Inverse Quadratic Decay in Random Subset Sum

TL;DR

This paper introduces a new algorithm for the Random Subset Sum Problem that achieves efficient mesh construction and error decay, demonstrating practical robustness and establishing a new baseline for approximation.

Contribution

It presents a novel mesh-based algorithm and a beam search heuristic that improve subset sum approximation efficiency and robustness across various input distributions.

Findings

Algorithm constructs the same mesh as previous methods with trimmed elements in O(w log w) time.

Beam search heuristic achieves expected error decay of O(B / (n w^2)) in linearithmic time.

Empirically robust to multiple input distributions and extendable to variants with simple heuristic modifications.

Abstract

The Subset Sum Problem is a fundamental NP-complete problem in cryptography and combinatorial optimization, with many real-world applications. The Random Subset Sum Problem (RSSP) is a more applicable version of subset sum, where numbers are drawn from some i.i.d input distribution. We present an algorithm that, with probability $1-\delta$, constructs the same $O(B/w)$ mesh as Da Cunha et al. (2023), while trimming to $w$ elements throughout and running in $O(w\log w)$ time. Then, we present a novel beam search heuristic running in linearithmic time w.r.t list size $n$ and beam width $w$ using the mesh that gives an expected error of $O\!\left(\frac{B}{nw^2}\right)$ under a standard mean-field assumption with equal standard deviation, demonstrating the practical effectiveness of meshing to achieve error decay. The algorithm is empirically robust to multiple input distributions and can naturally extend to variants with simple changes to the scoring heuristic, establishing a new practical baseline for robust subset sum error decay and $\epsilon$-approximation theory.