  Kidney exchange is a barter market where patients trade willing but medically
incompatible donors. These trades occur via cycles, where each patient-donor
pair both gives and receives a kidney, and via chains, which begin with an
altruistic donor who does not require a kidney in return. For logistical
reasons, the maximum length of a cycle is typically limited to a small
constant, while chains can be much longer. Given a compatibility graph of
patient-donor pairs, altruists, and feasible potential transplants between
them, finding even a maximum-cardinality set of vertex-disjoint cycles and
chains is NP-hard. There has been much work on developing provably optimal
solvers that are efficient in practice. One of the leading techniques has been
branch and price, where column generation is used to incrementally bring cycles
and chains into the optimization model on an as-needed basis. In particular,
only positive-price columns need to be brought into the model. We prove that
finding a positive-price chain is NP-complete. This shows incorrectness of two
leading branch-and-price solvers that suggested polynomial-time chain pricing
algorithms.
