A FPTAS for the Subset Sum Problem with Real Numbers

Abstract

In this paper we study the subset sum problem with real numbers. Starting from the given problem, we formulate a quadratic maximization problem over a polytope which is eventually written as a distance maximization to a fixed point. For solving this, we provide a polynomial algorithm which maximizes the distance to a fixed point over a certain convex set. This convex set is obtained by intersecting the unit hypercube with two relevant half spaces. We show that in case the subset sum problem has a solution, our algorithm gives the correct maximum distance up to an arbitrary chosen precision. In such a case, we show that the obtained maximizer is a solution to the subset sum problem. Therefore, we compute the maximizer and upon analyzing it we can assert the feasibility of the subset sum problem.

arXiv commentIt relies on maximizing the distance over an intersection of balls to a given point. The used algorithm for this however, is not able to solve the class of problem the SSP generates