# Covering max flow arcs by arc disjoint paths

Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it always possible to find a collection of arc disjoint, flow-carrying $s$-$t$-paths that cover all arcs $a$ with flow $x_a=T$?

This question came up in work on the scheduling of arc outages to maximize the total flow over a time horizon.

Update: To avoid Brendan's counterexample let's assume that every node can be reached from the source $s$, and from every node the sink $t$ can be reached. Fixing a flow $x$ we can reduce the capacity of every arc to its flow value (and delete zero capacity arcs) without really changing the problem. Now the question is as follows:

Given a network with the property that for every node $v\in N\setminus\{s,t\}$ the sum of the capacities of the incoming arcs equals the sum of the capacities of the outgoing arcs, can the arcs of maximum capacity be covered by a collection of arc-disjoint $s$-$t$-paths?

In this formulation it seems possible that someone has looked at this problem before.

• It isn't true for any maximal flow. For example, to your example add an unreachable cycle of capacity $T$ and put a flow of magnitude $T$ around it. However, it might be true of acyclic maximal flows. If not, it still might be true of flows that can be constructed by augmenting paths. May 12, 2012 at 16:38
• Thanks Brendan. From the application point it would be actually quite natural to add the assumption that the network is acyclic. May 12, 2012 at 22:55

It looks like the following argument works. Consider a binary program to maximize $\sum_{a\in A^*}\xi_a$ subject to the constraints
where $A^*=\{a\in A\ :\ x_a=T\}$ is the set of arcs that have to be covered, and $\delta^+(v)$ and $\delta^-(v)$ are the sets of outgoing and incoming arcs of node $v$, respectively. The desired covering exists if and only if the optimal objective value for this problem is $\lvert A^*\rvert$. The constraint matrix is totally unimodular, so we don't lose anything by relaxing the integrality constraint to $0\leqslant \xi_a\leqslant 1$. It is sufficient to show that $\lvert A^*\rvert$ is a lower bound for the dual problem which is to minimize $\sum_{a\in A}\eta_A$ subject to \begin{align} \pi_{v}-\pi_w+\eta_a &\geqslant 0 && a=(v,w)\in A\setminus A^*, &&(1)\\\\ \pi_{v}-\pi_w+\eta_a &\geqslant 1 && a=(v,w)\in A^*,&&(2)\\\\ \pi_s=\pi_t&=0,\\\\ \eta_a &\geqslant 0 && a\in A. \end{align}
The given flow $x=(x_a)_{a\in A}$ can be decomposed into $s$-$t$-paths such that arc $a$ is on exactly $x_a$ paths. Let $\mathcal P$ be the set of paths in a fixed decomposition. Now adding constraints (1) and (2) along any path $P\in\mathcal P$ gives $\sum_{a\in P}\eta_a\geqslant\lvert P\cap A^*\rvert$. Summing over all paths $P\in\mathcal P$ we obtain $\sum_{a\in A}x_a\eta_a\geqslant T\lvert A^*\rvert$, and finally, $$\sum_{a\in A}\eta_a\geqslant\sum_{a\in A}\frac{x_a}{T}\eta_a\geqslant\lvert A^*\rvert.$$