Enhanced instance space analysis for the maximum flow problem

The Maximum Flow Problem (MFP) is a fundamental network flow theory problem, for which many algorithms, supported by strong theoretical worst-case analyses, have been proposed. However, their practical efficiency depends on the network structure, making it unclear which algorithm is best for a particular instance or a class of MFP. Instance Space Analysis (ISA) is a methodology that provides insights into such per-instance analysis. In this paper, the instance space of MFP is constructed and analysed for the first time. Novel features from the networks are extracted, capturing the performance of MFP algorithms. Additionally, this paper expands the ISA methodology by addressing the issue of how benchmark instances should be selected to reduce bias in the analysis. Using the enhanced ISA methodology with MFP as the case study, this paper demonstrates that the most important features can be detected, and machine learning methods can identify their impact on algorithm performance, whilst reducing the bias caused by over-representation within the selected sample of test instances. The enhanced methodology enables new insights into the performance of state-of-the-art general purpose MFP algorithms, as well as recommendations for the construction of comprehensive and unbiased benchmark test suites for MFP algorithm testing.