We study the \emph{Subset Balancing} problem: given $\mathbf{x} \in \mathbb{Z}^n$ and a coefficient set $C \subseteq \mathbb{Z}$, find a nonzero vector $\mathbf{c} \in C^n$ such that $\mathbf{c}\cdot\mathbf{x} = 0$. The standard meet-in-the-middle algorithm runs in time $\tilde{O}(|C|^{n/2})=\tilde{O}(2^{n\log |C|/2})$, 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 $\mathrm{SVP}_{\infty}$ in dimension $n+1$, which yields a deterministic algorithm with running time $\tilde{O}((6\sqrt{2πe})^n) \approx \tilde{O}(2^{4.632n})$, and a randomized algorithm with running time $\tilde{O}(2^{2.443n})$ (here $\tilde{O}$ suppresses $\operatorname{poly}(n)$ factors). We also show that for sufficiently large $d$, Subset Balancing is solvable in polynomial time. More generally, we extend the box constraint $[-d,d]^n$ to an arbitrary centrally symmetric convex body $K \subseteq \mathbb{R}^n$ with a deterministic $\tilde{O}(2^{c_K n})$-time algorithm, where $c_K$ depends only on the shape of $K$. We further study the \emph{Generalized Subset Sum} problem of finding $\mathbf{c} \in C^n$ such that $\mathbf{c} \cdot \mathbf{x} = τ$. For $C = \{-d, \dots, d\}$, we reduce the worst-case problem to a single instance of $\mathrm{CVP}_{\infty}$. Although no general single exponential time algorithm is known for exact $\mathrm{CVP}_{\infty}$, we show that in the average-case setting, for both $C = \{-d, \dots, d\}$ and $C = \{-d, \dots, d\} \setminus \{0\}$, the embedded instance satisfies a bounded-distance promise with high probability. This yields a deterministic algorithm running in time $\tilde{O}((18\sqrt{2πe})^n) \approx \tilde{O}(2^{6.217n})$.

翻译：我们研究\emph{子集平衡}问题：给定$\mathbf{x} \in \mathbb{Z}^n$和一个系数集合$C \subseteq \mathbb{Z}$，寻找非零向量$\mathbf{c} \in C^n$使得$\mathbf{c}\cdot\mathbf{x} = 0$。标准的中点相遇算法运行时间为$\tilde{O}(|C|^{n/2})=\tilde{O}(2^{n\log |C|/2})$，而近期（SODA~2022, Chen, Jin, Randolph, and Servedio; STOC~2026, Randolph and Węgrzycki）在此障碍之上的改进主要适用于$d$为常数的情况。我们给出从$C = \{-d, \dots, d\}$的子集平衡问题到单一$\mathrm{SVP}_{\infty}$实例（维度为$n+1$）的归约，由此产生一个运行时间为$\tilde{O}((6\sqrt{2πe})^n) \approx \tilde{O}(2^{4.632n})$的确定性算法，以及一个运行时间为$\tilde{O}(2^{2.443n})$的随机化算法（此处$\tilde{O}$省略$\operatorname{poly}(n)$因子）。我们还证明对于充分大的$d$，子集平衡问题可在多项式时间内求解。更一般地，我们将盒子约束$[-d,d]^n$推广到任意中心对称凸体$K \subseteq \mathbb{R}^n$，并给出一个运行时间为$\tilde{O}(2^{c_K n})$的确定性算法，其中$c_K$仅取决于$K$的形状。我们进一步研究\emph{广义子集和}问题，即寻找$\mathbf{c} \in C^n$使得$\mathbf{c} \cdot \mathbf{x} = τ$。对于$C = \{-d, \dots, d\}$，我们将最坏情况问题归约到单一$\mathrm{CVP}_{\infty}$实例。尽管已知无针对精确$\mathrm{CVP}_{\infty}$的通用单指数时间算法，我们证明在平均情形设置下，对于$C = \{-d, \dots, d\}$和$C = \{-d, \dots, d\} \setminus \{0\}$，嵌入的实例以高概率满足有界距离承诺。由此产生一个运行时间为$\tilde{O}((18\sqrt{2πe})^n) \approx \tilde{O}(2^{6.217n})$的确定性算法。