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.

翻译：我们考虑在可加估值下不可分割杂务的公平分配问题。假设杂务分为两种类型，在此场景下我们给出若干结果。第一个结果是对帕累托最优分配的新刻画，以及一种多项式时间算法，用于计算满足至多一项物品无嫉妒（EF1）且帕累托最优的分配。随后我们探讨是否能够实现更强的公平概念——任意物品无嫉妒（EFX）。我们给出一种返回EFX分配的多项式时间算法。最后，我们证明在此场景下，可以在多项式时间内判断是否存在无嫉妒分配。