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Register a new userMinmax Regret 1-Sink for Aggregate Evacuation Time on Path Networks
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Added by
Tsunehiko Kameda
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Evacuation in emergency situations can be modeled by a dynamic flow network. Two criteria have been used before: one is the evacuation completion time and the other is the aggregate evacuation time of individual evacuees. The aim of this paper is to optimize the aggregate evacuation time in the simplest case, where the network is a path and only one evacuation center (called a sink) is to be introduced. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute the sink location that minimizes the maximum regret. We present an $O(n^2log n)$ time algorithm to solve this problem, improving upon the previously fastest $O(n^3)$ time algorithm, where $n$ is the number of vertices.
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We consider the problem of locating a set of $k$ sinks on a path network with general edge capacities that minimizes the sum of the evacuation times of all evacuees. We first present an $O(knlog^4n)$ time algorithm when the edge capacities are non-uniform, where $n$ is the number of vertices. We then present an $O(knlog^3 n)$ time algorithm when the edge capacities are uniform. We also present an $O(nlog n)$ time algorithm for the special case where $k=1$ and the edge capacities are non-uniform.
We present a novel approach to finding the $k$-sink on dynamic path networks with general edge capacities. Our first algorithm runs in $O(n log n + k^2 log^4 n)$ time, where $n$ is the number of vertices on the given path, and our second algorithm runs in $O(n log^3 n)$ time. Together, they improve upon the previously most efficient $O(kn log^2 n)$ time algorithm due to Arumugam et al. for all values of $k$. In the case where all the edges have the same capacity, we again present two algorithms that run in $O(n + k^2 log^2n)$ time and $O(n log n)$ time, respectively, and they together improve upon the previously best $O(kn)$ time algorithm due to Higashikawa et al. for all values of $k$.
This paper considers the minimax regret 1-median problem in dynamic path networks. In our model, we are given a dynamic path network consisting of an undirected path with positive edge lengths, uniform positive edge capacity, and nonnegative vertex supplies. Here, each vertex supply is unknown but only an interval of supply is known. A particular assignment of supply to each vertex is called a scenario. Given a scenario s and a sink location x in a dynamic path network, let us consider the evacuation time to x of a unit supply given on a vertex by s. The cost of x under s is defined as the sum of evacuation times to x for all supplies given by s, and the median under s is defined as a sink location which minimizes this cost. The regret for x under s is defined as the cost of x under s minus the cost of the median under s. Then, the problem is to find a sink location such that the maximum regret for all possible scenarios is minimized. We propose an O(n^3) time algorithm for the minimax regret 1-median problem in dynamic path networks with uniform capacity, where n is the number of vertices in the network.
This paper considers the k-sink location problem in dynamic path networks. In our model, a dynamic path network consists of an undirected path with positive edge lengths, uniform edge capacity, and positive vertex supplies. Here, each vertex supply corresponds to a set of evacuees. Then, the problem requires to find the optimal location of $k$ sinks in a given path so that each evacuee is sent to one of k sinks. Let x denote a k-sink location. Under the optimal evacuation for a given x, there exists a (k-1)-dimensional vector d, called (k-1)-divider, such that each component represents the boundary dividing all evacuees between adjacent two sinks into two groups, i.e., all supplies in one group evacuate to the left sink and all supplies in the other group evacuate to the right sink. Therefore, the goal is to find x and d which minimize the maximum cost or the total cost, which are denoted by the minimax problem and the minisum problem, respectively. We study the k-sink location problem in dynamic path networks with continuous model, and prove that the minimax problem can be solved in O(kn) time and the minisum problem can be solved in O(n^2 min{k, 2^{sqrt{log k log log n}}}) time, where n is the number of vertices in the given network. Note that these improve the previous results by [6].
A dynamic network ${cal N} = (G,c,tau,S)$ where $G=(V,E)$ is a graph, integers $tau(e)$ and $c(e)$ represent, for each edge $ein E$, the time required to traverse edge $e$ and its nonnegative capacity, and the set $Ssubseteq V$ is a set of sources. In the $k$-{sc Sink Location} problem, one is given as input a dynamic network ${cal N}$ where every source $uin S$ is given a nonnegative supply value $sigma(u)$. The task is then to find a set of sinks $X = {x_1,ldots,x_k}$ in $G$ that minimizes the routing time of all supply to $X$. Note that, in the case where $G$ is an undirected graph, the optimal position of the sinks in $X$ needs not be at vertices, and can be located along edges. Hoppe and Tardos showed that, given an instance of $k$-{sc Sink Location} and a set of $k$ vertices $Xsubseteq V$, one can find an optimal routing scheme of all the supply in $G$ to $X$ in polynomial time, in the case where graph $G$ is directed. Note that when $G$ is directed, this suffices to obtain polynomial-time solvability of the $k$-{sc Sink Location} problem, since any optimal position will be located at vertices of $G$. However, the computational complexity of the $k$-{sc Sink Location} problem on general undirected graphs is still open. In this paper, we show that the $k$-{sc Sink Location} problem admits a fully polynomial-time approximation scheme (FPTAS) for every fixed $k$, and that the problem is $W[1]$-hard when parameterized by $k$.
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