Quantum algorithm speeds up maximum-weight perfect matching in graphs
A Quantum Scaling Algorithm for Maximum-Weight Perfect Matching in General Graphs
Data Structures and Algorithms
Summary
The problem of finding the best perfect matching in a network with weighted connections is very important but challenging, especially for general graphs. The authors present a quantum algorithm that solves this problem faster than all known classical methods in certain cases. Their approach builds on a classical method but replaces key steps with quantum techniques and special data structures. This work demonstrates the first known quantum speed-up for this difficult matching problem.
What this means in practice
- •For network schedulers: Optimize pairing tasks in large communication or transportation networks faster by using quantum-accelerated weighted perfect matching algorithms.
- •For quantum algorithm engineers: Develop more efficient quantum software tools for solving combinatorial optimization problems involving weighted matchings in graphs.
Authors
Kourosh Mirsohi, Sandy Irani, Michael T. Goodrich
Abstract
Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in \( \widetilde{O}(n m^{2/3}\log W) \) time, where $W$ is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of \( \widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.