带边约束的多拟阵上的线性最优化--排序问题与极小化次模函数
Linear Optimization over a Polymatroid with Side Constraints -- Scheduling Queues and Minimizing Submodular Functions
摘要：
多类调度系统的调度和最小化次模函数这两个看似不相关的问题，通过一方面以次模集函数为特征，另一方面又代表调度系统性能多胞形的多拟阵有着相当深的联系.我们首先开发了一个我们称之为{ \it grouping}的算法，它解决了边约束下的排序问题，计算工作量为$O(n^3LP(n))$，$n$是作业类的数量，LP（n）是求解不超过$n$变量和$n$约束的线性规划的计算工作量。该算法将作业类组织成组，并将最优策略确定为跨组的优先级规则和每个组内的随机规则（以强制执行侧约束）。然后，我们应用分组算法的次模函数最小化，后者映射到一个调度问题的侧约束。%每次应用该算法时，它识别一个子集;并且我们表明可以通过应用分组算法$n$次来识别最小化子集。因此，这导致一个算法，最小化一个次模函数的努力$O(n^4LP(n))$。
英文摘要：
Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a {\it grouping} algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of $O(n^3LP(n))$, $n$ being the number of job classes, and LP(n) being the computational efforts of solving a linear program with no more than $n$ variables and $n$ constraints. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. %Each time the algorithm is applied, it identifies a subset; and We show the minimizing subset can be identified by applying the grouping algorithm $n$ times. Hence, this results in a algorithm that minimizes a submodular function with an effort of $O(n^4LP(n))$.
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。
