Average-case solutions to hard problems
The first example deals with the problem of assigning medical graduates to residency programs in the presence of married couples. The problem of finding a stable matching between residency programs and doctors without couples is solved by the famous Gale-Shapley algorithm. In the presence of couples, such a matching may not exist, and it is NP-hard to determine whether it exists or not. Still, in a large random market we show how to find a good stable outcome with high probability.
The second example deals with aggregating noisy comparison signals, such as sport competition results, into a ranking most consistent with the observed signals. The worst-case version of the problem, known as the Minimum Feedback Arcset problem is NP-hard. We give an efficient algorithm for the average-case version of the problem.