Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives. Finally, we apply our algorithm to obtain an approximate leximin solution for the problem of \emph{stochastic allocations of indivisible goods}. For this problem, assuming sub-modular objectives functions, the single-objective egalitarian welfare can be approximated, with only a multiplicative error, to an optimal $1-\frac{1}{e}\approx 0.632$ factor w.h.p. We show how to extend the approximation to leximin, over all the objective functions, to a multiplicative factor of $\frac{(e-1)^2}{e^2-e+1} \approx 0.52$ w.h.p or $\frac{1}{3}$ deterministically.

翻译：Leximin是多目标优化中的一种常见方法，常用于公平分配问题。在Leximin优化中，首先力求最大化最小目标值；在此基础上，最大化第二小的目标值；依此类推。通常，即使最小化值最大化的单目标问题也无法精确求解。在这种情况下，我们对于Leximin优化能实现什么目标？最近，Henzinger等人（2022）定义了“近似”Leximin最优性的概念。然而，他们的定义仅考虑了加法近似。在本文中，我们首先定义了近似Leximin最优性的概念，允许同时存在乘法和加法误差。接着，我们展示了如何使用一个求解单目标问题近似的预言机，在多项式时间内计算出这种近似Leximin解。算法的近似因子密切相关：单目标问题的$(\alpha,\epsilon)$-近似（其中$\alpha \in (0,1]$和$\epsilon \geq 0$分别为乘法和加法因子）转化为多目标Leximin问题的$\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-近似，且与目标数量无关。最后，我们将算法应用于求解“不可分物品的随机分配”问题的近似Leximin解。对于此问题，假设次模目标函数，单目标的平等福利可以仅以乘法误差近似到最优的$1-\frac{1}{e}\approx 0.632$因子（高概率）。我们展示了如何将该近似扩展到Leximin，使得在所有目标函数上，高概率下乘法因子达到$\frac{(e-1)^2}{e^2-e+1} \approx 0.52$，或确定性达到$\frac{1}{3}$。