A heuristic algorithm for rainbow matchings and its application in rainbow Ramsey number for matchings

It is well known that a maximum matching in a given graph can be found in polynomial time. The maximum rainbow matching problem is to find a rainbow matching of maximum size in an edge-colored graph. This problem is equivalent to the multiple choice matching problem which is $NP$-Complete. Moreover, it is surprising that the rainbow matching problem is even $APX$-Complete for paths. So far, there is few efficient algorithm for rainbow matchings. The only positive result is to reduce it to the maximum independent sets in $K_{1,4}$-free graphs, which can be approximated by a polynomial algorithm with approximation ratio $\frac{2}{3}-ϵ$ for every $ϵ>0$. In this paper, we give a heuristic polynomial algorithm to find a large rainbow matching in an edge-colored $K_n$. For any given integer $k$, we can find either a rainbow $k{K}_2$, or a $K_{n-3i}$ with at most $k-i-1$ colors for some $0\le i\le k-2$. It is interesting that our result is useful for the existence of a monochromatic $G$ against a rainbow matching in $K_n$. We give applications of the algorithm and, based on it, we generalize the previous results about the rainbow Ramsey number for matchings.