Vertex coloring of graphs via phase dynamics of coupled oscillatory networks

TL;DR

This paper introduces a physics-inspired analog computing method using coupled oscillators and VO2 phase transitions to efficiently approximate solutions for the vertex coloring problem in graphs, a classically hard combinatorial optimization task.

Contribution

It presents a novel dynamical system leveraging VO2 insulator-metal transitions for graph coloring, connecting physical energy minimization with spectral algorithms.

Findings

Demonstrates efficient approximation of graph coloring using oscillator networks.

Establishes a link between dynamical system eigenproperties and spectral algorithms.

Proposes a physics-based approach for solving hard combinatorial problems.

Abstract

While Boolean logic has been the backbone of digital information processing, there are classes of computationally hard problems wherein this conventional paradigm is fundamentally inefficient. Vertex coloring of graphs, belonging to the class of combinatorial optimization represents such a problem; and is well studied for its wide spectrum of applications in data sciences, life sciences, social sciences and engineering and technology. This motivates alternate, and more efficient non-Boolean pathways to their solution. Here, we demonstrate a coupled relaxation oscillator based dynamical system that exploits the insulator-metal transition in vanadium dioxide (VO2), to efficiently solve the vertex coloring of graphs. By harnessing the natural analogue between optimization, pertinent to graph coloring solutions, and energy minimization processes in highly parallel, interconnected dynamical systems, we harness the physical manifestation of the latter process to approximate the optimal coloring of k-partite graphs. We further indicate a fundamental connection between the eigen properties of a linear dynamical system and the spectral algorithms that can solve approximate graph coloring. Our work not only elucidates a physics-based computing approach but also presents tantalizing opportunities for building customized analog co-processors for solving hard problems efficiently.