We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives such as makespan and weighted completion time. For many problems, we develop nearly-linear time approximation algorithms with approximation ratios matching the current best ones achieved in polynomial time. Our main technique is linear programming relaxation. For the unrelated machine setting, we formulate mixed packing and covering LP relaxations of nearly-linear size, and solve them approximately using the nearly-linear time solver of Young. For the makespan objective, we develop a rounding algorithm with $(2+\epsilon)$-approximation ratio. For the weighted completion time objective, we prove the LP is as strong as the rectangle LP used by Im and Li, leading to a nearly-linear time $(1.45 + \epsilon)$-approximation for the problem. For problems in the identical machine with precedence constraints setting, the precedence constraints can not be formulated as packing or covering constraints. To achieve the nearly-linear running time, we define a polytope for the constraints, and leverage the multiplicative weight update (MWU) method with an oracle which always returns solutions in the polytope.

翻译：我们研究了非抢占式调度问题在两个场景中的近线性时间近似算法：无关联机器设置，以及具有作业优先约束的相同机器设置，目标为经典的大时间跨度与加权完成时间。针对许多问题，我们开发了具有近线性时间的近似算法，其近似比与当前多项式时间内达成的最佳结果相匹配。我们的主要技术是线性规划松弛。对于无关联机器设置，我们构建了规模近线性的混合打包与覆盖LP松弛，并利用Young的近线性时间求解器对其进行近似求解。针对大时间跨度目标，我们开发了具有$(2+\epsilon)$-近似比的舍入算法。针对加权完成时间目标，我们证明该LP与Im和Li使用的矩形LP具有等价强度，从而为该问题导出了近线性时间的$(1.45+\epsilon)$-近似算法。对于具有优先约束的相同机器设置，优先约束无法被表述为打包或覆盖约束。为实现近线性运行时间，我们为这些约束定义了一个多面体，并借助带oracle的乘性权重更新（MWU）方法，其中oracle始终返回该多面体内的解。