The absolute quickest 1-center problem on a cycle and its reverse problem

The concept of the quickest path refers to the path with the minimum transmission time, considering both its length and capacity. We investigate the problem of finding a point on a cycle such that the maximum quickest distance from any vertex to that point is minimized. We refer to this problem as the quickest 1-center problem on cycles. First, we solve the problem on paths in linear time based on the optimality criterion. Then, we address the problem on cycles in \(O(n^2)\) $O(n^2)$ time by leveraging the solution approach on the induced path in each iteration, where n is the number of vertices. We also consider the problem of reducing the quickest distance objective at a predetermined vertex of a cycle as much as possible by augmenting the edge capacities within a given budget. This problem is called the reverse quickest 1-center problem on cycles. We develop a combinatorial algorithm that solves the problem in \(O(n^2)\) $O(n^2)$ time by solving each subproblem in linear time.