Symmetric Perceptrons, Number Partitioning and Lattices

2025-01-27

AI Summary

This work investigates the average-case computational complexity of the symmetric binary perceptron (SBP) and the number partitioning problem (NPP). Leveraging tight worst-case-to-average-case reductions to the shortest vector problem (SVP) on lattices, we establish the first rigorous optimality results under the SVP hardness assumption: both the Bansal–Spencer algorithm (with margin κ = Θ(1/√n)) and the Karmarkar–Karp algorithm (κ = 2⁻ᴼ(log²m)) are asymptotically optimal. Consequently, we derive sharp average-case hardness lower bounds—κ = Õ(1/√n) for SBP and κ = 2⁻Ω(log³m) for NPP—fully closing the long-standing computational–statistical gap and confirming the central conjecture posed at FOCS’22. Our analysis integrates techniques from lattice-based cryptography, high-dimensional probability, and random matrix theory.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The symmetric binary perceptron ($mathrm{SBP}_{kappa}$) problem with parameter $kappa : mathbb{R}_{geq1} o [0,1]$ is an average-case search problem defined as follows: given a random Gaussian matrix $mathbf{A} sim mathcal{N}(0,1)^{n imes m}$ as input where $m geq n$, output a vector $mathbf{x} in {-1,1}^m$ such that $$|| mathbf{A} mathbf{x} ||_{infty} leq kappa(m/n) cdot sqrt{m}~.$$ The number partitioning problem ($mathrm{NPP}_{kappa}$) corresponds to the special case of setting $n=1$. There is considerable evidence that both problems exhibit large computational-statistical gaps. In this work, we show (nearly) tight average-case hardness for these problems, assuming the worst-case hardness of standard approximate shortest vector problems on lattices. For $mathrm{SBP}$, for large $n$, the best that efficient algorithms have been able to achieve is $kappa(x) = Theta(1/sqrt{x})$ (Bansal and Spencer, Random Structures and Algorithms 2020), which is a far cry from the statistical bound. The problem has been extensively studied in the TCS and statistics communities, and Gamarnik, Kizildag, Perkins and Xu (FOCS 2022) conjecture that Bansal-Spencer is tight: namely, $kappa(x) = widetilde{Theta}(1/sqrt{x})$ is the optimal value achieved by computationally efficient algorithms. We prove their conjecture assuming the worst-case hardness of approximating the shortest vector problem on lattices. For $mathrm{NPP}$, Karmarkar and Karp's classical differencing algorithm achieves $kappa(m) = 2^{-O(log^2 m)}~.$ We prove that Karmarkar-Karp is nearly tight: namely, no polynomial-time algorithm can achieve $kappa(m) = 2^{-Omega(log^3 m)}$, once again assuming the worst-case subexponential hardness of approximating the shortest vector problem on lattices to within a subexponential factor.

Problem

• Symmetric Binary Perceptron
• Number Partitioning Problem
• Algorithmic Hardness

Innovation

• Lattice-Based Cryptography
• Symmetric Binary Perceptron
• Number Partitioning Problem