Hardness of CONTIGUOUS SAT and Visibility with Uncertain Obstacles

Consider SAT with the following restrictions. An input formula is in CNF form, and, there exists an ordering of the clauses in which clauses containing any fixed literal appear contiguously. We call this restricted SAT CONTIGUOUS SAT. This problem arises naturally while attempting to compute visibility between a point and a line in the plane in the presence of uncertain obstacles. The simplest case is when each obstacle is randomly chosen from two possible line segments. We define the problem SEGMENT COVER capturing the situation. In SEGMENT COVER, we are given a set of uncertain segments, each uncertain segment defined to be a random interval, chosen from two possible sub-intervals of the unit interval. Then, SEGMENT COVER asks if the probability of covering the entire unit interval is non-zero. We prove that SEGMENT COVER and hence CONTIGUOUS SAT are NP-hard. We further show that we can assume all intervals of SEGMENT COVER to be of equal lengths, this proves another problem called BCU to be NP-hard in dimension 1. Moreover, we also deduce hardness of approximation for CONTIGUOUS SAT and show it cannot have a PTAS, unless P=NP

On-line Viewpoint and Motion Planning for Efficient Visual Navigation under Uncertainty

This paper proposes an on-line viewpoint planning method for a mobile robot to reach a goal position safely and quickly. When a robot passes through a narrow space, it moves slowly while carefully observing surrounding objects and estimating the distances to the objects precisely. On the other hand, the robot moves quickly in a widely open space. To realize such a behavior of the robot, by considering both the predicted positional uncertainty and the configuration of obstacles, the viewpoint is adaptively determined according to the narrowness of the nearby environment. The planner works on-line to cope with actual errors. The robot motion is continuously performed with speed control between the planned viewpoints during visual processing. An experimental result using a mobile robot with stereo vision shows the validity of the proposed method