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 an important special case of the Santa Claus problem. Obtaining a polylogarithmic approximation for MMDA in polynomial time is of high interest as it is a necessary condition to improve upon the well-known 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only way we have to solve the MMDA problem within a polylogarithmic factor is via an elegant recursive rounding of the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, which needs time $n^{O(\ell)}$ to solve. However, it remains plausible that one could obtain a polylogarithmic approximation in polynomial time by using the same rounding with only $1$ round of the Sherali-Adams hierarchy. As a main result, we rule out this possibility by constructing 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 tight since it is known that after only $2$ rounds the gap is at most polylogarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in \Omega(1)\cap o(\log n/\log \log n)$ (the whole range of interest), 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重要特例。在多项式时间内获得MMDA的多对数近似具有高度重要性，因为这是改进Lenstra、Shmoys和Tardos [FOCS'87] 提出的无关机环境下完工时间调度问题的著名2-近似算法的必要条件。目前获得多对数因子近似的唯一途径是通过对Sherali-Adams层次结构第 $(\ell-1)$ 层进行优雅的递归舍入，这需要 $n^{O(\ell)}$ 的时间求解。然而，通过仅使用Sherali-Adams层次结构的 $1$ 轮舍入，仍有可能在多项式时间内获得多对数近似。作为主要结果，我们通过构造一个深度为 $3$ 的MMDA实例排除了这种可能性，该实例在经历 $1$ 轮Sherali-Adams层次结构后仍保持 $n^{\Omega(1)}$ 的积分间隙。该结果是紧的，因为已知在仅进行 $2$ 轮后，深度-3图上的间隙至多为多对数级别。其次，我们证明可以通过简单技巧将我们的实例“提升”至任意深度 $\ell\in \Omega(1)\cap o(\log n/\log \log n)$（整个感兴趣的范围）的MMDA实例，我们推测对于这些实例，在经历 $\Omega(\ell)$ 轮Sherali-Adams后仍会保持 $n^{\Omega(1/\ell)}$ 的积分间隙。我们展示了若干支持该猜想的中间结果，这些结果也表明我们的构造对目前用于圣诞老人问题的技术构成了重大挑战。