Abstract
Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ for $k\ge 5$. The list coloring problem asks the following more general question: given a list of available colors for each vertex in a graph, does it admit a proper coloring? We propose a quantum algorithm based on Grover search 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.
Highlights
We propose a simple Grover search-based approach to obtain a quadratic speedup on exhaustive search for the list coloring problem
APPLICATIONS AND CONCLUDING REMARKS The list coloring problem is ubiquitous in real life, as it generalizes an already well-appearing problem of graph coloring, but is applicable to several other scenarios, such as:
Suppose that radios in close proximity cannot operate on the same frequency due to interference
Summary
G RAPH coloring problems provide for a rich family of NP-complete problems in theoretical computer science. All of these algorithms use an oracle design which uses binary comparators, and provide solutions where almost all binary strings have positive probabilities of being selected, including those that do not represent valid colorings Our approach circumvents this problem via a modified initialization and diffusion operator that restricts the evolution of the quantum algorithm to the only v∈Lv |Lv| plausible states. Note that this is the total number of valid colorings when the underlying graph is empty. RESTRICTED SEARCH SPACE Let us consider a search space S {0, 1}n In this case, the algorithm is designed to only evolve over the states of S, and this is achieved via an initialization operator A such that. They do not modify their diffusion operator, leading to states outside the search space having positive probabilities of being measured
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