Packing cycles exactly in polynomial time

Let G=(V,E) be an undirected graph in which every vertex v∈V is assigned a nonnegative integer w(v). A w‐packing is a collection 𝒫 of cycles (repetition allowed) in G such that every v∈V is contained at most w(v) times by the members of 𝒫. Let ⟨w⟩=2|V|+Σ v∈V ⌈log (w(v)+1)⌉ denote the binary encoding length (input size) of the vector (w(v): v∈V) ^T . We present an efficient algorithm which finds in O(|V|⁸⟨w⟩²+|V|¹⁴) time a w‐packing of maximum cardinality in G provided packing and covering cycles in G satisfy the ℤ+‐max‐flow min‐cut property.