We consider a quadratic programming (QP) problem (Π) of the form min x T Cx subject to Ax ≥ b where C ∈ R n×n + , rank(C) = 1 and A ∈ R m×n , b ∈ R m . We present an FPTAS for this problem by reformulating the QP (Π) as a parameterized LP and "rounding" the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0-1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions. *

We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovative data structure controls the memory requirements of the state space and enables solution of problems of realistic size. The algorithm finds optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must not interfere with each other, although one may yield to the other. The state space representation permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a cartesian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be considered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning four hours). It can also be used to create benchmarks for tuning heuristic algorithms that solve larger instances.