Lecture Notes Discrete Optimization - Applied Mathematics

augmenting f along P, the flow on edge (u,v) was decreased. That means that the flow
on the reverse edge(v,u) was increased and thus(v,u) must be part of P.
The next lemma shows that for every node v ∈ V, the shortest path distance from s to v
does not decrease.
Lemma 5.5. For each v∈V, δ ′ (s,v)≥δ(s,v).
Proof. Suppose there exists a node v with δ ′ (s,v) < δ(s,v). Among all such nodes, let
v be one with δ ′ (s,v) being smallest. Note that v ≠ s because δ(s,s) = δ ′ (s,s) = 0 by
definition. Let P ′ be a shortest s,v-path in G f ′ of distance δ ′ (s,v) and let u be the secondlast
node of P ′ . Because P ′ is a shortest path and by the choice of v, we have
δ(s,v)>δ ′ (s,v)=δ ′ (s,u)+1≥δ(s,u)+1. (5)
Note that the distance function δ satisfies the triangle inequality. Therefore, edge (u,v)
cannot be part of G f because otherwise we would have δ(s,v)≤δ(s,u)+1. That is,(u,v)
is contained in G f ′ but not contained in G f . Using our observation above, we conclude
that there is some i (1≤i≤k) such that u=v i and v=v i−1 . But then δ(s,v)=i−1 and
δ(s,u)=i which is a contradiction to (5).
Consider an augmentation of the current flow f along path P. We say that an edge e =
(u,v) is critical with respect to f and P if (u,v) is part of the augmenting path P and its
residual capacity coincides with the amount of flow that is pushed along P, i.e., r f (u,v)=
r f (P). Note that after the augmentation of f along P, every critical edge on P will be
saturated and thus vanishes from the residual network.
Lemma 5.6. The number of times an edge e=(u,v) is critical throughout the execution
of the algorithm is bounded by O(n).
Proof. Suppose edge e=(u,v) is critical with respect to flow f and path P. Let δ refer
to the shortest path distances in G f . We have
δ(s,v)=δ(s,u)+1.
After the augmentation of f along P, edge e is saturated and thus disappears from the
residual graph. It can only reappear in the residual graph when in a successive iteration
some positive flow is pushed over the reverse edge(v,u). Suppose edge(v,u) is part of an
augmenting path P ′ that is used to augment the current flow, say f ′ . Let δ ′ be the distance
function with respect to G f ′. We have
By Lemma 5.5, δ(s,v)≤δ ′ (s,v) and thus
δ ′ (s,u)=δ ′ (s,v)+1.
δ ′ (s,u)=δ ′ (s,v)+1≥δ(s,v)+1=δ(s,u)+2.
That is, between any two augmentations for which edge e=(u,v) is critical, the distance
of u from s must increases by at least 2.
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