Subset Balancing and Generalized Subset Sum via Lattices

TL;DR

This paper introduces lattice-based reductions for the Subset Balancing and Generalized Subset Sum problems, achieving improved deterministic and randomized algorithms with exponential time complexity, extending to convex bodies and average-case scenarios.

Contribution

It provides a novel reduction from Subset Balancing to SVP$_{ty}$, leading to faster algorithms and extending applicability to convex bodies and average-case instances.

Findings

Deterministic $ ilde{O}((6 ext{--}2.4)^n)$-time algorithm for Subset Balancing.

Randomized $ ilde{O}(2^{2.443n})$-time algorithm surpassing meet-in-the-middle for large $d$.

Efficient algorithms for Generalized Subset Sum via lattice reductions, with improved average-case performance.

Abstract

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 W\k{e}grzycki) 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 $\tilde{O}((6\sqrt{2\pi e})^n)$-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 $c \cdot x = \tau$. 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 $\tilde{O}((18\sqrt{2\pi e})^n)$.