Theoretical Approach for planning public facilities in shrinking era

在设施基数固定的条件下，给出了贪心法、吝啬法和交换法等次模最大化方法的最坏情况界。我们证明了这些边界比已有的结果更小，从而得到了具有基数不等式约束的次模函数最大化问题。其次，推导出与相邻点和线之间距离的相关系数；假设点和线是随机均匀分布的。距离是用连续平面上的欧氏距离来测量的。第k和第j个最近邻距离之间的相关系数是方差系数的比值。第三，我们使用简单的直线定位模型来表征投票结果与最优定位之间的差距。提出了投票结果不有效的等效条件。为了揭示投票方案的无效率性，我们给出了损害投票方案的简单例子。我们将我们的模型应用到日本一个实际的市政厅选址模型中。

We presented the worst-case bound of myopic approaches such as greedy, stingy and interchange methods for submodular maximization subject to the requirement that the cardinality of facilities is fixed. We prove that these bounds are smaller than existing results, which is derived he submodular function maximization problem with cardinality inequality constraint. Secondly, we derived the correlation coefficients between the distances to the neighbor points and lines. It is supposed that the points and the lines are distributed randomly and uniformly. Distance is measured as the Euclidean distances on a continuous plane. The correlation coefficient between the k-th and j-th nearest neighbor distances is obtained as ratio of the coefficient of variance. Thirdly, we characterized the gap between voting outcome and optimal location by use a simple line location model. We presented the equivalent conditions that the voting outcome is not efficient solution. Then we give simple examples which harm the voting solution in order to reveal the inefficiency of the voting solutions. And we applied our model to an actual city hall location model in Japan.