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

TL;DR

This paper proves that the maximum matching problem and related linear programming problems are definable within fixed-point logic with counting (FPC), resolving an open question about the expressive power of FPC for polynomial-time problems.

Contribution

It demonstrates that maximum matchings and linear programming can be expressed in FPC by implementing the ellipsoid method within this logic, including defining maximum flows and minimum cuts.

Findings

Maximum matching is definable in FPC.

Linear programming optimization can be performed in FPC.

FPC formulas for maximum flows and minimum cuts are provided.

Abstract

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, 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 showing that the ellipsoid method for solving linear programs can be implemented in FPC. This allows us to prove that linear programs 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.