On an algorithm to decide whether a free group is a free factor of   another
  We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change. 
Pedro Silva (CMUP), Pascal Weil (LaBRI)