algorithmstable-marriage

What is the largest number of solutions to an instance of Stable Marriage?


Every Stable Marriage problem has at least one solution. However, how to know the largest number of solutions to a stable marriage problem? The Gale-Shapley algorithm can only find one man-optimal solution. If we use woman-optimal method to run Gale-Shapley, there may be another solution. Are there at most two solutions to an instance of stable marriage, or can Gale-Shapley only find two solutions, and there are some other solutions beyond them?


Solution

  • See https://cstheory.stackexchange.com/questions/5619/what-is-the-maximum-number-of-stable-marriages-for-an-instance-of-the-stable-mar where it is discussed that an exponential worst-case lower bound of Omega(2.28^n) stable marriages is given for all n for n men/women (by giving a family of examples where this many stable marriages are possible).