SIAM Journal on Computing, Ahead of Print.
In a classical problem in scheduling, one has $n$ unit size jobs with a precedence order and the goal is to find a schedule of those jobs on $m$ identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey and Johnson whether or not this problem is $\mathbf{NP}$-hard for $m=3$. We prove that for any fixed $\varepsilon$ and $m$, an LP-hierarchy lift of the time-indexed LP with a slightly super poly-logarithmic number of $r = (\log(n))^{\Theta(\log \log n)}$ rounds provides a $(1 + \varepsilon)$-approximation. For example, Sherali--Adams suffices as hierarchy. This implies an algorithm that yields a $(1+\varepsilon)$-approximation in time $n^{O(r)}$. The previously best approximation algorithms guarantee a $2 - \frac{7}{3m+1}$-approximation in polynomial time for $m \geq 4$ and $\frac{4}{3}$ for $m=3$. Our algorithm is based on a recursive scheduling approach where in each step we reduce the correlation in form of long chains. Our method adds to the rather short list of examples where hierarchies are actually useful to obtain better approximation algorithms.

中文翻译：

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使用LP层次结构优先约束的Makespan调度的（1 +ε）逼近

《 SIAM计算杂志》，预印本。
在调度中的经典问题中，有一个具有优先顺序的$ n $个单位大小的作业，目标是在$ m $个相同的机器上找到这些作业的调度，以最大程度地减少工期。无论这个问题是否是$ \ mathbf {NP} $-hard for $ m = 3 $，这都是Garey和Johnson著作中剩下的四个悬而未决的问题之一。我们证明对于任何固定的$ \ varepsilon $和$ m $，时间索引LP的LP层次提升都具有$ r =（\ log（n））^ {\ Theta（ \ log \ log n）} $轮次提供了$（1 + \ varepsilon）$近似值。例如，Sherali-Adams作为层次结构就足够了。这暗示了一种算法，该算法在时间$ n ^ {O（r）} $中产生$（1+ \ varepsilon）$近似值。先前最佳的近似算法可保证$ m \ geq 4 $和$ \ frac {4} {3} $的多项式时间中$ 2-\ frac {7} {3m + 1} $的近似值（$ m = 3 $）。我们的算法基于递归调度方法，在每一步中，我们以长链形式减少相关性。我们的方法增加了示例的简短列表，其中层次结构实际上对于获得更好的近似算法很有用。