A Quasi-Polynomial Algorithm for Subset-Sum Problems with At Most One Solution

TL;DR

This paper introduces a quasi-polynomial algorithm for the Subset Sum Problem when there is at most one solution, leveraging a novel projection method to efficiently narrow down the search space.

Contribution

It presents a new quasi-polynomial algorithm for SSP with at most one solution, extending geometric approaches to this specific case.

Findings

Algorithm effectively reduces the search radius in SSP cases with at most one solution.

Numerical tests demonstrate the practical efficiency of the proposed method.

The approach bridges geometric optimization and combinatorial problem solving.

Abstract

In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of the balls centers. The cases where the given point is in the convex hull of the balls centers include all NP-complete problems as we show. Some novel results are given in this area. A novel projection algorithm is developed then applied in the context of the Subset Sum Problem (SSP). Under the assumption that the SSP has at most one solution, we provide a quasi-polynomial algorithm, which decreases the radius of an initial ball containing the solution to the SSP. We perform some numerical tests which show the effectiveness of the proposed algorithm.