Efficient reductions and algorithms for variants of Subset Sum

TL;DR

This paper introduces new efficient algorithms for variants of the Subset Sum and Subset Product problems, focusing on parameterized solutions based on the number of solutions, and employs advanced mathematical techniques for improved performance.

Contribution

It presents parameterized search algorithms for Subset Sum variants, including NP-hardness results, and offers both randomized and deterministic algorithms with improved time and space complexities.

Findings

NP-hardness of Subset Sum with a unique solution

Deterministic algorithm with (k \u00b7 (n+t)) time for realisable sets

Randomized (n + t) time algorithm for Subset Product

Abstract

Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. There is a close variant of the $\mathsf{SSUM}$, called $\mathsf{Subset~Product}$. Given positive integers $a_1, ..., a_n$ and a target integer $t$, the $\mathsf{Subset~Product}$ problem asks to determine whether there exists a subset $S \subseteq [n]$ such that $\prod_{i \in S} a_i=t$. There is a pseudopolynomial time dynamic programming algorithm, due to Bellman (1957) which solves the $\mathsf{SSUM}$ and $\mathsf{Subset~Product}$ in $O(nt)$ time and $O(t)$ space. In the first part, we present {\em search} algorithms for variants of the Subset Sum problem. Our algorithms are parameterized by $k$, which is a given upper bound on the number of realisable sets (i.e.,~number of solutions, summing exactly $t$). We show that $\mathsf{SSUM}$ with a unique solution is already NP-hard, under randomized reduction. This makes the regime of parametrized algorithms, in terms of $k$, very interesting. Subsequently, we present an $\tilde{O}(k\cdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a poly$(knt)$-time and $O(\log(knt))$-space deterministic algorithm that finds all the realisable sets for a subset sum instance. In the latter part, we present a simple and elegant randomized $\tilde{O}(n + t)$ time algorithm for $\mathsf{Subset~Product}$. Moreover, we also present a poly$(nt)$ time and $O(\log^2 (nt))$ space deterministic algorithm for the same. We study these problems in the unbounded setting as well. Our algorithms use multivariate FFT, power series and number-theoretic techniques, introduced by Jin and Wu (SOSA'19) and Kane (2010).