Minimum Flow Decomposition in Graphs with Cycles using Integer Linear Programming

TL;DR

This paper introduces the first exact ILP-based method for solving the minimum flow decomposition problem in graphs with cycles, applicable to bioinformatics and transportation, efficiently handling complex instances.

Contribution

It presents novel ILP formulations for three variants of MFD in cyclic graphs, enabling exact solutions where only heuristics existed before.

Findings

Solves instances in under 10 minutes

Applicable to bioinformatics and transportation datasets

First exact solutions for cyclic graphs in MFD

Abstract

Minimum flow decomposition (MFD) -- the problem of finding a minimum set of weighted source-to-sink paths that perfectly decomposes a flow -- is a classical problem in Computer Science, and variants of it are powerful models in different fields such as Bioinformatics and Transportation. Even on acyclic graphs, the problem is NP-hard, and most practical solutions have been via heuristics or approximations. While there is an extensive body of research on acyclic graphs, currently, there is no \emph{exact} solution on graphs with cycles. In this paper, we present the first ILP formulation for three natural variants of the MFD problem in graphs with cycles, asking for a decomposition consisting only of weighted source-to-sink paths or cycles, trails, and walks, respectively. On three datasets of increasing levels of complexity from both Bioinformatics and Transportation, our approaches solve any instance in under 10 minutes. Our implementations are freely available at github.com/algbio/MFD-ILP.