The Freeze-Tag Problem: How to Wake Up a Swarm of Robots

TL;DR

This paper studies the Freeze-Tag Problem in swarm robotics, analyzing its computational complexity, hardness of approximation, and providing algorithms with provable guarantees for various graph and geometric scenarios.

Contribution

It proves NP-hardness and approximation hardness of FTP, and introduces algorithms including a PTAS for star graphs and geometric environments, with competitive online strategies.

Findings

FTP is NP-hard even on star graphs.

Approximation factor cannot be better than 5/3 in general.

PTAS exists for star graphs and geometric instances.

Abstract

An optimization problem that naturally arises in the study of swarm robotics is the Freeze-Tag Problem (FTP) of how to awaken a set of ``asleep'' robots, by having an awakened robot move to their locations. Once a robot is awake, it can assist in awakening other slumbering robots.The objective is to have all robots awake as early as possible. While the FTP bears some resemblance to problems from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its algorithmic characteristics are surprisingly different. We consider both scenarios on graphs and in geometric environments.In graphs, robots sleep at vertices and there is a length function on the edges. Awake robots travel along edges, with time depending on edge length. For most scenarios, we consider the offline version of the problem, in which each awake robot knows the position of all other robots. We prove that the problem is NP-hard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NP-hard to obtain an approximation factor better than 5/3, even for graphs of bounded degree.These lower bounds are complemented with several positive algorithmic results, including: (1) We show that the natural greedy strategy on star graphs has a tight worst-case performance of 7/3 and give a polynomial-time approximation scheme (PTAS) for star graphs. (2) We give a simple O(log D)-competitive online algorithm for graphs with maximum degree D and locally bounded edge weights. (3) We give a PTAS, running in nearly linear time, for geometrically embedded instances.