An FPTAS for minimizing the product of two non-negative linear cost functions
We consider a quadratic programming (QP) problem (Pi) of the form min x(T) Cx subject to Ax >= b, x >= 0 where C is an element of R(+)(nxn), rank(C) = 1 and A is an element of R(mxn), b is an element of R(m). We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Pi) 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 Pi models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.
quadratic programming approximation schemes combinatorial optimization