Is IBP in QFT a good mathematical problem?

In real world perturbative quantum field theory (QFT), integrations by parts (IBP) is the bottleneck
of most calculations.

In the physics community, IBP is presented in a very physical language with the Laporta algorithm being its most popular tool. However, the Laporta algorithm is brute force and scales very badly with the loop order. Other approaches also exist but seem not very well studied. I try to give a somewhat more mathematical presentation of IBP (without presenting a solution) in the hope to interest the mathematical community for this problem.

In order to also present a somewhat substantial result, I also briefly report on a new $\phi^4$ identity: the five vertex twist.