Online Dual Edge Coloring of Paths and Trees
We study a dual version of online edge coloring, where the goal is to color as many edges as possible using only a given number, \(k\), of available colors. All of our results are with regard to competitive analysis. For paths, we consider \(k=2\), and for trees, we consider any \(k \ge 2\). We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms on paths as well as trees. This is the first time that an optimal algorithm for online dual edge coloring has been identified for a class of graphs. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. Again, it is the first time that this has been done for a class of graphs. For trees, we also show that even randomized algorithms cannot be much better than First-Fit.
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