Move tokens from s to t as fast as possible

Question

Let $G=(V, E)$ be an unweighted and undirected graph, and $s, t \in E$.

The problems starts with $n$ tokens on $s$.

The goal is to move theses tokens to $t$ in a minimum of rounds with these rules :

• Each token can be moved up to once per round (a movement being when you transfer the token from $v$ the vertex that holds it, to $w \in N_G(v)$).
• Each vertex $v \in V\backslash \left\{s, t\right\}$ can hold at most one token ($s$ and $t$ are unconstrained).

So far, what I've thought to is :

• If $N_G(s)=0$ or $N_G(t)=0$, it is not possible.
• If $N_G(s)=1$ or $N_G(t)=1$, you can just apply a BFS from $s$ to find the shortest path, then transfer every token one by one trough this path.
• If $N_G(s) \geq 2$ and $N_G(t) \geq 2$ you can apply a maximum flow algorithm on $G$ with $1$ as capacity for every edge. If the maximum flow is $1$ then you can apply a BFS and transfer every token through this path again. However if the maximum flow is $\geq 2$, then I don't really see what to do (having a maximum flow $\geq 2$ doesn't even guarantee there are multiple paths as nodes can hold at most $1$ token, while max flow tests only for edges, and even if there are multiple paths in some cases it can be better to use only the shortest path - e. g. for $2$ tokens if you have to paths but one is at least to edges longer -).

Does this problem have a name? What topic of graph theory does it belongs to? Are there efficient algorithms to solve it?

Answer 1

Menger's theorem states that the maximum number of $s$-$t$ (internally) vertex-disjoint paths is equal to the minimum size of an $s$-$t$ vertex cut. Let the common value be $m$. Then it can be shown that the asymptotic number of rounds required is $n/m + O(1)$.

Using the vertex disjoint paths, we can route $n$ tokens in $n/m + O(1)$ rounds, where the constant depends on the graph $G$ (but not on $n$). This gives an upper bound.

On the other hand, every token must pass through the vertex cut, and at most $m$ can do so at any given round, so routing $n$ tokens takes at least $n/m$ rounds. (For each token, there must be a first time at which it reaches a vertex in the cut. Each such time can repeat at most $m$ times, since at most $m$ vertices of the cut can be occupied at any given time. So the number of rounds is at least $n/m$.) This gives a matching lower bound.

From these two bounds, we deduce that the asymptotic number of rounds required is $n/m + O(1)$.