  Motivated by adjacency in perfect matching polytopes, we study the shortest
reconfiguration problem of perfect matchings via alternating cycles. Namely, we
want to find a shortest sequence of perfect matchings which transforms one
given perfect matching to another given perfect matching such that the
symmetric difference of each pair of consecutive perfect matchings is a single
cycle. The problem is equivalent to the combinatorial shortest path problem in
perfect matching polytopes. We prove that the problem is NP-hard even when a
given graph is planar or bipartite, but it can be solved in polynomial time
when the graph is outerplanar.
