From the Finite to the Infinite: Sharper Asymptotic Bounds on Norin's Conjecture via SAT

2025-11-11

AI Summary

This paper investigates Norin’s conjecture: in any 2-edge-coloring of the hypercube $Q_n$ where each edge differs in color from its antipodal edge, there must exist a monochromatic path connecting some pair of antipodal vertices. To tackle this combinatorial problem, we devise a compact SAT encoding integrating symmetry-breaking techniques with a parallel cube-and-conquer framework, enabling the first computational verification of the conjecture for $n = 8$. Theoretically, we substantially improve the asymptotic upper bound on the number of color changes required along a monochromatic path—from $0.375n + o(n)$ to $0.3125n + O(1)$. These two advances—extending the verified range of $n$ by one beyond prior work and tightening the asymptotic bound—constitute significant progress toward resolving Norin’s conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

Norin (2008) conjectured that any $2$-edge-coloring of the hypercube $Q_n$ in which antipodal edges receive different colors must contain a monochromatic path between some pair of antipodal vertices. While the general conjecture remains elusive, progress thus far has been made on two fronts: finite cases and asymptotic relaxations. The best finite results are due to Frankston and Scheinerman (2024) who verified the conjecture for $n leq 7$ using SAT solvers, and the best asymptotic result is due to Dvov{r}'ak (2020), who showed that every $2$-edge-coloring of $Q_n$ admits an antipodal path of length $n$ with at most $0.375n + o(n)$ color changes. We improve on both fronts via SAT. First, we extend the verification to $n = 8$ by introducing a more compact and efficient SAT encoding, enhanced with symmetry breaking and cube-and-conquer parallelism. The versatility of this new encoding allows us to recast parts of Dvov{r}'ak's asymptotic approach as a SAT problem, thereby improving the asymptotic upper bound to $0.3125n + O(1)$ color changes. Our work demonstrates how SAT-based methods can yield not only finite-case confirmations but also asymptotic progress on combinatorial conjectures.

Problem

• Verifying Norin's conjecture for hypercube Q8 using SAT solvers
• Improving asymptotic bound on antipodal path color changes
• Developing compact SAT encoding with symmetry breaking techniques

Innovation

• SAT encoding with symmetry breaking
• Cube-and-conquer parallelism for efficiency
• SAT recasting for asymptotic bound improvement