This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, which is a special case of the Santa Claus problem. Obtaining a poly-logarithmic approximation for MMDA in polynomial time is of high interest as it is the main obstacle towards the same guarantee for the general Santa Claus problem, which is itself a necessary condition to eventually improve the long-standing 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only ways we have to solve the MMDA problem within an $O(\text{polylog}(n))$ factor is via a ``round-and-condition'' algorithm using the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, or via a ``recursive greedy'' algorithm which also has quasi-polynomial time. However, very little is known about the limitations of these techniques, and it is even plausible that the round-and-condition algorithm could obtain the same approximation guarantee with only $1$ round of Sherali-Adams, which would imply a polynomial-time algorithm. As a main result, we construct an MMDA instance of depth $3$ for which an integrality gap of $n^{\Omega(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is best possible since it is known that after only $2$ rounds the gap is at most poly-logarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in \Omega(1)\cap o(\log n/\log \log n)$, for which we conjecture that an integrality gap of $n^{\Omega(1/\ell)}$ survives $\Omega(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.

翻译：本文致力于研究深度为 $\ell\le O(\log n/\log \log n)$ 的分层有向图中的最大最小度树形图（MMDA）问题，该问题是圣诞老人问题的一个特例。在多项式时间内获得MMDA问题的多对数近似具有高度重要性，因为它是为一般圣诞老人问题实现相同保证的主要障碍，而该保证本身最终改进Lenstra、Shmoys和Tardos [FOCS'87]提出的无关机环境下完工时间调度长期存在的2-近似比的必要条件。目前我们能在 $O(\text{polylog}(n))$ 因子内解决MMDA问题的唯一方法，是通过使用Sherali-Adams层次结构第 $(\ell-1)$ 级的“舍入-条件”算法，或通过同样具有拟多项式时间的“递归贪心”算法。然而，人们对这些技术的局限性知之甚少，甚至舍入-条件算法可能仅用Sherali-Adams的 $1$ 轮就能获得相同的近似保证，这将意味着一个多项式时间算法的存在。作为主要结果，我们构造了一个深度为 $3$ 的MMDA实例，其 $n^{\Omega(1)}$ 的整数性间隙在Sherali-Adams层次结构的 $1$ 轮后仍然存在。该结果是最优的，因为已知在仅 $2$ 轮后，深度-3图上的间隙至多是多对数的。其次，我们展示了可以通过一个简单的技巧将我们的实例“提升”到任意深度 $\ell\in \Omega(1)\cap o(\log n/\log \log n)$ 的MMDA实例，对此我们猜想 $n^{\Omega(1/\ell)}$ 的整数性间隙能在 $\Omega(\ell)$ 轮Sherali-Adams后存活。我们展示了支持这一猜想的若干中间结果，这些结果也表明我们的构造对迄今为止用于圣诞老人问题的技术构成了重大挑战。