Beating Meet-in-the-Middle for Subset Balancing Problems

2025-11-13

AI Summary

This paper studies the Subset Balancing problem: given an integer vector and a constant-size coefficient set (C), find a nonzero integer solution vector (x) such that the dot product of (x) and the input vector is zero. The problem unifies classical NP-hard problems including Subset Sum, Partition, and Equal Subset Sum. Addressing the long-standing worst-case time complexity bottleneck—the Meet-in-the-Middle lower bound of (O(|C|^{n/2}))—this work achieves the first worst-case improvement. We introduce a novel hybrid representation technique, extend the Howgrave-Graham-Joux framework, incorporate flexible input encoding and pseudosolution cancellation, and design an efficient compatible solution-pair recovery algorithm. Our main result is a deterministic (O(|C|^{(0.5-varepsilon)n})) time algorithm for some constant (varepsilon > 0). This yields exponential speedups for Equal Subset Sum and related problems, significantly surpassing prior state-of-the-art algorithms, which only achieve such bounds in average-case settings.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

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 $O(|C|^{(0.5 - epsilon)n})$ for a constant $epsilon>0$ 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 Wk{e}grzycki (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.

Problem

• Developing faster exact algorithms for Subset Balancing problems
• Breaking the Meet-in-the-Middle barrier for worst-case inputs
• Improving exponential time bounds for Equal Subset Sum

Innovation

• Applied representation technique from average-case to worst-case inputs
• Developed strategies for achieving good mixing with worst-case inputs
• Created flexible input representations for coefficient sets without zero