New Algorithms for Subset Sum Problem

TL;DR

This paper introduces new deterministic and probabilistic algorithms for the subset sum problem, offering efficient solutions and the ability to generate all solutions with a novel data arrangement.

Contribution

It presents a novel deterministic algorithm that produces universal, lightweight code and a probabilistic variant with one-sided error, advancing subset sum solution methods.

Findings

Deterministic algorithm can generate all solutions efficiently for small n.

Probabilistic algorithm with one-sided error offers faster approximate solutions.

Greedy algorithm minimizes variance in subset sum solutions.

Abstract

Given a set (or multiset) S of n numbers and a target number t, the subset sum problem is to decide if there is a subset of S that sums up to t. There are several methods for solving this problem, including exhaustive search, divide-and-conquer method, and Bellman's dynamic programming method. However, none of them could generate universal and light code. In this paper, we present a new deterministic algorithm based on a novel data arrangement, which could generate such code and return all solutions. If n is small enough, it is efficient for usual purpose. We also present a probabilistic version with one-sided error and a greedy algorithm which could generate a solution with minimized variance.