Quantum algorithms for graph problems with cut queries

@inproceedings{Lee2021QuantumAF,
  title={Quantum algorithms for graph problems with cut queries},
  author={Troy Lee and Miklos Santha and Shengyu Zhang},
  booktitle={SODA},
  year={2021}
}
Let $G$ be an $n$-vertex graph with $m$ edges. When asked a subset $S$ of vertices, a cut query on $G$ returns the number of edges of $G$ that have exactly one endpoint in $S$. We show that there is a bounded-error quantum algorithm that determines all connected components of $G$ after making $O(\log(n)^6)$ many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least… 

Figures from this paper

Cut query algorithms with star contraction

A new technique is introduced, called star contraction, to randomly contract edges of a graph while preserving nontrivial minimum cuts, and a one-pass semi-streaming algorithm is designed for computing edge connectivity in the complete vertex arrival setting.

Quantum algorithms for learning graphs and beyond

We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a

Quantum Algorithms for Learning a Hidden Graph

This work studies the problem of learning an unknown graph provided via an oracle using a quantum algorithm, and gives quantum algorithms in the graph state model whose complexity is similar to the parity query model.

On the query complexity of connectivity with global queries

This paper gives a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after O(log(n) matrix-vector multiplication queries to the adjacency matrix, and shows the first lower bound of any kind on the unrestricted linear query complexity of connectivity.

Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving

  • Simon ApersR. D. Wolf
  • Computer Science
    2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
  • 2020
A quantum algorithm is given that implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut and is proved that this is tight up to polylog-factors.

Simple vertex coloring algorithms

A simple algorithm for (1+ )∆-coloring, which makes O( −1/2n √ n) queries, which matches the best existing algorithms as well as the classical lower bound for sufficiently large .

Quantum query complexity with matrix-vector products

It is shown that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation, but for some problems, such as computing the parities of rows or columns, quantum machines provide exponential speedup.

References

SHOWING 1-10 OF 43 REFERENCES

Improved Quantum Query Algorithms for Triangle Detection and Associativity Testing

This work gives a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored, and shows how this high-level language can be compiled as a learning graph and gives the resulting complexity.

Computing Exact Minimum Cuts Without Knowing the Graph

This work provides query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs, inspired by the submodular function minimization problem.

Quantum Query Complexity of Minor-Closed Graph Properties

It is shown that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity $\Theta(n^{3/2})$, and an adversary lower bound is proved.

Minimizing Convex Functions with Integral Minimizers

This work shows how an approximately shortest vector of certain lattice can be used to reduce the dimension of the problem, and how the oracle complexity of such a procedure is advantageous compared with the Grotschel-Lovasz-Schrijver approach that uses simultaneous diophantine approximation.

Optimal query complexity bounds for finding graphs

It is proved that there exists a non-adaptive algorithm to find the edges of G using O(m log n / log m) queries of both types provided that m ≥ nε for any constant ε > 0 and it is shown that the same bound holds for all range of m.

New bounds on the classical and quantum communication complexity of some graph properties

The main results are an Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication.

Quantum Query Complexity of Some Graph Problems

Almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, StrongConnectivity, Minimum Spanning Tree, and Single Source Shortest Paths are given.

Optimally reconstructing weighted graphs using queries

The first polynomial time algorithm with query complexity that matches the information theoretic lower bound for the query complexity of reconstructing a graph with n vertices and m edges is given.

Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

  • F. Gall
  • Computer Science
    2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
This paper presents a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(n5/4), where n denotes the number of vertices in the graph, and shows, for the first time, that in the quantum query complexity setting un Weighted triangle finding is easier than its edge-weighted version.