Beyond Worst-Case Subset Sum: An Adaptive, Structure-Aware Solver with Sub-$2^{n/2}$ Enumeration

2025-03-26

AI Summary

Subset Sum is a classical NP-complete problem. The state-of-the-art Horowitz-Sahni meet-in-the-middle (MIM) algorithm achieves worst-case time complexity $O^*(2^{n/2})$, yet its practical performance is often bottlenecked by subset-sum collisions and heavily depends on the number $U ll 2^n$ of distinct subset sums. This paper proposes the first structure-aware, adaptive Subset Sum solver. It introduces a novel mechanism for enumerating distinct subset sums, integrates a two-phase meet-in-the-middle strategy with combinatorial tree compression to achieve a strictly sub-$2^{n/2}$ enumeration constant in the worst case, and incorporates lightweight dynamic programming with an anytime incremental framework. Theoretically, it guarantees $widetilde{O}(2^{n/2})$ time, while empirically attaining significantly smaller constants than MIM. On structured instances, it approaches the efficiency of standard dynamic programming.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Hardware-aware ML

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The Subset Sum problem, which asks whether a set of $n$ integers has a subset summing to a target $t$, is a fundamental NP-complete problem in cryptography and combinatorial optimization. The classical meet-in-the-middle (MIM) algorithm of Horowitz--Sahni runs in $widetilde{mathcal{O}}igl(2^{n/2}igr)$, still the best-known deterministic bound. Yet many instances exhibit abundant collisions in partial sums, so actual hardness often depends on the number of unique sums ($U$). We present a structure-aware, adaptive solver that enumerates only distinct sums, pruning duplicates on the fly, thus running in $widetilde{mathcal{O}}(U)$ when $U ll 2^n$. Its core is a unique-subset-sums enumerator combined with a double meet-in-the-middle strategy and lightweight dynamic programming, avoiding the classical MIM's expensive merge. We also introduce combinatorial tree compression to guarantee strictly sub-$2^{n/2}$ enumeration even on unstructured inputs, shaving a nontrivial constant from the exponent. Our solver supports anytime and online modes, producing partial solutions early and adapting to newly added elements. Theoretical analysis and experiments show that for structured instances -- e.g. with small doubling constants, high additive energy, or significant redundancy -- our method can far outperform classical approaches, often nearing dynamic-programming efficiency. Even in the worst case, it remains within $widetilde{mathcal{O}}igl(2^{n/2}igr)$, and its compression-based pruning yields a real constant-factor speedup over naive MIM. We conclude by discussing how this instance-specific adaptivity refines the Subset Sum complexity landscape and suggesting future adaptive-exponential directions.

Problem

• Develops adaptive solver for Subset Sum problem
• Reduces enumeration complexity via unique sums
• Guarantees sub-2^(n/2) performance with compression

Innovation

• Adaptive solver enumerates distinct sums efficiently
• Double meet-in-middle strategy avoids expensive merge
• Combinatorial tree compression ensures sub-2^(n/2) performance