After 8 years of struggle I finally managed to make certain ideas work, and as a result, I have an algorithm that approximates the Shortest Superstring problem with a ratio (slightly) smaller than the long standing 2 1/2. The algorithm is surprisingly simple, since it chooses the best solution out of the following: (1) returned by one of the known 2 1/2 approximation algorithms (2) returned by a particularly naive cycle-breaking greedy algorithm. More details here.