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Parameterized Quantum Query Algorithms for Graph Problems

Authors Tatsuya Terao , Ryuhei Mori

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Author Details

Tatsuya Terao
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Ryuhei Mori
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

  • Terao, Tatsuya: KAKENHI Grant Number JP24KJ1494.
  • Mori, Ryuhei: JST FOREST Program Grant Number JPMJFR216V and KAKENHI Grant Numbers JP20H04138, JP20H05966, and JP22H00522.

Acknowledgements

The authors thank Yusuke Kobayashi and Kazuhisa Makino for their insightful comments.

Cite AsGet BibTex

Tatsuya Terao and Ryuhei Mori. Parameterized Quantum Query Algorithms for Graph Problems. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 99:1-99:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.99

Abstract

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum query complexity
  • parameterized algorithms
  • vertex cover
  • matching
  • kernelization

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