We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a $2$-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of $13/8 = 1.625$ improving upon an earlier $5/3$-approximation. The improvement builds on a new $\alpha$-approximation preserving reduction for any $\alpha\geq 3/2$ from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted.

翻译：我们考虑了匹配的扩增问题(MAP),在这个问题上,需要通过增加最小边缘数,将图表的匹配扩展为2美元与边缘数相连接的直线覆盖的子集层。我们提出了一个多元时间算法,其近似比率为13/8=1.625美元,比先前的5/3美元接近率提高1.625美元。这一改进建立在一个新的 $\ alpha$-oporum 的基础上,从任意的MAP 3/2美元到结构完善的不包含平行边缘、小分隔器和可承包子集成等某些被禁止的结构。我们进一步引入了重复同时收缩的技术,并为无法订约的情况提供了更好的下限。