Stability vs. Cost of Matchings

This paper studies the classic minimum-cost perfect matching problem in complete graphs under certain stability constraints. Given some real α ≥ 1, a matching M is said to be α-stable if there does not exist any edge (x, y) / ∈ M such that α · w(x, y) < min{w(x, x′), w(y, y′)}, where x′ and y′ are the vertices matched to x and y, respectively, under M . Given some real β ≥ 1, the perfect matching M is said to be β-minimum if the total cost of M is at most β times larger than that of the minimumcost perfect matching. We present a simple greedy algorithm that transforms a minimum-cost perfect matching on 2n vertices into an α-stable and β-minimum matching where β = O(nlog(1+1/(2α))). In particular, this means that we can obtain a constant approximation for the minimum-cost perfect matching by choosing α = O(log n). On the negative side, we show that for any α ≥ 1, there exists a metric graph such that no α-stable perfect matching can be β-minimum unless β = Ω(n). Together, our findings establish an asymptotically tight trade-off between the (local) stability and the (global) cost of perfect matchings.