Successive shortest paths with reduced costs

Abstract view

Algorithmic problem: Min-cost flow problem

Type of algorithm: loop

Auxiliary data: For each node [math]\displaystyle{ v\in V }[/math], there is a real number [math]\displaystyle{ \pi(v) }[/math].

Invariant:

1. All points of the invariant of the successive shortest paths algorithm.
2. For each arc [math]\displaystyle{ a=(v,w)\in A_f }[/math], the reduced cost [math]\displaystyle{ c^\pi(a):=c(a)-\pi(v)+\pi(w) }[/math] is nonnegative.

Variant: The total imbalance strictly decreases.

Definition: Such a node labeling [math]\displaystyle{ \pi }[/math] is called consistent with [math]\displaystyle{ f }[/math].

Induction basis

Abstract view: Start with the zero flow [math]\displaystyle{ f }[/math] and with the zero node labeling [math]\displaystyle{ \pi }[/math].

Proof: The assumption that all cost values are nonnegative implies that [math]\displaystyle{ \pi\equiv 0 }[/math] is consistent with [math]\displaystyle{ f\equiv 0 }[/math].

Induction step

Abstract view: The induction step is a variation and extension of the induction step of the successive shortest paths algorithm. The essential modification is that [math]\displaystyle{ c }[/math] is replaced by [math]\displaystyle{ c^\pi }[/math]. Due to point 2 of the invariant, it is always [math]\displaystyle{ c^\pi\geq 0 }[/math]. Therefore, an efficient algorithm such as Dijkstra's may be applied. The extension is that, after each augmentation of the flow, the reduced cost values must be updated to maintain consistency.

Implementation of the update of the reduced cost values: For all nodes [math]\displaystyle{ v\in V }[/math], set [math]\displaystyle{ \pi(v):=\pi(v)-\delta(v) }[/math], where [math]\displaystyle{ \delta(v) }[/math] is the distance of [math]\displaystyle{ v }[/math] from [math]\displaystyle{ s }[/math] as computed in the induction step of the successive shortest paths algorithm.

Proof: The variant and points 1 and 3 of the invariant of the successive shortest paths algorithm are obviously maintained. For point 2 of that invariant, it suffices to show that the shortest paths with respect to the arc lengths [math]\displaystyle{ c^\pi }[/math] are also shortest paths with respect to the arc lengths [math]\displaystyle{ c }[/math]; however, that results immediately from this statement.

It remains to show that the updated node potentials are consistent with the augmented flow. Let [math]\displaystyle{ \pi_\mathrm{before} }[/math] and [math]\displaystyle{ \pi_\mathrm{after} }[/math] denote the values of [math]\displaystyle{ \pi }[/math] immediately before and after the current iteration, respectively. Let [math]\displaystyle{ (v,w) }[/math] be an arc in the residual network of the augmented flow, that is, the flow immediately after the current iteration. We make a case distinction:

1. If [math]\displaystyle{ (v,w) }[/math] was also in the residual network immediately before the current iteration, the new reduced cost of [math]\displaystyle{ (v,w) }[/math] is [math]\displaystyle{ c^{\pi_\mathrm{after}}(v,w) }[/math] [math]\displaystyle{ =c(v,w)-\pi_\mathrm{after}(v)+\pi_\mathrm{after}(w) }[/math] [math]\displaystyle{ =c(v,w)-\left[\pi_\mathrm{before}(v)-\delta(v)\right]+\left[\pi_\mathrm{before}(w)-\delta(w)\right] }[/math] [math]\displaystyle{ =c^{\pi_\mathrm{before}}(v,w)-\delta(w)+\delta(v) }[/math]. As the [math]\displaystyle{ \delta }[/math]-values are shortest-path distances with respect to arc lengths [math]\displaystyle{ c^{\pi_\mathrm{before}} }[/math], the valid distance property yields [math]\displaystyle{ \delta(w)\leq\delta(v)+c^{\pi_\mathrm{before}}(v,w) }[/math], which proves the claim.
2. Otherwise, [math]\displaystyle{ (w,v) }[/math] is on the shortest path computed in the current iteration. Recall [math]\displaystyle{ c(v,w)=-c(w,v) }[/math], so we obtain [math]\displaystyle{ c^{\pi_\mathrm{after}}(v,w) }[/math] [math]\displaystyle{ =c(v,w)-\pi_\mathrm{after}(v)+\pi_\mathrm{after}(w) }[/math] [math]\displaystyle{ =-\left[c(w,v)-\pi_\mathrm{after}(w)+\pi_\mathrm{after}(v)\right] }[/math] [math]\displaystyle{ =-\left[c(w,v)-(\pi_\mathrm{before}(w)-\delta(w))+(\pi_\mathrm{before}(v)-\delta(v))\right] }[/math] [math]\displaystyle{ =-\left[c^{\pi_\mathrm{before}}(w,v)-\delta(v)+\delta(w)\right] }[/math]. This statement proves that the last expression is zero.

Correctness

Cf. here.

Complexity

The asymptotic complexity and its proof are identical to the complexity considerations of the successive shortest paths algorithm. Note that [math]\displaystyle{ T }[/math] may now be the asymptotic complexity of an algorithm that cannot handle negative arc weights.