Dynamic Subset Sum with Truly Sublinear Processing Time

TL;DR

This paper introduces a dynamic subset sum algorithm with truly sublinear amortized processing time in terms of the maximum value, significantly improving efficiency for large-scale, dynamic instances.

Contribution

It presents the first algorithm achieving truly sublinear amortized time per operation in the dynamic subset sum problem under certain conditions.

Findings

Achieves truly sublinear amortized processing time for dynamic subset sum.

Establishes a lower bound based on SETH for algorithms with certain capabilities.

Provides insights into the computational limits of dynamic subset sum algorithms.

Abstract

Subset sum is a very old and fundamental problem in theoretical computer science. In this problem, $n$ items with weights $w_1, w_2, w_3, \ldots, w_n$ are given as input and the goal is to find out if there is a subset of them whose weights sum up to a given value $t$. While the problem is NP-hard in general, when the values are non-negative integer, subset sum can be solved in pseudo-polynomial time $~\widetilde O(n+t)$. In this work, we consider the dynamic variant of subset sum. In this setting, an upper bound $\tmax$ is provided in advance to the algorithm and in each operation, either a new item is added to the problem or for a given integer value $t \leq \tmax$, the algorithm is required to output whether there is a subset of items whose sum of weights is equal to $t$. Unfortunately, none of the existing subset sum algorithms is able to process these operations in truly sublinear time\footnote{Truly sublinear means $n^{1-\Omega(1)}$.} in terms of $\tmax$. Our main contribution is an algorithm whose amortized processing time\footnote{Since the runtimes are amortized, we do not use separate terms update time and query time for different operations and use processing time for all types of operations.} for each operation is truly sublinear in $\tmax$ when the number of operations is at least $\tmax^{2/3+\Omega(1)}$. We also show that when both element addition and element removal are allowed, there is no algorithm that can process each operation in time $\tmax^{1-\Omega(1)}$ on average unless \textsf{SETH}\footnote{The \textit{strong exponential time hypothesis} states that no algorithm can solve the satisfiability problem in time $2^{n(1-\Omega(1))}$.} fails.