  The purpose of this paper is two-fold. The first is to give a tutorial
introduction to the Sarkisov program, a 3-dimensional generalization of
Castelnuovo-N\"other Theorem ``untwisting" birational maps between Mori fiber
spaces, which was recently established by Corti after Sarkisov and Reid.
  The second is an attempt to give a logarithmic generalization of the program,
introducing the notion of the Sarkisov relation. We give a simple description
of the structure of the program and clarify the mechanism of termination
connected to the boundedness conjecture of (log) Q-Fano n-folds by Borisov.
