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.
Highlights
Given a graph G = (V, E) with edge cost function c, the minimum-cost perfect matching problem is to find a perfect matching E ⊆ E so that the sum of the costs of E is minimized
As mentioned in [4], the minimum-cost perfect matching problem is a classical problem in combinatorial optimization with numerous and varied applications
To overcome the potential numerical difficulties caused by perturbation, we present a variant of the algorithm which does not require an explicit perturbation to ensure uniqueness
Summary
2016, when Chandrasekaran et al [2] gave a cutting-plane algorithm which uses only a polynomial number of linear programs Their approach involves carefully selecting the blossom inequalities to be included and dropping ones that are not helpful at each iteration and requires that the optimal solution to the linear program be unique. As this uniqueness property does not always hold in general, their method introduces an edge ordering and a perturbation on the edge costs. We first explain this in a general case and point out its connection with lexicographic linear goal programming (Section 6) and apply it to the specific problem of finding perfect matchings (Section 7)
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