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Approximation Algorithms for the Freeze Tag Problem inside Polygons

Fatemeh Rajabi-Alni, Alireza Bagheri, Behrouz Minaei-Bidgoli

TL;DR

This work studies this problem inside a polygonal domain and presents approximation algorithms for it, which can assist in awakening other sleeping robots.

Abstract

The freeze tag problem (FTP) aims to awaken a swarm of robots with one or more initial awake robots as soon as possible. Each awake robot must touch a sleeping robot to wake it up. Once a robot is awakened, it can assist in awakening other sleeping robots. We study this problem inside a polygonal domain and present approximation algorithms for it.

Approximation Algorithms for the Freeze Tag Problem inside Polygons

TL;DR

This work studies this problem inside a polygonal domain and presents approximation algorithms for it, which can assist in awakening other sleeping robots.

Abstract

The freeze tag problem (FTP) aims to awaken a swarm of robots with one or more initial awake robots as soon as possible. Each awake robot must touch a sleeping robot to wake it up. Once a robot is awakened, it can assist in awakening other sleeping robots. We study this problem inside a polygonal domain and present approximation algorithms for it.

Paper Structure

This paper contains 4 sections, 6 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Approximation Algorithms
  3. Constant Factor Approximation Algorithm
  4. A PTAS for the GFTP

Key Result

Theorem 1

Let $S$ be a set of robots inside a polygonal domain $P$ with $m$ reflex vertices and $h$ polygonal hole vertices. There is an $O(1)$-approximation algorithm with the makespan $O(diam(S,P))$ for the GFTP on $S$ using $m'$ Steiner robots, where $m'\leq h+m$.

Figures (3)

  • Figure 1: A polygonal domain with two holes; the dashed lines represent the geodesic path between the filled circles.
  • Figure 2: An example for illustration of the GVD; $l_a$ (resp. $l_b$) denotes the GVD from $a$ to $b$ (resp. from $b$ to $a$).
  • Figure 3: An example for decomposing the polygonal domain $P$ into convex partitions (pixels).

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Theorem 4