二维的网格中最小凹费用流的计算复杂性
On the computational complexity of minimum-concave-cost flow in a two-dimensional grid
摘要：
研究了二维的【形】网格上的最小凹费用流问题。我们根据网格的行数和列数、所有弧上不同容量的数量以及源和汇的位置来表征这个问题的计算复杂性。假设每个弧上的凹成本通过oracle机器来评估，即，在给定流值和弧索引的情况下，Oracle机器在单个计算步骤中返回弧上的成本。我们提出了一个算法，它的运行时间是关于网格列数的多项式，对于下列情况：（1）网格具有恒定的行数，所有弧上的不同容量的数目恒定，并且源和汇最多在两行中：（2）网格具有两行，连接行的所有弧上的不同容量的数目恒定;（3）网格具有恒定数量的行和在一行中的所有源，在每个弧上具有无限容量。这些大概是最一般的多项式可解的情况下，因为我们表明，当这些情况下的任何条件被删除的问题成为NP-困难。我们的结果适用于丰富的变种和推广的动态批量模型，并回答了串行供应链优化中提出的几个问题。
英文摘要：
We study the minimum-concave-cost flow problem on a two-dimensional grid. We characterize the computational complexity of this problem based on the number of rows and columns of the grid, the number of different capacities over all arcs, and the location of sources and sinks. The concave cost over each arc is assumed to be evaluated through an oracle machine, i.e., the oracle machine returns the cost over an arc in a single computational step, given the flow value and the arc index. We propose an algorithm whose running time is polynomial in the number of columns of the grid, for the following cases: (1) the grid has a constant number of rows, a constant number of different capacities over all arcs, and sources and sinks in at most two rows; (2) the grid has two rows and a constant number of different capacities over all arcs connecting rows; (3) the grid has a constant number of rows and all sources in one row, with infinite capacity over each arc. These are presumably the most general polynomially solvable cases, since we show the problem becomes NP-hard when any condition in these cases is removed. Our results apply to abundant variants and generalizations of the dynamic lot sizing model, and answer several questions raised in serial supply chain optimization.
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