License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.28
URN: urn:nbn:de:0030-drops-143026
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14302/
Apers, Simon ;
Lee, Troy
Quantum Complexity of Minimum Cut
Abstract
The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using Õ(n^{3/2}√{τ}) queries and time. Moreover, for every integer 1 ≤ τ ≤ n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires Ω(n^{3/2}√{τ}) queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n²) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not.
In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is ̃ Θ(m). We show that the quantum query and time complexity are Õ(√{mnτ}) and Õ(√{mnτ} + n^{3/2}), respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n^{3/2}) for τ > 1 and Ω(τ n) for any 1 ≤ τ ≤ n.
Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger’s tree packing technique (STOC 1996).
BibTeX - Entry
@InProceedings{apers_et_al:LIPIcs.CCC.2021.28,
author = {Apers, Simon and Lee, Troy},
title = {{Quantum Complexity of Minimum Cut}},
booktitle = {36th Computational Complexity Conference (CCC 2021)},
pages = {28:1--28:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-193-1},
ISSN = {1868-8969},
year = {2021},
volume = {200},
editor = {Kabanets, Valentine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14302},
URN = {urn:nbn:de:0030-drops-143026},
doi = {10.4230/LIPIcs.CCC.2021.28},
annote = {Keywords: Quantum algorithms, quantum query complexity, minimum cut}
}
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Keywords: |
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Quantum algorithms, quantum query complexity, minimum cut |
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Collection: |
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36th Computational Complexity Conference (CCC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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08.07.2021 |
