  We study an elementary path problem which appears in the pricing step of a
column generation scheme solving the kidney exchange problem. The latter aims
at finding exchanges of donations in a pool of patients and donors of kidney
transplantations. Informally, the problem is to determine a set of cycles and
chains of limited length maximizing a medical benefit in a directed graph. The
cycle formulation, a large-scale model of the problem restricted to cycles of
donation, is efficiently solved via branch-and-price. When including chains of
donation however, the pricing subproblem becomes NP-hard. This article proposes
a new complete column generation scheme that takes into account these chains
initiated by altruistic donors. The development of non-exact dynamic approaches
for the pricing problem, the NG-route relaxation and the color coding
heuristic, leads to an efficient column generation process.
