从家务分配到作业调度的归约
A Reduction from Chores Allocation to Job Scheduling
AI总结：
本文通过将家务分配算法 HFFD 归约到作业调度算法 MultiFit，证明其近似比等于 MultiFit 的已知近似比，并提出多项式时间算法达到 13/11+epsilon 及 n=3 时 15/13 的最佳近似。
摘要：
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家务，使得所有智能体承担的成本至多为其最大最小份额（maximin share）的常数倍。现有最佳结果由 Huang 和 Lu 在 EC 2021 上提出。他们给出了一个非多项式时间算法 HFFD，达到 11/9 的近似比，以及一个多项式时间算法，达到 5/4 的近似比。本文中，我们证明 HFFD 可以归约到 Coffman、Garey 和 Johnson 于 1978 年为作业调度中的完工时间最小化而提出的 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，后者在一般情况下为 13/11，当 n 至多为 7 时为 20/17，当 n=3 时为 15/13。此外，对于任意 epsilon>0，我们提出了一个 (13/11+epsilon)-最大最小份额分配算法，其运行时间关于问题规模和 1/epsilon 是多项式的。对于 n=3，我们可以改进该算法，以关于问题规模多项式的运行时间找到 15/13-最大最小份额分配。因此，我们得到了达到最大最小份额家务分配已知最佳近似的实用算法。
英文摘要：
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.
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