We present a new algorithm for solving the dense linear (sum) assignment problem and an efficient, parallel implementation that is based on the successive shortest path algorithm. More specifically, we introduce the well-known epsilon scaling approach used in the Auction algorithm to approximate the dual variables of the successive shortest path algorithm prior to solving the assignment problem to limit the complexity of the path search. This improves the runtime by several orders of magnitude for hard-to-solve real-world problems, making the runtime virtually independent of how hard the assignment is to find. In addition, our approach allows for using accelerators and/or external compute resources to calculate individual rows of the cost matrix. This enables us to solve problems that are larger than what has been reported in the past, including the ability to efficiently solve problems whose cost matrix exceeds the available systems memory. To our knowledge, this is the first implementation that is able to solve problems with more than one trillion arcs in less than 100 hours on a single machine.

中文翻译：

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算法 1015

我们提出了一种解决密集线性（和）分配问题的新算法，以及一种基于连续最短路径算法的高效并行实现。更具体地说，我们介绍了拍卖算法中众所周知的 epsilon 缩放方法，以在解决分配问题之前逼近连续最短路径算法的对偶变量，以限制路径搜索的复杂性。对于难以解决的实际问题，这将运行时间提高了几个数量级，使运行时间几乎与查找分配的难易程度无关。此外，我们的方法允许使用加速器和/或外部计算资源来计算成本矩阵的各个行。这使我们能够解决比过去报告的问题更大的问题，包括有效解决成本矩阵超出可用系统内存的问题的能力。据我们所知，这是第一个能够在单台机器上在不到 100 小时内解决超过 1 万亿弧的问题的实现。