Improved Algorithms for Computing Minmax Regret 1-Sink and 2-Sink on Path Network

Abstract

Suppose that in an emergency, such as an earthquake or fire, a number of people need to be evacuated to a safe “sink” from every vertex of a network. The $k$-sink problem seeks to minimize the evacuation time of all the evacuees to the sinks. In the minmax regret version of this problem, the exact number of evacuees at each vertex is unknown, but only an interval of possible numbers is given. We want to minimize the evacuation time in the worst case, where the actual numbers of evacuees are most unfavorable to the chosen sink locations. We present an optimal $O(n)$ time algorithm for finding the minmax regret 1-sink on a path network, improving the previously best time complexity of $O(n\log n)$ [6, 12]. Some ideas we conceived for the new algorithm have other useful applications. For example, we demonstrate that it leads to an algorithm for computing the minmax regret 2-sink in path networks in $O(n\log ^4 n)$ time, which is a fairly significant improvement over an $O(n^2\log ^2 n)$ time algorithm [1]. Moreover, the two sinks that our algorithm finds are not restricted to vertices.