Hardness of uncertain segment cover, contiguous SAT and visibility with uncertain obstacles

Abstract

In this paper, we prove that the uncertain segment cover problem is NP-hard. The uncertain segment cover problem, which appears in solving the question of visibility of a segment in the plane from a given point in the presence of uncertain obstacles, asks if a given interval can be covered by a collection of uncertain sub-intervals having two possible positions. This is done by showing that this problem is equivalent to a special version of the SAT problem that we will call contiguous SAT. In a contiguous SAT problem a CNF formula with a given ordering on its clauses is given such that any literal appears contiguously. We prove that contiguous SAT problem is NP-hard and as a consequence it follows that the uncertain segment cover problem is NP-hard. The approximation solution of this problem and its hardness are also studied.