Exploring and Triangulating a Region by a Swarm of Robots

We consider online and offline problems related to exploring and surveying a region by a swarm of robots with limited communication range. The minimum relay triangulation problem (MRTP) asks for placing a minimum number of robots, such that their communication graph is a triangulated cover of the region. The maximum area triangulation problem (MATP) aims at finding a placement of n robots such that their communication graph contains a root and forms a triangulated cover of a maximum possible amount of area. Both problems are geometric versions of natural graph optimization problems. The offline version of both problems share a decision problem, which we prove to be NP-hard. For the online version of the MRTP, we give a lower bound of 6/5 for the competitive ratio, and a strategy that achieves a ratio of 3; for different offline versions, we describe polynomial-time approximation schemes. For the MATP we show that no competitive ratio exists for the online problem, and give polynomial-time approximation schemes for offline versions.

[1]  Santosh S. Vempala,et al.  New Approximation Guarantees for Minimum-Weight k-Trees and Prize-Collecting Salesmen , 1999, SIAM J. Comput..

[2]  James McLurkin,et al.  Distributed Algorithms for Dispersion in Indoor Environments Using a Swarm of Autonomous Mobile Robots , 2004, DARS.

[3]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[4]  Steve Fisk,et al.  A short proof of Chvátal's Watchman Theorem , 1978, J. Comb. Theory, Ser. B.

[5]  D. Du,et al.  Computing in Euclidean Geometry , 1995 .

[6]  Subhash Suri,et al.  Simple Robots with Minimal Sensing: From Local Visibility to Global Geometry , 2007, Int. J. Robotics Res..

[7]  Joseph S. B. Mitchell,et al.  Minimum covering with travel cost , 2009, J. Comb. Optim..

[8]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[9]  David Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[10]  Rolf Klein,et al.  The Polygon Exploration Problem , 2001, SIAM J. Comput..

[11]  Friedhelm Meyer auf der Heide,et al.  A survey on relay placement with runtime and approximation guarantees , 2011, Comput. Sci. Rev..

[12]  Sándor P. Fekete,et al.  Polygon exploration with time-discrete vision , 2008, Comput. Geom..

[13]  Rolf Klein,et al.  Exploring Simple Grid Polygons , 2005, COCOON.

[14]  Esther M. Arkin,et al.  Algorithms for Rapidly Dispersing Robot Swarms in Unknown Environments , 2002, WAFR.

[15]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[16]  Donald E. Knuth,et al.  The Problem of Compatible Representatives , 1992, SIAM J. Discret. Math..

[17]  Subhash Suri,et al.  Simple Robots in Polygonal Environments: A Hierarchy , 2008, ALGOSENSORS.

[18]  Sándor P. Fekete,et al.  Robot Swarms for Exploration and Triangulation of Unknown Environments , 2010 .

[19]  David R. Karger,et al.  Approximation algorithms for orienteering and discounted-reward TSP , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[20]  Joseph S. B. Mitchell,et al.  Improved Approximation Algorithms for Relay Placement , 2008, ESA.

[21]  V. Chvátal.  A combinatorial theorem in plane geometry , 1975 .