图中预算分配的优化
Optimizing Budget Allocation in Graphs
摘要：
在经典的设施选址问题中，我们考虑一个图$G$，其中$G$的边具有固定的权。然后，目标是找到图上的一组设施相对于某个目标函数的最佳定位。我们引入了一个新的框架设施选址问题，在图边缘的权重是不固定的，而是应该分配。我们的目标是找到一个有效的分配，由此产生的加权图优化设施位置目标函数。我们提出的算法，找到最佳的{ \em预算分配}的中心点问题和树的中点问题。我们的算法在线性时间内运行，无论是对于候选顶点作为输入的一部分给出的情况，还是对于找到优化解决方案的顶点是问题的一部分的情况。我们还提出了一个硬度的中心点问题的一般图形的情况下，其次是一个$O(\log^2(n))$近似算法的图形-一般度量空间。
英文摘要：
In the classical facility location problem we consider a graph $G$ with fixed weights on the edges of $G$. The goal is then to find an optimal positioning for a set of facilities on the graph with respect to some objective function. We introduce a new framework for facility location problems, where the weights on the graph edges are not fixed, but rather should be assigned. The goal is to find a valid assignment for which the resulting weighted graph optimizes the facility location objective function. We present algorithms for finding the optimal {\em budget allocation} for the center point problem and for the median point problem on trees. Our algorithms run in linear time, both for the case where a candidate vertex is given as part of the input, and for the case where finding a vertex that optimizes the solution is part of the problem. We also present a hardness result for the general graph case of the center point problem, followed by an $O(\log^2(n))$ approximation algorithm on graphs - with general metric spaces.
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