Certificate-Sensitive Subset Sum: Realizing Instance Complexity

TL;DR

This paper introduces a new instance-sensitive algorithm for Subset Sum that scales with the number of distinct subset sums and improves the worst-case deterministic runtime over classical bounds, linking complexity to certificate size.

Contribution

It presents the first deterministic algorithm whose runtime depends on the certificate size and achieves an unconditional runtime improvement over classical bounds.

Findings

Runtime scales with the number of distinct subset sums U

Deterministic solver improves over classical O*(2^{n/2}) bound

Provides structural insights into Subset Sum hardness regimes

Abstract

The Subset Sum problem is a classical NP-complete problem with a long-standing $O^*(2^{n/2})$ deterministic bound due to Horowitz and Sahni. We present results at two distinct levels of generality. First (instance-sensitive bound), we introduce, to our knowledge, the first deterministic algorithm whose runtime provably scales with the certificate size $U = |\Sigma(S)|$, the number of distinct subset sums. Our enumerator constructs all such sums in time $O(U \cdot n^2)$, with a randomized variant achieving expected time $O(U \cdot n)$. This provides a constructive link to Instance Complexity by tying runtime to the size of an information-theoretically minimal certificate. Second (unconditional worst-case bound), by combining this enumerator with a double meet-in-the-middle strategy and a Controlled Aliasing technique that enforces a simple canonical-normal-form (CNF) expansion policy on aliased states, we obtain a deterministic solver running in $O^*(2^{n/2-\varepsilon})$ time with $\varepsilon=\log_2(\frac{4}{3})$ - the first unconditional deterministic improvement over the classical $O^*(2^{n/2})$ bound for all sufficiently large $n$. Finally, we refine fine-grained hardness for Subset Sum by making explicit the structural regime (high collision entropy / near collision-free) implicitly assumed by SETH-based reductions, i.e., instances with near-maximal $U$.