We study the Subset Balancing problem: given $x \in \mathbb{Z}^n$ and a coefficient set $C \subseteq \mathbb{Z}$, find a nonzero vector $c \in C^n$ such that $c\cdot x = 0$. The standard meet-in-the-middle algorithm runs in time $\tilde{O}(|C|^{n/2})$, and recent improvements (SODA 2022, Chen, Jin, Randolph, and Servedio; STOC 2026, Randolph and Węgrzycki) beyond this barrier apply mainly when $d$ is constant. We give a reduction from Subset Balancing with $C = \{-d, \dots, d\}$ to a single instance of SVP$_{\infty}$ in dimension $n+1$. Instantiating this reduction with the best known $\ell_\infty$-SVP algorithms yields a deterministic $\tilde{O}((6\sqrt{2πe})^n)$-time algorithm and a randomized $\tilde{O}(2^{2.443n})$-time algorithm. The exponent depends only on $n$, improving on meet-in-the-middle for all $d\ge 15$. For sufficiently large $d$ we also obtain a polynomial-time algorithm. The reduction extends from the box constraint $[-d,d]^n$ to any centrally symmetric convex body $K\subseteq\mathbb{R}^n$, giving deterministic time $\tilde{O}(2^{c_K n})$ for a constant $c_K$ depending only on the shape of $K$. We further study the Generalized Subset Sum problem of finding $c \in C^n$ such that $c \cdot x = τ$. For $C = \{-d, \dots, d\}$ or $C = \{-d,\dots,d\}\setminus\{0\}$, we reduce the worst-case problem to CVP$_{\infty}$ in dimension $n+1$. We observe that distances in our lattice take only integer values, so an approximate CVP$_{\infty}$ oracle still suffices, yielding a deterministic worst-case algorithm running in time $2^{O(n\log\log d)}$. In the average-case setting, we demonstrate that for both coefficient sets the embedded CVP$_{\infty}$ instance satisfies a bounded-distance promise with high probability, removing the $\log\log d$ factor altogether and obtaining a deterministic algorithm running in time $\tilde{O}((18\sqrt{2πe})^n)$.

翻译：我们研究了子集平衡问题：给定 $x \in \mathbb{Z}^n$ 和系数集合 $C \subseteq \mathbb{Z}$，寻找非零向量 $c \in C^n$ 使得 $c\cdot x = 0$。标准的中间相遇算法运行时间为 $\tilde{O}(|C|^{n/2})$，而近期突破该障碍的改进（SODA 2022, Chen, Jin, Randolph, and Servedio; STOC 2026, Randolph and Węgrzycki）主要适用于 $d$ 为常数的情形。我们给出了从 $C = \{-d, \dots, d\}$ 的子集平衡问题到 $n+1$ 维空间中单个 SVP$_{\infty}$ 实例的归约。将该归约与目前已知的最优 $\ell_\infty$-SVP 算法相结合，可得到一个确定性的 $\tilde{O}((6\sqrt{2πe})^n)$ 时间算法和一个随机化的 $\tilde{O}(2^{2.443n})$ 时间算法。该指数仅与 $n$ 相关，对于所有 $d\ge 15$ 均优于中间相遇算法。对于足够大的 $d$，我们还可得到多项式时间算法。该归约从盒约束 $[-d,d]^n$ 推广至任意中心对称凸体 $K\subseteq\mathbb{R}^n$，给出确定性时间 $\tilde{O}(2^{c_K n})$，其中常数 $c_K$ 仅取决于 $K$ 的形状。我们进一步研究了广义子集和问题：寻找 $c \in C^n$ 使得 $c \cdot x = τ$。对于 $C = \{-d, \dots, d\}$ 或 $C = \{-d,\dots,d\}\setminus\{0\}$，我们将最坏情况问题归约到 $n+1$ 维空间中的 CVP$_{\infty}$。我们观察到格距离仅取整数值，因此近似 CVP$_{\infty}$ 预言机仍然足够，从而得到一个运行时间为 $2^{O(n\log\log d)}$ 的确定性最坏情况算法。在平均情况设定下，我们证明对于两种系数集合，嵌入的 CVP$_{\infty}$ 实例以高概率满足有界距离承诺，从而完全去除了 $\log\log d$ 因子，并得到一个运行时间为 $\tilde{O}((18\sqrt{2πe})^n)$ 的确定性算法。