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：我们研究了在具有不同成本函数的智能体间分配不可分割杂务的问题,使得所有智能体所承担的成本至多为其最大最小份额的常数倍。该问题的最新进展见于黄和卢在2021年EC会议上提出的工作。他们提出了一种名为HFFD的非多项式时间算法,达到了11/9的近似比,以及一种达到5/4近似比的多项式时间算法。本文表明,HFFD可归约为Coffman、Garey和Johnson于1978年为作业调度中完工时间最小化而开发的MultiFit算法。利用这一归约,我们证明HFFD的近似比实际上等于MultiFit的近似比,后者已知为一般情形下13/11,当n≤7时为20/17,当n=3时为15/13。此外,我们针对任意ε>0开发了一种(13/11+ε)-最大最小份额分配算法,其运行时间关于问题规模和1/ε呈多项式级。当n=3时,我们可改进算法,在多项式时间内找到15/13-最大最小份额分配。因此,我们获得了在杂务分配中达到已知最佳最大最小份额近似比的可实践算法。