Lower Bounds for Shortest Path and Related Problems

We present the rst lower bounds for shortest path problems (including euclidean shortest path) in three dimensions, and for some constrained motion planning problems in two and three dimensions. Our proofs are based a technique called free path encoding and use homotopy equivalence classes of paths to encode state. We rst apply the method to the shortest path problem in three dimensions. The problem is to nd the shortest path under an L metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Speci cally, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may di er from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of nding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the rst provably intractable problem in robotics. Acknowledgements. John Canny was supported by an IBM fellowship. The research by John Reif was sponsored in part by the NSF under contract NSF{DCR{ 85{03251 and the OÆce of Naval Research under OÆce of Naval Research contracts N00014{81{K{0494 and N00014{80{C{0647 and in part by the Advanced Research Projects Agency under OÆce of Naval Research contracts N00014{80{C{0505 and N00014{82{K{0334; it was subsequently supported by National Science Foundation Grants CCF{0432038, CCF{0432047, ITR 0326157, EIA{0218376, EIA{0218359, and EIA{0086015. A preliminary version of this paper appeared as J. Canny and J.H. Reif, New Lower Bound Techniques for Robot Motion Planning Problems. 28th Annual IEEE Symposium on Foundations of Computer Science, Los Angeles, CA, October 1987, pp. 49-60.