Formalizing Push-Relabel Algorithms

We present a formalization of push-relabel algorithms for computing the maximum flow in a network. We start with Goldberg's et al.~generic push-relabel algorithm, for which we show correctness and the time complexity bound of O(V^2E). We then derive the relabel-to-front and FIFO implementation. Using stepwise refinement techniques, we derive an efficient verified implementation. Our formal proof of the abstract algorithms closely follows a standard textbook proof. It is accessible even without being an expert in Isabelle/HOL, the interactive theorem prover used for the formalization.