I am preparing a presentation about a version of Edmond's Blossom Algorithm as it has been described by William Pulleyblank. The algorithm finds a maximum weight b-matching for a graph $G = (V, E)$ with $n = |V|$ and $b \in \mathbb{N}^n$. The running time of the algorithm is in $\mathcal{O}(b(V)n^2)$ where $b(V) = \sum_{v_i \in V}b_i$. So $b$ is a vector of length $n$ of arbitrary positive integers and the running time of the algorithm depends on the sum over the entries of $b$.
Now my question is: what is the order of $b(V)$? I would say it's in $\mathcal{O}(Bn)$ where $B = max\{b_i: v_i \in V\}$. But $B$ can get arbitrarily large. Could this algorithm then be said to run in polynomial time? If not, what is the bound then?
In general I don't quite understand how an algorithm whose running time depends on more than one variable can be said to have a polynomial running time. I would appreciate any explanation.