We consider the problem of fair allocation of indivisible chores under
additive valuations. We assume that the chores are divided into two types and
under this scenario, we present several results. Our first result is a new
characterization of Pareto optimal allocations in our setting, and a
polynomial-time algorithm to compute an envy-free up to one item (EF1) and
Pareto optimal allocation. We then turn to the question of whether we can
achieve a stronger fairness concept called envy-free up any item (EFX). We
present a polynomial-time algorithm that returns an EFX allocation. Finally, we
show that for our setting, it can be checked in polynomial time whether an
envy-free allocation exists or not