Minimax regret 1-sink location problem in dynamic path networks

Algorithmic Aspects in Information and Management

Golin

Katoh

2014

Self Cite

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 √ 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].

Section: Known Properties Of 1-sink Location Problemmentioning

confidence: 69%

Multiple Sink Location Problems in Dynamic Path Networks

Algorithmic Aspects in Information and Management

Golin

Katoh

2014

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“…This formula was developed by [33,35] which has also been shown in [18,29]. Notice that this formula holds also for the case of general edge capacities.…”

Section: K-facility Location In Pathsmentioning

confidence: 82%

“…[Case 1]: By (25), (27) and the condition of A ≤ B , we have C ≤ G ≤ H ≤ D , which contradicts (29).…”

Section: Claim 1 For Integers P Satisfyingmentioning

confidence: 99%

See 1 more Smart Citation

A Survey on Facility Location Problems in Dynamic Flow Networks

Katoh

2019

Rev Socionetwork Strat

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This paper surveys the facility location problems in dynamic flow networks that have been actively studied in recent years. These problems have been motivated by evacuation planning which has become increasingly important in Japan. The evacuation planning problem is formulated using a dynamic flow network consisting of a graph in which a capacity as well as a transit time is associated with each edge. The goal of the problem is to find a way to send evacuees originally existing at vertices to facilities (evacuation centers) as quickly as possible. The problem can be viewed as a generalization of the classical k-center and k-median problems. In this paper we show recent results about the difficulty and approximability of a single facility location for general networks and polynomial time algorithms for k-facility location problems in path and tree networks. We also mention the minimax regret version of these problems. Fig. 2 A time-expanded network N(5) for the dynamic flow network N illustrated in Fig. 1. A number attached to a vertex represents its supply. A number attached to an edge represents its capacity buildings to reduce the loss of human lives from a tsunami triggered by the earthquake. In this respect, we hope that the methods developed for facility location problems will help to reduce the budget used for such disaster prevention and help reduce the loss of human lives.

“…Obviously the number of such scenarios is O(n). The authors of [8,13] treated the minimax regret 1-center problem in dynamic path networks, which requires to find a sink location in a path that minimizes the maximum regret similarly defined as (7) where the completion time criterion is adopted instead of the total time one. They proved that for any point in an input path, at least one worst case scenario is left-bipartite or right-bipartite.…”

Section: Bipartite Scenariomentioning

confidence: 99%

Minimax Regret 1-Median Problem in Dynamic Path Networks

Lecture Notes in Computer Science

Cheng

Kameda

et al. 2016

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Abstract. 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.