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 .
We define the problem segment cover as follows. We are given a set of pairs of sub-intervals of the unit interval. The problem asks if there is a choice of a single interval from each pair such that the union of the chosen intervals covers the entire unit interval. This problem arises naturally while attempting to compute visibility between a point and a line segment in the plane in the presence of uncertain obstacles. Segment cover is equivalent to a restricted version of SAT which we call contiguous SAT. Consider a SAT with the following restrictions. An input formula is in CNF form and an ordering of the clauses is given in which clauses containing any fixed literal appear contiguously. We call this restricted problem contiguous SAT. Our main result is that the problems segment cover and contiguous SAT are NP-hard. We also discuss hardness of approximation for these problems .
R
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.
We define the problem segment cover as follows. We are given a set of pairs of sub-intervals of the unit interval. The problem asks if there is a choice of a single interval from each pair such that the union of the chosen intervals covers the entire unit interval.
A
segment
R
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
Segment cover is equivalent to a restricted version of SAT which we call contiguous SAT. Consider a SAT with the following restrictions. An input formula is in CNF form and an ordering of the clauses is given in which clauses containing any fixed literal appear contiguously. We call this restricted problem contiguous SAT. Our main result is that the problems segment cover and contiguous
R
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
also discuss
R
CONTIGUOUS SAT and show it cannot have a PTAS, unless P=NP
these problems