Simple vertex coloring algorithms
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Given a graph G with n vertices and maximum degree Δ, it is known that G admits a vertex coloring with Δ + 1 colors such that no edge of G is monochromatic. This can be seen constructively by a simple greedy algorithm, which runs in time O(nΔ). Very recently, a sequence of results (e.g., [Assadi et. al. SODA'19, Bera et. al. ICALP'20, Alon Assadi Approx/Random'20]) show randomized algorithms for (ϵ + 1)Δ-coloring in the query model making Õ(n√(n)) queries, improving over the greedy strategy on dense graphs. In addition, a lower bound of Ω(n√(n)) for any O(Δ)-coloring is established on general graphs. In this work, we give a simple algorithm for (1 + ϵ)Δ-coloring. This algorithm makes O(ϵ^-1/2n√(n)) queries, which matches the best existing algorithms as well as the classical lower bound for sufficiently large ϵ. Additionally, it can be readily adapted to a quantum query algorithm making Õ(ϵ^-1n^4/3) queries, bypassing the classical lower bound. Complementary to these algorithmic results, we show a quantum lower bound of Ω(n) for O(Δ)-coloring.
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