Input to the Load Balanced Demand Distribution (LBDD) consists of the following: (a) a set of public service centers (e.g., schools); (b) a set of demand (people) units and; (c) a cost matrix containing the cost of assignment for all demand unit-service center pairs. In addition, each service center is also associated with a notion of capacity and a penalty which is incurred if it gets overloaded. Given the input, the LBDD problem determines a mapping from the set of demand units to the set of service centers. The objective is to determine a mapping that minimizes the sum of the following two terms: (i) the total assignment cost between demand units and their allotted service centers and, (ii) total of penalties incurred. The problem of LBDD finds its application in the domain of urban planning. An instance of the LBDD problem can be reduced to an instance of the min-cost bi-partite matching problem. However, this approach cannot scale up to the real world large problem instances. The current state of the art related to LBDD makes simplifying assumptions such as infinite capacity or total capacity being equal to the total demand. This paper proposes a novel allotment subspace re-adjustment based approach (ASRAL) for the LBDD problem. We analyze ASRAL theoretically and present its asymptotic time complexity. We also evaluate ASRAL experimentally on large problem instances and compare with alternative approaches. Our results indicate that ASRAL is able to scale-up while maintaining significantly better solution quality over the alternative approaches. In addition, we also extend ASRAL to para-ASRAL which uses the GPU and CPU cores to speed-up the execution while maintaining the same solution quality as ASRAL.

翻译：负载均衡需求分配（LBDD）问题的输入包括：(a) 一组公共服务设施（如学校）；(b) 一组需求单元（人口）；(c) 包含所有需求单元-设施对所对应分配成本的矩阵。此外，每个设施还关联容量概念及超载时产生的惩罚值。给定上述输入，LBDD问题需确定从需求单元集合到设施集合的映射，其目标是最小化以下两项之和：(i) 需求单元与其分配设施之间的总分配成本；(ii) 总惩罚值。LBDD问题在城市规划领域具有重要应用。虽然可将LBDD实例转化为最小成本二分匹配问题，但该方法无法扩展到现实中的大规模问题。当前LBDD相关研究成果常采用理想化假设，如设施容量无限或总容量等于总需求。本文针对LBDD问题提出了一种基于分配子空间重调整的新方法（ASRAL）。我们从理论上分析了ASRAL方法，给出了其渐近时间复杂度，并通过大规模实例实验与替代方法进行对比评估。结果表明，ASRAL在保持可扩展性的同时，能显著优于替代方法的解质量。此外，我们将ASRAL扩展为para-ASRAL，该变体利用GPU和CPU核心加速执行，同时保持与ASRAL相同的解质量。