Degree branching with a one-tree bound
Past exam of the integer programming course of the University of Oxford 2010 1 c Solution 2026-10-31
Write for the undirected edge . At the root of the branch and bound search, the minimum spanning tree on is of cost . The cheapest edges incident to vertex one are , of cost . The root lower bound is therefore twenty, attained byIt is not a tour: vertex four has degree three and vertex three degree one.
A tour must omit at least one edge from , since its degree at vertex four is two. Partition tours according to the first omitted edge in that ordered list. This gives three disjoint, exhaustive children, retaining included and excluded edges when recomputing the one-tree relaxation:
- Exclude . The incident pair is now of cost twelve, and the other spanning tree still costs ten. The resulting one-tree is a tour, so it supplies an incumbent solution of cost .
- Include and exclude . The other spanning tree must be of cost fourteen; the cheapest permitted incident pair costs ten. The lower bound is , so prune this child.
- Include and exclude . The spanning tree must be of cost sixteen; the incident pair again costs ten. The lower bound is , so prune this child too.
All tours belong to one of the three children, and no remaining child can improve the incumbent. ThereforeThe branching rule is motivated by a violated tour degree equation, rather than by an arbitrary costly edge. This is degree branching with a one-tree bound.