Abstract The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time $$\varOmega (2^n)$$ Ω ( 2 n ) for $$k\ge 5$$ k ≥ 5 . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $$O(1.9140^n)$$ O ( 1 . 9140 n ) using quantum random access memory (QRAM). Our approach is based on Ambainis et al’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time $$O(1.9575^n)$$ O ( 1 . 9575 n ) . For the polynomial-space quantum algorithm, we essentially develop $$(4-\epsilon )^n$$ ( 4 - ϵ ) n -time classical algorithms that can be improved quadratically by Grover’s search.

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Exponential-Time Quantum Algorithms for Graph Coloring Problems

Kazuya Shimizu, Ryuhei Mori

Published Jan 1, 2020• 8 citations

Publisher

DOI: 10.1007/978-3-030-61792-9_31

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The fastest known classical algorithm deciding the

k

-colorability of

n

-vertex graph requires running time

Ω(2^n)

for

k\ge 5

. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time

O(1.9140^n)

using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph

20

-coloring problem with running time

O(1.9575^n)

. In the polynomial-space quantum algorithm, we essentially show

(4-ε)^n

-time classical algorithms that can be improved quadratically by Grover's search.

Abstract The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time

\varOmega (2^n)

Ω ( 2 n ) for

k\ge 5

k ≥ 5 . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time

O(1.9140^n)

O ( 1 . 9140 n ) using quantum random access memory (QRAM). Our approach is based on Ambainis et al’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time

O(1.9575^n)

O ( 1 . 9575 n ) . For the polynomial-space quantum algorithm, we essentially develop

(4-\epsilon )^n

( 4 - ϵ ) n -time classical algorithms that can be improved quadratically by Grover’s search.