Minmax regret 1-sink location problems on dynamic flow path networks with parametric weights

This paper addresses the minmax regret 1-sink location problem on a dynamic flow path network with parametric weights. A dynamic flow path network consists 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 where all the people evacuate quickly. In our model, each weight is represented by a linear function of a common parameter t, and the decision maker who determines the sink location does not know the value of t. We formulate the problem under such uncertainty as the minmax regret problem. Given t and sink location x, the cost 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 this cost and the optimal cost under t. The problem is to find the sink location minimizing the maximum regret over all t. For the problem, we propose an \(O(n^4 2^{\alpha (n)} \alpha (n)^2 \log n)\) time algorithm, where n is the number of vertices in the network and \(\alpha (\cdot )\) 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 \(O(n^3 2^{\alpha (n)} \alpha (n) \log n)\).