Lower Bounds for Subset Sum in Resolution with Modular Counting

TL;DR

This paper establishes exponential lower bounds on the size of refutations for unsatisfiable Subset Sum instances in a modular resolution proof system, linking complexity to error-correcting code properties.

Contribution

It introduces robustness-based criteria for Subset Sum instances and proves new exponential lower bounds for both dag-like and tree-like Res(lin) refutations.

Findings

Random instances are robust with high minimal distance

Lower bounds depend on the minimal distance of associated error-correcting codes

Specific algebraic geometry codes can achieve the robustness bounds

Abstract

In this paper we prove lower bounds for sizes of refutations of unsatisfiable vector Subset Sum instances $\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$ in the proof system Res(lin$_{\mathbb{F}_q}$) where $char(\mathbb{F}_{q})\geq 5$. As a basis for the hardness criterion for such instances we choose the property of the matrix $A$ with columns $(\overrightarrow{a}_1, \ldots, \overrightarrow{a}_n)$ to be (the transpose of) the generating matrix for a good error-correcting code $C_{A} := \{x\cdot A\, |\, x \in \mathbb{F}_{q}^k\}\subset \mathbb{F}_{q}^n$ and prove the following lower bounds: 1) For a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We introduce the notion of $(s,r)$-robustness for Subset Sum instances, which in particular implies that $A$ defines an error-correcting code with the minimal distance $s\geq r$. For $(s,r)$-robust instances we prove $2^{\Omega(r)}$ lower bound for sizes of refutations in a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We show that random instances are $(n / 3, \Omega\left((n/(q + 1)\ln q))^{1/3}\right))$-robust and that specific examples achieving these bounds can be constructed using algebraic geometry codes. 2) For tree-like Res(lin$_{\mathbb{F}_q}$) refutations we show the size lower bound $2^{\Omega({((q+1)\ln q)^{-1/3}}d^{1/5})}$ for any Subset Sum instance where $d$ is the minimal distance of $C_{A}$.