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