Minimum cost flow decomposition on arc-coloured networks

2025-03-07

AI Summary

This paper investigates the minimum color-cost flow decomposition problem in arc-colored networks: given a flow value, decompose it into a set of paths such that the sum of the numbers of distinct arc colors on each path is minimized. We formally define the problem and prove its NP-hardness for general flows. For λ-uniform flows, we identify a complexity phase transition: the problem admits a polynomial-time algorithm for two colors but becomes NP-hard for three or more colors. Moreover, we establish NP-hardness for acyclic networks with as few as five colors, tightening the boundary of computational intractability. Our approach integrates graph-theoretic modeling, computational complexity analysis, and combinatorial optimization algorithm design. The core contributions are (i) establishing the first theoretical framework for color-driven flow decomposition, (ii) precisely characterizing the tractability threshold in terms of color number and flow structure, and (iii) providing the first exact complexity classification alongside an efficient algorithm for the solvable case.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

Abstract

A network $mathcal{N}$ is formed by a (multi)digraph $D$ together with a emph{capacity function} $u : A(D) o R_+$, and it is denoted by $mathcal{N} = (D,u)$. A flow on $mathcal{N}$ is a function $x: A(D) o R_+$ 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 $lambda$-uniform if its value on each arc of the network with positive flow value is exactly $lambda$, for some $lambda in R_+^*$. 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 $lambda$-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.

Problem

• Decompose flow on arc-coloured networks with minimum cost.
• Determine complexity for λ-uniform flows in different network types.
• Apply findings to communication, transportation, and molecular biology.

Innovation

• Decomposes flow on arc-coloured networks
• Minimizes cost based on path colours
• Polynomial time for λ-uniform flows