Exponential-time quantum algorithms for graph coloring problems
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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> 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.
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