A 2/3-Approximation Algorithm for the Residual Maximum Matching Problem in General Graphs

The residual maximum matching problem asks for a maximum matching whose deletion leaves a graph with the largest possible matching number. For unweighted graphs, previous work provides a $1/2$-approximation guarantee in general graphs and a $2/3$ guarantee in graphs with a perfect matching. We present a $2/3$-approximation algorithm running in $O(n^5)$ time, extending the latter guarantee to general graphs. The algorithm improves the residual matching number through single-edge exchanges involving free vertices. We prove that, for any maximum matching that admits no further improvement by such exchanges, the residual matching number is at least $2/3$ of the optimum, and we give an example showing that this bound is tight.