Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [1], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by noting that the ellipsoid method for solving linear programs of full dimension can be implemented in FPC. This allows us to prove that linear programs of full dimension can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.