Interpersonal comparison of utility What are the criteria which have been proposed to deal with the problem of interpersonal comparison of utility?
There are not just rules but whole theories. Namely cardinal utility allows you to make interpersonal utility comparisons (as opposed to ordinal utility). A common cardinal utility framework is the one based on  von Neumann-Morgenstern.
Von Neumann-Morgenstern cardinal utility requires that axioms of completeness, transitivity, continuity and independence are satisfied (see von Neumann-Morgenstern (1944) Theory of Games and Economic Behavior), so you can view them as 'conditions' for interpersonal utility comparison (although there are also other approaches to cardinal utility that might have different conditions).
Furthermore, note these rather necessary conditions for interpersonal utility comparisons not sufficient ones, as mentioned in the comments a good book that discusses this in greater detail is Roemer Theories of Distributive Justice.