We consider exact algorithms for Subset Balancing, a family of related problems that generalizes Subset Sum, Partition, and Equal Subset Sum. Specifically, given as input an integer vector $\vec{x} \in \mathbb{Z}^n$ and a constant-size coefficient set $C \subset \mathbb{Z}$, we seek a nonzero solution vector $\vec{c} \in C^n$ satisfying $\vec{c} \cdot \vec{x} = 0$. For $C = \{-d,\ldots,d\}$, $d > 1$ and $C = \{-d,\ldots,d\}\setminus\{0\}$, $d > 2$, we present algorithms that run in time $O(|C|^{(0.5 - ε)n})$ for a constant $ε> 0$ that depends only on $C$. These are the first algorithms that break the $O(|C|^{n/2})$-time ``Meet-in-the-Middle barrier'' for these coefficient sets in the worst case. This improves on the result of Chen, Jin, Randolph and Servedio (SODA 2022), who broke the Meet-in-the-Middle barrier on these coefficient sets in the average-case setting. We also improve the best exact algorithm for Equal Subset Sum (Subset Balancing with $C = \{-1,0,1\}$), due to Mucha, Nederlof, Pawlewicz, and Węgrzycki (ESA 2019), by an exponential margin. This positively answers an open question of Jin, Williams, and Zhang (ESA 2025). Our results leave two natural cases in which we cannot yet break the Meet-in-the-Middle barrier: $C = \{-2, -1, 1, 2\}$ and $C = \{-1, 1\}$ (Partition). Our results bring the representation technique of Howgrave-Graham and Joux (CRYPTO 2010) from average-case to worst-case inputs for many $C$. This requires a variety of new techniques: we present strategies for (1) achieving good ``mixing'' with worst-case inputs, (2) creating flexible input representations for coefficient sets without 0, and (3) quickly recovering compatible solution pairs from sets of vectors containing ``pseudosolution pairs''. These techniques may find application to other algorithmic problems on integer sums or be of independent interest.

翻译：我们研究子集平衡问题的精确算法，该问题族推广了子集和、划分及等子集和问题。具体而言，给定整数向量 $\vec{x} \in \mathbb{Z}^n$ 和常数规模系数集 $C \subset \mathbb{Z}$ 作为输入，我们寻求满足 $\vec{c} \cdot \vec{x} = 0$ 的非零解向量 $\vec{c} \in C^n$。对于 $C = \{-d,\ldots,d\}$（$d > 1$）及 $C = \{-d,\ldots,d\}\setminus\{0\}$（$d > 2$）的情形，我们提出了在 $O(|C|^{(0.5 - ε)n})$ 时间内运行的算法，其中常数 $ε> 0$ 仅依赖于 $C$。这些是首批在最坏情况下突破 $O(|C|^{n/2})$ 时间“中间相遇算法界限”的算法。该结果改进了 Chen、Jin、Randolph 和 Servedio（SODA 2022）在平均情况下突破这些系数集中间相遇界限的研究成果。我们还以指数级优势改进了由 Mucha、Nederlof、Pawlewicz 和 Węgrzycki（ESA 2019）提出的等子集和问题（即 $C = \{-1,0,1\}$ 的子集平衡问题）的最佳精确算法，这正面回答了 Jin、Williams 和 Zhang（ESA 2025）提出的开放性问题。我们的研究仍存在两个尚未突破中间相遇界限的自然情形：$C = \{-2, -1, 1, 2\}$ 和 $C = \{-1, 1\}$（划分问题）。本成果将 Howgrave-Graham 和 Joux（CRYPTO 2010）的表征技术从平均情况推广到多种 $C$ 的最坏情况输入，这需要一系列新技术：我们提出了（1）在最坏情况输入下实现良好“混合”的策略，（2）为不含 0 的系数集创建灵活输入表征的方法，以及（3）从包含“伪解对”的向量集合中快速恢复兼容解对的方案。这些技术可能应用于其他整数和相关的算法问题，或具有独立的研究价值。