An O(n^2 log^2 n) Time Algorithm for Minmax Regret Minsum Sink on Path Networks

We model evacuation in emergency situations by dynamic flow in a network. We want to minimize the aggregate evacuation time to an evacuation center (called a sink) on a path network with uniform edge capacities. 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 a sink location that minimizes the maximum "regret." We present the first sub-cubic time algorithm in n to solve this problem, where n is the number of vertices. Although we cast our problem as evacuation, our result is accurate if the "evacuees" are fluid-like continuous material, but is a good approximation for discrete evacuees

On the capacity provisioning on dynamic networks

In this thesis, we consider the development of algorithms suitable for designing evacuation procedures in sparse or remote communities. The works are extensions of sink location problems on dynamic networks, which are motivated by real-life disaster events such as the Tohoku Japanese Tsunami, the Australian wildfire and many more. The available algorithms in this context consider the location of the sinks (safe-havens) with the assumptions that the evacuation by foot is possible, which is reasonable when immediate evacuation is needed in urban settings. However, for remote communities, emergency vehicles may need to be dispatched or situated strategically for an efficient evacuation process. With the assumption removed, our problems transform to the task of allocating capacities on the edges of dynamic networks given a budget capacity c. We first of all consider this problem on a dynamic path network of n vertices with the objective of minimizing the completion time (minmax criterion) given that the position of the sink is known. This leads to an O(nlogn + nlog(c/ξ)) time, where ξ is a refinement or precision parameter for an additional binary search in the worst case scenario. Next, we extend the problem to star topologies. The case where the sink is located at the middle of the star network follows the same approach for the path network. However, when the sink is located on a leaf node, the problem becomes more complicated when the number of links (edges) exceeds three. The second phase of this thesis focuses on allocating capacities on the edges of dynamic path networks with the objective of minimizing the total evacuation time (minsum criterion) given the position of the sink and the budget (fixed) capacity. In general, minsum problems are more difficult than minmax problems in the context of sink location problems. Due to few combinatorial properties discovered together with the possibility of changing objective. function configuration in the course of the optimization process, we consider the development of numerical procedure which involves the use of sequential quadratic programming (SQP). The sequential quadratic programming employed allows the specification of an arbitrary initial capacities and also helps in monitoring the changing configuration of the objective function. We propose to consider these problems on more complex topolgies such as trees and general graph in future.NSERC Discovery Grants program. University of Lethbridge Graduate Research Award. Alberta Innovates Awar

Minmax sink location problem on dynamic cycle networks

We address both 1 and k sink location problems on dynamic cycle networks. Our 1-sink algorithms run in O(n) and O(nlogn) time for uniform and general edge capacity cases, respectively. We improve the previously best known O(nlogn) time algorithm for single sink introduced by Xu et al. [Xu et al. 2015] with uniform capacities. When k¿1, we improve two results [Benkoczi et al. 2017] for both with uniform and arbitrary capacities by a factor of O(logn). Using the same sorted matrices optimization framework originally devised by Frederickson and Johnson and employed by [Benkoczi et al. 2017], our algorithms for the k-sink problems have time complexities of O(nlogn) for uniform, and O(nlog3 n) for arbitrary capacities. Key to our results is a novel data structure called a cluster head forest, which allows one to compute batches of queries for evacuation time efficiently