A Grover Search-Based Algorithm for the List Coloring Problem
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Abstract
Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-coloring of graphs on <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> vertices has runtimes <inline-formula><tex-math notation="LaTeX">$\Omega (2^n)$</tex-math></inline-formula> for <inline-formula><tex-math notation="LaTeX">$k\geq 5$</tex-math></inline-formula>. The list coloring problem asks the following more general question: given a <italic>list</italic> of available colors for each vertex in a graph, does it admit a proper coloring? We propose a hybrid classical-quantum algorithm based on Grover search 12 to quadratically speed up exhaustive search. Our algorithm loses in complexity to classical ones in specific restricted cases, but improves exhaustive search for cases, where the lists and graphs considered are arbitrary in nature.