Tight Approximation Ratio For Minimum Maximal Matching

We study a combinatorial problem called Minimum Maximal Matching, where we are asked to find in a general graph the smallest matching that can not be extended. We show that this problem is hard to approximate with a constant smaller than 2, assuming the Unique Games Conjecture.As a corollary we show, that Minimum Maximal Matching in bipartite graphs is hard to approximate with constant smaller than 4/3, with the same assumption. With a stronger variant of the Unique Games Conjecture-that is Small Set Expansion Hypothesis-we are able to improve the hardness result up to the factor of 3/2.