We consider allocating indivisible chores among agents with different cost functions, such that all agents receive a cost of at most a constant factor times their maximin share. The state-of-the-art was presented in In EC 2021 by Huang and Lu. They presented a non-polynomial-time algorithm, called HFFD, that attains an 11/9 approximation, and a polynomial-time algorithm that attains a 5/4 approximation. In this paper, we show that HFFD can be reduced to an algorithm called MultiFit, developed by Coffman, Garey and Johnson in 1978 for makespan minimization in job scheduling. Using this reduction, we prove that the approximation ratio of HFFD is in fact equal to that of MultiFit, which is known to be 13/11 in general, 20/17 for n at most 7, and 15/13 for n=3. Moreover, we develop an algorithm for (13/11+epsilon)-maximin-share allocation for any epsilon>0, with run-time polynomial in the problem size and 1/epsilon. For n=3, we can improve the algorithm to find a 15/13-maximin-share allocation with run-time polynomial in the problem size. Thus, we have practical algorithms that attain the best known approximation to maximin-share chore allocation.

翻译：我们考虑在具有不同成本函数的智能体之间分配不可分割的杂务，使得所有智能体获得的成本最多为其最大最小份额的常数倍。目前最先进的方法由Huang和Lu在EC 2021中提出。他们提出了一种名为HFFD的非多项式时间算法，能达到11/9近似比，以及一种多项式时间算法，能达到5/4近似比。在本文中，我们证明HFFD可以归约为Coffman、Garey和Johnson在1978年提出的用于作业调度中最小化最大完工时间的MultiFit算法。利用这种归约，我们证明HFFD的近似比实际上等于MultiFit的近似比——已知一般情况下为13/11，当n≤7时为20/17，当n=3时为15/13。此外，我们开发了一种算法，对于任意ε>0可实现(13/11+ε)-最大最小份额分配，其运行时间在问题规模和1/ε上是多项式阶的。对于n=3的情况，我们可以改进该算法，以问题规模的多项式时间找到15/13-最大最小份额分配。因此，我们拥有了达到最大最小份额杂务分配已知最佳近似比的实用算法。