  The steady development of motor vehicle technology will enable cars of the
near future to assume an ever increasing role in the decision making and
control of the vehicle itself. In the foreseeable future, cars will have the
ability to communicate with one another in order to better coordinate their
motion. This motivates a number of interesting algorithmic problems. One of the
most challenging aspects of traffic coordination involves traffic
intersections. In this paper we consider two formulations of a simple and
fundamental geometric optimization problem involving coordinating the motion of
vehicles through an intersection.
  We are given a set of $n$ vehicles in the plane, each modeled as a unit
length line segment that moves monotonically, either horizontally or
vertically, subject to a maximum speed limit. Each vehicle is described by a
start and goal position and a start time and deadline. The question is whether,
subject to the speed limit, there exists a collision-free motion plan so that
each vehicle travels from its start position to its goal position prior to its
deadline.
  We present three results. We begin by showing that this problem is
NP-complete with a reduction from 3-SAT. Second, we consider a constrained
version in which cars traveling horizontally can alter their speeds while cars
traveling vertically cannot. We present a simple algorithm that solves this
problem in $O(n \log n)$ time. Finally, we provide a solution to the discrete
version of the problem and prove its asymptotic optimality in terms of the
maximum delay of a vehicle.
