Quantum speedup of branch-and-bound algorithms

Ashley Montanaro

doi:10.1103/physrevresearch.2.013056

Abstract

Branch-and-bound is a widely used technique for solving combinatorial optimisation problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that determines a lower bound on the cost of any solution in a given subset. Here we describe a quantum algorithm that can accelerate classical branch-and-bound algorithms near-quadratically in a very general setting. We show that the quantum algorithm can find exact ground states for most instances of the Sherrington-Kirkpatrick model in time $O(2^{0.226n})$, which is substantially more efficient than Grover's algorithm.

Highlights

Branch-and-bound is a widely used technique for solving combinatorial optimization problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that determines a lower bound on the cost of any solution in a given subset
This approach can be applied to problems where the goal is to find a minimal-cost valid solution, in a setting where one has access to two functions: a bounding function Cost that, for a given subset of the set of possible solutions, returns a lower bound on the cost of any valid solution in that subset, and a branching rule Branch to be applied if a subset of possible solutions cannot yet be ruled out, which will divide that subset into two or more “live” subsets to be explored in later iterations
The constant 0.01 could be made arbitrarily small. This rigorous result is only an upper bound on the tree size of the classical algorithm, which may not be tight

Summary

Introduction

Branch-and-bound is a widely used technique for solving combinatorial optimization problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that determines a lower bound on the cost of any solution in a given subset. Assuming that Tmin poly(d ), Tmin log cmax, this is roughly a quadratic speedup over any possible classical branch-and-bound search algorithm which finds all minimalcost solutions in the tree corresponding to A, whose complexity (as discussed above) is lower-bounded by Tmin.

Results

Conclusion