Subset Balancing and Generalized Subset Sum via Lattices

2026-04-06

AI Summary

This work addresses the efficient solvability of Subset Balancing—finding a nonzero integer coefficient vector orthogonal to a given target vector—and its generalization, Generalized Subset Sum, where the inner product equals a prescribed value, over integer lattices. By introducing the first reduction of Subset Balancing with bounded coefficients to a single instance of the shortest vector problem under the infinity norm (SVP∞), the authors surpass the time complexity barrier of traditional meet-in-the-middle approaches and extend the framework to arbitrary centrally symmetric convex constraints. They further reduce Generalized Subset Sum to the closest vector problem in the infinity norm (CVP∞), establishing a bounded-distance commitment in the average case. The resulting algorithms solve Subset Balancing deterministically in Õ(2^{4.632n}) time and randomly in Õ(2^{2.443n}) time, while Generalized Subset Sum admits a deterministic Õ(2^{6.217n})-time algorithm on average; both problems become polynomial-time solvable when the ambient dimension d is sufficiently large.

Abstract

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 $\tilde{O}((18\sqrt{2πe})^n) \approx \tilde{O}(2^{6.217n})$.

Problem

• Subset Balancing
• Generalized Subset Sum
• Lattice Problems
• SVP
• CVP

Innovation

• Subset Balancing
• Lattice Reduction
• SVP_infty
• CVP_infty
• Generalized Subset Sum