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 $1-δ$, 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 $ε$-approximation theory.

翻译：子集和问题是密码学与组合优化中的一个基本NP完全问题，具有广泛的实际应用。随机子集和问题（RSSP）是子集和问题更具普适性的版本，其中数字从独立同分布（i.i.d.）的输入分布中抽取。我们提出一种算法，能以概率$1-δ$构造与Da Cunha等人（2023）相同的$O(B/w)$网格，同时在整个过程中将元素修剪至$w$个，并具有$O(w\log w)$的运行时间复杂度。随后，我们提出一种新颖的束搜索启发式算法，该算法相对于列表大小$n$和束宽$w$以线性对数时间复杂度运行，利用网格在标准差相等的标准平均场假设下产生$O\!\left(\frac{B}{nw^2}\right)$的期望误差，证明了网格化在实现误差衰减方面的实际有效性。该算法对多种输入分布均具有经验鲁棒性，并能通过对评分启发式进行简单修改自然地扩展到变体问题，为鲁棒子集和误差衰减及$ε$-近似理论建立了新的实用基线。