Improved Approximation Schemes for (Un-)Bounded Subset-Sum and Partition

TL;DR

This paper advances approximation algorithms for the SUBSET SUM problem and its variants, focusing on weak approximation schemes that allow slight constraint violations, and improves their computational efficiency.

Contribution

The paper introduces improved approximation schemes for SUBSET SUM and its variants, surpassing previous time bounds for weak approximation algorithms.

Findings

New weak approximation schemes with faster running times

Improved algorithms for special cases like t = half of total sum

Enhanced understanding of the complexity bounds for approximate SUBSET SUM

Abstract

We consider the SUBSET SUM problem and its important variants in this paper. In the SUBSET SUM problem, a (multi-)set $X$ of $n$ positive numbers and a target number $t$ are given, and the task is to find a subset of $X$ with the maximal sum that does not exceed $t$. It is well known that this problem is NP-hard and admits fully polynomial-time approximation schemes (FPTASs). In recent years, it has been shown that there does not exist an FPTAS of running time $\tilde\OO( 1/\epsilon^{2-\delta})$ for arbitrary small $\delta>0$ assuming ($\min$,+)-convolution conjecture~\cite{bringmann2021fine}. However, the lower bound can be bypassed if we relax the constraint such that the task is to find a subset of $X$ that can slightly exceed the threshold $t$ by $\epsilon$ times, and the sum of numbers within the subset is at least $1-\tilde\OO(\epsilon)$ times the optimal objective value that respects the constraint. Approximation schemes that may violate the constraint are also known as weak approximation schemes. For the SUBSET SUM problem, there is a randomized weak approximation scheme running in time $\tilde\OO(n+ 1/\epsilon^{5/3})$ [Mucha et al.'19]. For the special case where the target $t$ is half of the summation of all input numbers, weak approximation schemes are equivalent to approximation schemes that do not violate the constraint, and the best-known algorithm runs in $\tilde\OO(n+1/\epsilon^{{3}/{2}})$ time [Bringmann and Nakos'21].