Revisiting a Cutting Plane Method for Perfect Matchings

TL;DR

This paper improves a polynomial-time algorithm for minimum-cost perfect matchings by removing the need for perturbations, enabling practical application on larger graphs while maintaining strong polynomial time complexity.

Contribution

The authors develop a new method that solves the perfect matching problem without perturbations, overcoming numerical limitations of previous approaches.

Findings

The new algorithm is strongly polynomial and does not require cost perturbations.

Counterexamples show perturbations are necessary for previous methods.

The method is practical for larger graphs due to numerical stability.

Abstract

In 2016, Chandrasekaran, V\'egh, and Vempala published a method to solve the minimum-cost perfect matching problem on an arbitrary graph by solving a strictly polynomial number of linear programs. However, their method requires a strong uniqueness condition, which they imposed by using perturbations of the form $c(i)=c_0(i)+2^{-i}$. On large graphs (roughly $m>100$), these perturbations lead to cost values that exceed the precision of floating-point formats used by typical linear programming solvers for numerical calculations. We demonstrate, by a sequence of counterexamples, that perturbations are required for the algorithm to work, motivating our formulation of a general method that arrives at the same solution to the problem as Chandrasekaran et al. but overcomes the limitations described above by solving multiple linear programs without using perturbations. We then give an explicit algorithm that exploits are method, and show that this new algorithm still runs in strongly polynomial time.