Improved approximation algorithms for the freeze-tag problem

Esther M. Arkin, Michael A. Bender, Dongdong Ge

Abstract: In the Freeze-Tag Problem, the objective is to awaken a set of "asleep" robots, starting with only one "awake" robot. A robot awakens a sleeping robot by moving to the sleeping robot's position. When a robot awakens, it is available to assist in awakening other slumbering robots. The objective is to compute an optimal awakening schedule/ such that all robots are awake by time t*, for the smallest possible value of t*. Because of its resemblance to the children's game of freeze-tag, this problem has been called Freeze-Tag Problem (FTP).A particularly intriguing aspect of the FTP is that any algorithm that is not purposely unproductive yields an O(log n)-approximation, while no o(log n)-approximation algorithms are known for general metric spaces.This paper presents an O(1)-approximation algorithm for the FTP in unweighted graphs, in which there is one asleep robot at each node. We show that this version of the FTP is NP-hard.We generalize our methods to the case in which there are multiple robots at each node and edges are unweighted; we obtain a θ(∗log n)-approximation in this case. In the case of weighted edges, our methods yield an O((L/d)log n)-approximation algorithm, where L is the length of the longest edge and d is the diameter of the graph.