A Polynomial-Time Algorithm for Special Cases of the Unbounded Subset-Sum Problem

TL;DR

This paper introduces polynomial-time algorithms for specific cases of the NP-complete Unbounded Subset-Sum Problem, improving efficiency for certain input conditions and providing probabilistic solutions for larger instances.

Contribution

The paper presents new polynomial-time algorithms for special cases of USSP and a probabilistic method for larger inputs, advancing solution approaches for this NP-complete problem.

Findings

Efficient polynomial-time algorithm with O(n) complexity for specific USSP cases.

Probabilistic algorithm with O(n^2) complexity for larger inputs, with success probability up to 1.

Algorithms can solve large-scale USSP instances under certain conditions, despite NP-completeness.

Abstract

The Unbounded Subset-Sum Problem (USSP) is defined as: given sum $s$ and a set of integers $W\leftarrow \{p_1,\dots,p_n\}$ output a set of non-negative integers $\{y_1,\dots,y_n\}$ such that $p_1y_1+\dots+p_ny_n=s$. The USSP is an NP-complete problem that does not have any known polynomial-time solution. There is a pseudo-polynomial algorithm for the USSP problem with $O((p_{1})^{2}+n)$ time complexity and $O(p_{1})$ memory complexity, where $p_{1}$ is the smallest element of $W$ \cite{PH}. This algorithm is polynomial in term of the number of inputs, but exponential in the size of $p_1$. Therefore, this solution is impractical for the large-scale problems. In this paper, first we propose an efficient polynomial-time algorithm with $O(n)$ computational complexity for solving the specific case of the USSP where $ s> \sum_{i=1}^{k-1}q_iq_{i+1}-q_i-q_{i+1}$, $q_i$'s are the elements of a small subset of $W$ in which $gcd$ of its elements divides $s$ and $2\le k \le n$. Second, we present another algorithm for smaller values of $s$ with $O(n^2)$ computational complexity that finds the answer for some inputs with a probability between $0.5$ to $1$. Its success probability is directly related to the number of subsets of $W$ in which $gcd$ of their elements divides $s$. This algorithm can solve the USSP problem with large inputs in the polynomial-time, no matter how big inputs are, but, in some special cases where $s$ is small, it cannot find the answer.