A FPTAS for the Subset Sum Problem with Real Numbers

TL;DR

This paper introduces a polynomial-time approximation scheme (FPTAS) for the subset sum problem with real numbers by transforming it into a convex optimization problem and solving it with a distance maximization approach.

Contribution

The paper presents the first FPTAS for the subset sum problem with real numbers, using a novel convex optimization formulation and a distance maximization algorithm.

Findings

Provides a polynomial algorithm for the problem

Achieves arbitrary precision in solution verification

Ensures correctness if the subset sum has a solution

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.