ONLINE COLORING CO-INTERVAL GRAPHS

We study the problem of online coloring co-interval graphs. In this problem, a set of intervals on the real line is presented to the algorithm one at a time, and upon receiving each interval I, the algorithm must assign I a color different from the colors of all previously presented intervals not intersecting I. The objective is to use as few colors as possible. It is known that the competitive ratio of the simple FirstFit algorithm on the class of co-interval graphs is at most 2. We show that for the class of unit co-interval graphs, where all intervals have equal length, the 2-bound on the competitive ratio of First-Fit is tight. On the other hand, we show that no deterministic online algorithm for coloring unit co-interval graphs can be better than 3/2-competitive. We then study the effect of randomization on our problem, and show a lower bound of 4/3 on the competitive ratio of any randomized algorithm for the unit co-interval coloring problem. We also prove that for the class of general co-interval graphs no randomized algorithm has competitive ratio better than 3/2.

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