Minimum cost flow decomposition on arc-coloured networks

TL;DR

The paper defines MinCostCFD, the problem of decomposing an $(s,t)$-flow on an arc-coloured network into path flows to minimize the sum of path-colour costs. It maps the complexity landscape, proving NP-hardness in general while identifying precise polynomial-time solvable cases for $\lambda$-uniform flows with a small number of colours, and detailing NP-hardness in monochromatic, bichromatic, and multicolour settings. Key contributions include reductions from 3-Partition and 1-in-3SAT to establish NP-hardness (via KCostCFD) and a hierarchy of tractable versus intractable instances across colour counts and acyclicity. The findings have practical implications for routing and reliability in communications and multimodal transport, guiding when exact decompositions are feasible and highlighting directions for approximation and alternative cost formulations.

Abstract

A network $\mathcal{N}$ is formed by a (multi)digraph $D$ together with a \emph{capacity function} $u : A(D) \to R_+$, and it is denoted by $\mathcal{N} = (D,u)$. A flow on $\mathcal{N}$ is a function $x: A(D) \to R_+$ such that $x(a) \leq u(a)$ for all $a \in A(D)$, and it is said to be $k$-splittable if it can be decomposed into up to $k$ paths. We say that a flow is $λ$-uniform if its value on each arc of the network with positive flow value is exactly $λ$, for some $λ\in R_+^*$. Arc-coloured networks are used to model qualitative differences among different regions through which the flow will be sent. They have applications in several areas such as communication networks, multimodal transportation, molecular biology, packing etc. We consider the problem of decomposing a flow over an arc-coloured network with minimum cost, that is, with minimum sum of the cost of its paths, where the cost of each path is given by its number of colours. We show that this problem is NP-Hard for general flows. When we restrict the problem to $λ$-uniform flows, we show that it can be solved in polynomial time for networks with at most two colours, and it is NP-Hard for general networks with three colours and for acyclic networks with at least five colours.

Figures (10)

• Figure 1: Influence of cycles on a decomposition of a $\lambda$-uniform flow.
• Figure 2: Reduction from 3-Partition to kCostCFD.
• Figure 3: Example of decomposition produced by \ref{['alg:decfluxo2v']}.
• Figure 4: Network with 3 distinct flow values on the arcs.
• Figure 5: Bichromatic network with two distinct flow values in the arcs.

Theorems & Definitions (16)

• Theorem 2.1: Flow Decomposition Theorem FordFulkerson
• Theorem 3.1
• Lemma 3.2: Hartman
• Theorem 3.3: Hartman
• Lemma 3.4
• Theorem 3.5
• Theorem 3.6: Hartman
• Theorem 3.7
• Theorem 3.8
• proof