Decomposition of flow into paths to minimize their length

This paper deals with the decompositions of a flow into paths. Given a weighted directed acyclic graph and a compatible flow, we look for a decomposition of this flow such that the longest path stemming from this decomposition is the shortest possible. The complexity of this problem and some of its particular cases are first addressed, then a pseudo-polynomial algorithm and a fully polynomial time approximation scheme are proposed for the case where the value of the flow is bounded by a constant. 1 Definition and some notations Let G = ({s, t}, N, A) be a capacitated directed acyclic graph. We are given a cost (or length) function λ : A → Z≥0. Let s, t ∈ N be respectively the source and the sink of G. We are also given a compatible s-t flow, f : A → Z>0 and let F be the value of this flow. We number the nodes using a topological sort and, except stated otherwise, we denote by i → j an arc from node i to j , fi→j its flow and λi→j its cost. If there are more than one arc from i to j, we add an indexation: (i → j)k represents the arc from i to j with the knth largest cost. We denote by decomposition of the flow f into paths a set of F paths (in principle, not all different) such that by pushing a unit of flow on each of these paths, we obtain the flow f on G. Let Df be the set of all the decompositions of the flow f into paths and let Df ∈ Df . We denote by μ Df P the value of the flow along the s-t path P in the decomposition Df . If there is no ambiguity concerning the decomposition, we shorten it to μP . Then, a decomposition Df comes down to a set of pairs (P, μP ) such that ∑ μP = F . Let |Df | be the number of different paths (i.e. paths that differ for at least one arc) in the decomposition Df . For each path P , we denote by λ(P ) the cost of the path, that is the sum of costs of its arcs ( ∑ i→j∈P λi→j). Let Λ = maxP∈Df λ(P ). In this paper, we search for a decomposition Df such that Λ is minimum. We now denote this problem by SLP (for shortest longest path) and when there is no ambiguity concerning the flow function, SLP. On figure 1, the first number and the second number on each arc represent respectively its cost and its flow. On this example, F = 8. The following is a possible decomposition: