We consider three problems concerning alpha conversion of closed terms (combinators).
(1) Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.
(2) Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.
(3) Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.
We obtain the following results.
(1) There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.
(2) Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).
(3) At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.
There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.