Tightness of the 59/33 analysis for the LP-guided DFS pipeline

Determine whether the upper bound of 59/33 on the LP-relative approximation ratio of the three-step LP-guided DFS pipeline for Matching Augmentation is tight, either by proving that the analysis is tight or by obtaining a sharper analysis or a different LP-relative algorithm.

Background

The paper proves that the Bamas–Drygala–Svensson three-step algorithm—computing an optimal extreme point of the standard Matching Augmentation cut LP, performing LP-prioritized depth-first search, and optimally augmenting the resulting DFS tree with residual uplinks—returns a solution of cost at most $59/33$ times the LP optimum, up to an additive constant. The paper also recalls a $4/3$ integrality-gap example for the cut relaxation, so the possible worst-case LP-relative ratio of this pipeline lies between $4/3$ and $59/33$.

The authors explicitly leave unresolved whether their analysis achieves the true worst-case ratio. They identify two possible directions: sharpen the analysis of the same algorithm or design a different algorithm whose guarantee against the cut LP closes the gap.