Revisiting Mulmuley: Simple Proof That Maxflow Is Not in the Algebraic Version of NC

We give an alternate and simpler proof of the fact that PRAM without bit operations (shortened to iPRAM for integer PRAM) as considered in paper [Mulmuley, 1999] cannot solve the maxflow problem. To do so we consider the model of PRAM working over real number (rPRAM) which is at least as expressive as the iPRAM models when considering integer inputs. We then show that the rPRAM model is as expressive as the algebraic version of NC : algebraic circuits of fan-in 2 and of polylog depth noted NC^alg. We go on to show limitations of the NC^alg model using basic facts from real analysis : those circuits compute low degree piece wise polynomials. Then, using known results we show that the maxflow function is not a low-degree piece-wise polynomial. Finally we argue that NC^alg is actually a really limited class which limits our hope of extending our results to the boolean version of NC.