Randomized Lower Bounds for Online Path Coloring

We study the power of randomization in the design of on-line graph coloring algorithms. No specific network topology for which randomized online algorithms perform substantially better than deterministic algorithms is known until now. We present randomized lower bounds for online coloring of some well studied network topologies. We show that no randomized algorithm for online coloring of interval graphs achieves a competitive ratio strictly better than the best known deterministic algorithm [KT81]. We also present a first lower bound on the competitive ratio of randomized algorithms for path coloring on tree networks, then answering an open question posed in [BEY98]. We prove an Ω(log Δ) lower bound for trees of diameter Δ = O(log n) that compares with the known O(Δ)- competitive deterministic algorithm for the problem, then still leaving open the question if randomization helps for this specific topology.

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