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, e.g. due to computational hardness. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al (2022) defined a notion of approximate leximin optimality, and showed how it can be computed for the problem of representative cohort selection. However, their definition considers only an additive approximation in the single-objective problem. In this work, we define approximate leximin optimality allowing approximations that have multiplicative and additive errors. We show how to compute a solution that approximates leximin in polynomial time, using an oracle that finds an approximation to the single-objective problem. The approximation factors of the algorithms are closely related: a factor of $(\alpha,\epsilon)$ for the single-objective problem translates into a factor of $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\alpha(2-\alpha)\epsilon}{1-\alpha +\alpha^2}\right)$ for the multi-objective leximin problem, regardless of the number of objectives. As a usage example, we apply our algorithm to the problem of 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是多目标优化中的常见方法，常用于公平分配问题。在Leximin优化中，首先追求最大化最小值目标值；在此基础上，最大化次小目标值；依此类推。然而，即便仅针对最大化最小值的单目标问题，由于计算复杂度等原因，往往也无法精确求解。在这种情况下，我们对Leximin优化还能期待什么？近年来，Henzinger等人（2022）定义了近似Leximin最优性的概念，并展示了如何针对代表性队列选择问题计算该近似解。然而，他们的定义仅考虑了单目标问题中的加性近似。在本工作中，我们定义了允许同时存在乘性误差和加性误差的近似Leximin最优性。我们展示了如何利用一个能够求解单目标问题近似的预言机，在多项式时间内计算近似Leximin的解。算法的近似因子密切相关：单目标问题中的因子$(\alpha,\epsilon)$转化为多目标Leximin问题中的因子$\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\alpha(2-\alpha)\epsilon}{1-\alpha +\alpha^2}\right)$，且该转化与目标函数的数量无关。作为应用示例，我们将算法应用于不可分割商品的随机分配问题。针对该问题，假设目标函数为子模函数，单目标平等福利可以仅以乘性误差接近最优的$1-\frac{1}{e}\approx 0.632$因子（高概率）。我们展示了如何将这种近似扩展到所有目标函数的Leximin，在高概率下达到$\frac{(e-1)^2}{e^2-e+1}\approx 0.52$的乘性因子，或确定性达到$\frac{1}{3}$因子。