Computational Complexity of Quadratic Unconstrained Binary Optimization

TL;DR

This paper classifies the computational complexity of the QUBO problem under various coefficient constraints, revealing its FP^NP-completeness and DP-completeness, and connects these results to quantum annealing capabilities.

Contribution

It provides a detailed complexity classification of QUBO problems with different coefficient restrictions and links these classifications to quantum annealing solutions.

Findings

QUBO with all integer coefficients is FP^NP-complete.

QUBO with bounded coefficients is FP^NP[log]-complete.

Decision version of QUBO with integer coefficients is DP-complete.

Abstract

In this paper, we study the computational complexity of the quadratic unconstrained binary optimization (QUBO) problem under the functional problem FP^NP categorization. We focus on four sub-classes: (1) When all coefficients are integers QUBO is FP^NP-complete. (2) When every coefficient is an integer lower bounded by a constant k, QUBO is FP^NP[log]-complete. (3) When every coefficient is an integer upper bounded by a constant k, QUBO is again FP^NP[log]-complete. (4) When coefficients can only be in the set {1, 0, -1}, QUBO is FP^NP[log]-complete. With recent results in quantum annealing able to solve QUBO problems efficiently, our results provide a clear connection between quantum annealing algorithms and the FP^NP complexity class categorization. We also study the computational complexity of the decision version of the QUBO problem with integer coefficients. We prove that this problem is DP-complete.