带参数权重的动态流路网络上的Minmax遗憾1-Sink选址问题
Minmax Regret 1-Sink Location Problems on Dynamic Flow Path Networks with Parametric Weights
摘要：
研究了带参数权的动态流路网络上的极小极大遗憾1-汇点选址问题。我们给出了一个动态流网络组成的无向路径的正边长度，正边容量，和非负的顶点权重。路径可以被认为是一条道路，边的长度是沿着道路的距离，顶点的权重是站点上的人数。边缘容量限制了单位时间内可以进入边缘的人数。我们考虑在网络中定位一个汇点的问题，所有的人尽快从顶点撤离。在我们的模型中，每个权重由公共参数$t$中的线性函数表示，并且确定水槽位置的决策者不知道$t$的值。我们制定的汇的位置问题下，这样的不确定性的最小最大遗憾问题。给定$t$和宿位置$x$，在$t$下的$x$的成本是由$t$确定的所有人到达$x$的时间之和。在$t$下的$x$的遗憾是在$t$下的$x$的成本与在$t$下的最优成本之间的差距。该问题的任务是找到一个汇的位置，最大限度地减少所有$t$的最大遗憾。对于这个问题，我们提出了一个$O(n^4 2^{α(n)} α(n) \log n)$时间算法，其中$n$是网络中的顶点数，$α(\cdot)$是逆阿克曼函数.此外，对于特殊的情况下，每个边缘具有相同的容量，我们表明，复杂性可以降低到$O(n^3 2^{α(n)} α(n) \log n)$。
英文摘要：
This paper addresses the minmax regret 1-sink location problem on dynamic flow path networks with parametric weights. We are given a dynamic flow network consisting of an undirected path with positive edge lengths, positive edge capacities, and nonnegative vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. We consider the problem of locating a sink in the network, to which all the people evacuate from the vertices as quickly as possible. In our model, each weight is represented by a linear function in a common parameter $t$, and the decision maker who determines the location of a sink does not know the value of $t$. We formulate the sink location problem under such uncertainty as the minmax regret problem. Given $t$ and a sink location $x$, the cost of $x$ under $t$ is the sum of arrival times at $x$ for all the people determined by $t$. The regret for $x$ under $t$ is the gap between the cost of $x$ under $t$ and the optimal cost under $t$. The task of the problem is formulated as the one to find a sink location that minimizes the maximum regret over all $t$. For the problem, we propose an $O(n^4 2^{α(n)} α(n) \log n)$ time algorithm where $n$ is the number of vertices in the network and $α(\cdot)$ is the inverse Ackermann function. Also for the special case in which every edge has the same capacity, we show that the complexity can be reduced to $O(n^3 2^{α(n)} α(n) \log n)$.
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