A $59/33$ Cut-LP Guarantee for Matching Augmentation

Abstract: The Matching Augmentation Problem (MAP) asks for a minimum-cardinality set of unit-cost edges that, together with a zero-cost matching, forms a 2-edge-connected spanning multigraph. We study the standard cut relaxation. Bamas, Drygala, and Svensson proposed a particularly simple LP-guided algorithm: compute an extreme optimum, run a depth-first search that prioritizes large LP coordinates, and augment the resulting DFS tree optimally. We give a new structural analysis of the Bamas--Drygala--Svensson LP-guided DFS algorithm. The analysis combines an exact primal--dual identity for the residual uplink problem with a rank bound that measures fractional support relative to the unit-valued skeleton. The result is that for every root and every deterministic tie-breaking order consistent with the LP priorities, the algorithm returns a solution of cost at most $$\frac{59}{33}c(x<sup><em>)-\frac{25}{33}=\left(2-\frac7{33}\right)c(x^</em>)-\frac{25}{33}\approx1.788c(x<sup>*)-0.758$$, where $x<sup>*$ is an optimum of the cut LP. Consequently, the integrality gap of the relaxation is at most $59/33\approx1.788$. No new algorithmic step is required; the improvement is analytical. The exact packing certificate for the residual uplink problem yields a cost identity with a packing-slack term, while a rank theorem bounds fractional support relative to the unit-valued skeleton. A regional classification accounts for the non-tree edges, and a two-cut identity handles self-holes. As a direct corollary, the same $59/33\approx1.788$ bound holds for Forest Augmentation in the minimum-value regime. The proof is self-contained apart from one theorem on the dimension of minimum-cut vectors.