Minimax Regret 1-Median Problem in Dynamic Path Networks

TL;DR

This paper addresses the minimax regret 1-median problem in dynamic path networks with uncertain supplies, proposing an algorithm to find an optimal sink location that minimizes worst-case regret across all supply scenarios.

Contribution

The paper introduces the first polynomial-time algorithm for the minimax regret 1-median problem in dynamic path networks with uniform capacity and uncertain supplies.

Findings

Proposed an O(n^3) algorithm for the problem.

Established a formal model for supply uncertainty in dynamic path networks.

Provided theoretical analysis and correctness of the algorithm.

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.