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功能的代理商之间分配不可分割的杂务, 以便所有代理商都能以最高不变系数乘以其最大份额。 在EC 2021中, 黄和卢展示了最新技术。 他们展示了非线性时间算法, 称为HFFD, 达到11/ 9 近似值, 和多线性算法, 达到 5/4 近似值。 在本文中, 我们显示HFFD 可以降低为一种称为多线性算法, 由Coffman、 Garey和Johnson 于1978年开发, 以在工作时间安排中实现最小化。 我们使用这一减法, 我们证明HFFD的近似比率实际上等于多线性算法, 众所周知, 通常为13/11, 20/17 最多为 7, 15/13 和 13 13 13 。 此外, 我们为任何Eplasl- massional- share分配一个算法, 10, 由Coffrial- massion yalal am am asion asion 10, simalgrational simmission siquest probal problemissional lemissmissmissmissmissmissional