排序问题的紧下界
Towards Tight Lower Bounds for Scheduling Problems
摘要：
我们展示了$k$ -部图的结构硬度与优先约束调度问题的紧不可逼近结果之间的密切联系。假设一个自然的，但不平凡的推广的二分结构的硬度结果的Bansal和Khot，我们得到的硬度$2-ε$的问题，最小化的最大完工时间调度优先级约束的工件上相同的并行机抢占。这与该问题的最佳近似保证相匹配。假设相同的假设，我们也得到了一个超常数的不可逼近性结果的问题，调度优先级约束的工作相关的并行机，取得进展解决一个开放的问题，在两个列表中的10个开放的问题威廉姆森和Shmoys，和Schuurman和Woeginger。$k$ -部图的结构硬度的研究是独立的兴趣，因为它捕捉了一个大家庭的调度问题的内在硬度。除了已经提到的，这种推广也意味着紧不可逼近的问题，最小化加权完工时间的优先约束的工件在一台机器上，和最小化的问题的优先约束的工件在相同的并行机的完工时间，从而统一的结果Bansal和Khot，和Svensson，分别。
英文摘要：
We show a close connection between structural hardness for $k$-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. Assuming a natural but nontrivial generalisation of the bipartite structural hardness result of Bansal and Khot, we obtain a hardness of $2-ε$ for the problem of minimising the makespan for scheduling precedence-constrained jobs with preemption on identical parallel machines. This matches the best approximation guarantee for this problem. Assuming the same hypothesis, we also obtain a super constant inapproximability result for the problem of scheduling precedence-constrained jobs on related parallel machines, making progress towards settling an open question in both lists of ten open questions by Williamson and Shmoys, and by Schuurman and Woeginger. The study of structural hardness of $k$-partite graphs is of independent interest, as it captures the intrinsic hardness for a large family of scheduling problems. Other than the ones already mentioned, this generalisation also implies tight inapproximability to the problem of minimising the weighted completion time for precedence-constrained jobs on a single machine, and the problem of minimising the makespan of precedence-constrained jobs on identical parallel machine, and hence unifying the results of Bansal and Khot, and Svensson, respectively.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
