A Simple, but NP-Hard, Motion Planning Problem

Determining the existence of a collision-free path between two points is one of the most fundamental questions in robotics. However, in situations where crossing an obstacle is costly but not impossible, it may be more appropriate to ask for the path that crosses the fewest obstacles. This may arise in both autonomous outdoor navigation (where the obstacles are rough but not completely impassable terrain) or indoor navigation (where the obstacles are doors that can be opened if necessary). This problem, the minimum constraint removal problem, is at least as hard as the underlying path existence problem. In this paper, we demonstrate that the minimum constraint removal problem is NP-hard for navigation in the plane even when the obstacles are all convex polygons, a case where the path existence problem is very easy.

[1]  Bernhard Nebel,et al.  Coming up With Good Excuses: What to do When no Plan Can be Found , 2010, Cognitive Robotics.

[2]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[3]  Brigitte Jaumard,et al.  On the Complexity of the Maximum Satisfiability Problem for Horn Formulas , 1987, Inf. Process. Lett..

[4]  Timothy Bretl,et al.  Proving path non-existence using sampling and alpha shapes , 2012, 2012 IEEE International Conference on Robotics and Automation.

[5]  Dinesh Manocha,et al.  A Simple Path Non-existence Algorithm Using C-Obstacle Query , 2006, WAFR.

[6]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[7]  John H. Reif,et al.  Complexity of the mover's problem and generalizations , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[8]  Gregory S. Chirikjian,et al.  Editorial: Special Issue on the Eighth International Workshop on the Algorithmic Foundations of Robotics , 2010, Int. J. Robotics Res..

[9]  Kris K. Hauser,et al.  The minimum constraint removal problem with three robotics applications , 2014, Int. J. Robotics Res..

[10]  Viktor Schuppan,et al.  Diagnostic Information for Realizability , 2008, VMCAI.

[11]  Joseph Culberson,et al.  Sokoban is PSPACE-complete , 1997 .

[12]  John F. Canny,et al.  A new algebraic method for robot motion planning and real geometry , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[13]  Erik D. Demaine,et al.  PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation , 2002, Theor. Comput. Sci..