We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^{\epsilon})$-approximation algorithm with polynomial running time for any constant $\epsilon > 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.

翻译：我们考虑将不可分割的资源分配给参与者，以最大化任何参与者所获得的最小总价值的问题。该问题有时被称为圣诞老人问题，其不同变体在过去二十年中一直是近似算法广泛研究的对象。在每位参与者可能具有不同加性估值函数的情况下，Chakrabarty、Chuzhoy 和 Khanna [FOCS'09] 针对任意常数 $\epsilon > 0$ 给出了一种具有多项式运行时间的 $O(n^{\epsilon})$-近似算法，以及在拟多项式时间内的一种多对数近似算法。我们证明，对于单调子模估值函数，同样可以实现上述结果，这改进了先前由 Goemans、Harvey、Iwata 和 Mirrokni [SODA'09] 提出的最佳算法，该算法的近似比超过 $\sqrt{n}$。我们的结果建立在一个精密的线性规划松弛之上，该松弛具有递归的块结构，使我们能够求解它，尽管其变量和约束数量呈指数级增长。