Improved Results for Minimum Constraint Removal

摘要

Given a set of obstacles and two designated points in the plane, the Minimum Constraint Removal problem asks for a minimum number of obstacles that can be removed so that a collision-free path exists between the two designated points. It is a well-studied problem in both robotic motion planning and wireless computing that has been shown to be NP-hard in various settings. In this work, we extend the study of Minimum Constraint Removal. We start by presenting refined NP-hardness reductions for the two cases: (1) when all the obstacles are axes-parallel rectangles, and (2) when all the obstacles are line segments such that no three intersect at the same point. These results improve on existing results in the literature. As a byproduct of our NP-hardness reductions, we prove that, unless the Exponential-Time Hypothesis (ETH) fails, Minimum Constraint Removal cannot be solved in subexponential time 2o(n), where n is the number of obstacles in the instance. This shows that significant improvement on the brute-force 2O(n)-time algorithm is unlikely. We then present a subexponential-time algorithm for instances of Minimum Constraint Removal in which the number of obstacles that overlap at any point is constant; the algorithm runs in time 2O(√N), where N is the number of the vertices in the auxiliary graph associated with the instance of the problem. We show that significant improvement on this algorithm is unlikely by showing that, unless ETH fails, Minimum Constraint Removal with bounded overlap number cannot be solved in time 2o(√N). We describe several exact algorithms and approximation algorithms that leverage heuristics and discuss their performance in an extensive empirical simulation.