New improved bound on matching augmentation problem via lp guided dfs analysis
A $59/33$ Cut-LP Guarantee for Matching Augmentation
Data Structures and Algorithms
Summary
The matching augmentation problem is about adding the fewest possible edges to a graph so that it becomes 2-edge-connected, meaning it stays connected even if one edge fails. The authors studied a known linear programming (LP) based method that uses a depth-first search guided by the LP values to find these edges. They analyzed this method in detail and showed it produces better solutions than previously proven, with a specific cost guarantee of about 1.788 times the LP's optimal value. This is an improved understanding of the problem's approximation quality, and the result applies also to forest augmentation problems in certain cases.
What this means in practice
- •For network reliability engineers: Design more cost-effective edge sets to ensure network resilience against single link failures using tighter LP approximations.
- •For infrastructure planners: Optimize augmentations of spanning trees in transportation or utility networks to improve fault tolerance with improved theoretical cost bounds.
A theory result. No direct application yet.
Authors
Morteza Alimi, Tobias Mömke
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 $\frac{59}{33}c(x^*)-\frac{25}{33}=\left(2-\frac7{33}\right)c(x^*)-\frac{25}{33}\approx1.788c(x^*)-0.758$, where $x^*$ 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.