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. The objectives we study include makespan, weighted completion time, and $L_q$ norm of machine loads. We develop nearly-linear time approximation algorithms for the studied problems with $O(1)$-approximation ratios, many of which match the correspondent best known ratios achievable in polynomial time. Our main technique is linear programming relaxation. For problems in 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. We show the LP solutions can be rounded within $O(1)$-factor loss. 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 claimed 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. Along the way of designing the oracle, we encounter the single-commodity maximum flow problem over a directed acyclic graph $G = (V, E)$, where sources and sinks have limited supplies and demands, but edges have infinite capacities. We develop a $\frac{1}{1+\epsilon}$-approximation for the problem in time $O\left(\frac{|E|}{\epsilon}\log |V|\right)$, which may be of independent interest.
翻译:我们研究两种情况下的非先发制人时间安排问题的近线性时间近近近算算法:不相关的机器设置,和具有工作优先限制设置的相同机器。我们研究的目标包括 massan、加权完成时间和机器负载的$L_q美元标准。我们开发了用于处理研究问题的近线性时间近近算算法,使用美元(1)美元-准协调比率,其中许多比率与在多元时间里可以达到的最佳已知比率相符。我们的主要技术是线性编程放松。对于不相关的机器设置的问题,我们设计了混合包装,覆盖了近线性的LP宽度,并覆盖了近线性的LP宽度,并且大约用近线性Young的时间解答器解决问题。我们展示了LP的解决方案可以在O(1)美元-系数损失范围内四舍四舍五入四入。对于同一机器的问题,不能将偏差限制作为包装或覆盖制约。为了达到所声称的运行时间,我们定义了一个限制的聚度的聚度,并且利用多复制性重量更新的(MWU)方法,它总是在多线性时间大小内返回解决办法,但是在多线性时值1美元1美元1美元1美元的汇率中,我们会遇到一个方向的轨道上的流动源。