A Note on Multiflow Locking Theorem

This note addresses the undirected multiflow (multicommodity flow) theory. A multiflow in a network with terminal set T can be regarded as a single commodity (A, T \A)-flow for any nonempty proper subset A ⊂ T by ignoring flows not connecting A and T \ A. A set system A on T is said to be lockable if for every network having T as terminal set there exists a multiflow being simultaneously a maximum (A, T \A)-flow for every A ∈ A. The multiflow locking theorem, due to Karzanov and Lomonosov, says that A is lockable if and only if it is 3-cross-free. A multiflow can also be regarded as a single commodity (A,B)-flow for every partial cut (A,B) of terminals, where a partial cut is a pair of disjoint subsets (not necessarily a bipartition). Based on this observation, we study the locking property for partial cuts, and prove an analogous characterization for a lockable family of partial cuts.