Local-Minima-Preserving Continuous Relaxation of Ising Problems

2026-06-29

AI Summary

This work addresses combinatorial optimization problems arising in generalized Ising models—such as MAX-CUT, number partitioning, and maximum independent set—by introducing a polynomial continuous relaxation that preserves the structure of local minima. The core contribution is the establishment of a “landscape equivalence theorem,” which rigorously proves, for the first time, a one-to-one correspondence between the local minima of the relaxed smooth objective function and the single-spin-flip local minima of the original discrete problem. By leveraging gradient-based optimizers like ADAM, the proposed method demonstrates strong empirical performance and favorable scalability on benchmark instances including spin glasses, MAX-CUT, and number partitioning problems.

Abstract

The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove the landscape equivalence theorem: there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage gradient-based optimizers such as ADAM. We demonstrate that our method is scalable and it achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.

Problem

• Ising problem
• local minima
• continuous relaxation
• combinatorial optimization
• one-flip minima

Innovation

• Ising problem
• continuous relaxation
• local minima preservation
• gradient-based optimization
• combinatorial optimization