Maximum Independent Set

The Maximum Independent Set (MIS) problem is a well-known optimization problem in graph theory and computer science. Given an undirected graph, an independent set is a subset of vertices in which no two vertices are adjacent. The MIS problem seeks to find the largest such set, i.e., the independent set of maximum size.

The MIS problem is NP-hard, meaning that there is no known algorithm that can solve all instances of the problem in polynomial time (unless P = NP). It has applications in various fields, including network design, scheduling, resource allocation, and more. The complexity of the problem makes it a significant subject of study in both classical and quantum computing.

Quantum computing offers new avenues for tackling the MIS problem. Quantum algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA), have been applied to find approximate solutions to the MIS problem. While finding the exact maximum independent set remains a complex task, quantum algorithms may provide advantages in terms of speed or quality of approximations for specific instances of the problem.

The MIS problem is related to other combinatorial optimization problems, such as the Minimum Vertex Cover and Maximum Clique problems. Variants of the MIS problem, with additional constraints or objectives, are also studied, and the techniques developed for solving the MIS problem can often be adapted to these related problems.

Solving the MIS problem, especially for large and complex graphs, remains a challenging task. Research continues to explore new algorithms, heuristics, and quantum approaches to address the problem. Understanding the structure and properties of specific graphs can lead to more efficient solutions, and the interplay between classical and quantum methods is an area of ongoing exploration.

The Maximum Independent Set problem is a fundamental and challenging problem in computer science. Its complexity and wide-ranging applications make it a subject of interest in both classical and quantum computing. Quantum approaches to the MIS problem represent an exciting frontier, with potential to enhance our ability to solve this and related optimization problems.

This webinar demonstrates solving MIS and other optimization problems using the QuEra computer.

This application note provides an example of an MIS problem solved by quantum computers.