  Physarum polycephalum inspired algorithm (PPA), also known as the Physarum
Solver, has attracted great attention. By modelling real-world problems into a
graph with network flow and adopting proper equations to calculate the distance
between the nodes in the graph, PPA could be used to solve system optimization
problems or user equilibrium problems. However, some problems such as the
maximum flow (MF) problem, minimum-cost-maximum-flow (MCMF) problem, and
link-capacitated traffic assignment problem (CTAP), require the flow flowing
through links to follow capacity constraints. Motivated by the lack of related
PPA-based research, a novel framework, the capacitated physarum polycephalum
inspired algorithm (CPPA), is proposed to allow capacity constraints toward
link flow in the PPA. To prove the validity of the CPPA, we developed three
applications of the CPPA, i.e., the CPPA for the MF problem (CPPA-MF), the CPPA
for the MCFC problem, and the CPPA for the link-capacitated traffic assignment
problem (CPPA-CTAP). In the experiments, all the applications of the CPPA solve
the problems successfully. Some of them demonstrate efficiency compared to the
baseline algorithms. The experimental results prove the validation of using the
CPPA framework to control link flow in the PPA is valid. The CPPA is also very
robust and easy to implement since it could be successfully applied in three
different scenarios. The proposed method shows that: having the ability to
control the maximum among flow flowing through links in the PPA, the CPPA could
tackle more complex real-world problems in the future.
