On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w into w^1^4^l^g^w chains. They also observed that the problem of on-line chain partitioning of general posets of width w could be reduced to First-Fit chain partitioning of 2w^2+1-ladder-free posets of width w, where an m-ladder is the transitive closure of the union of two incomparable chains x"[email protected][email protected]?x"m, y"[email protected][email protected]?y"m and the set of comparabilities {x"[email protected]?y"1,...,x"[email protected]?y"m}. Here, we provide a subexponential upper bound (in terms of w with m fixed) for the performance of First-Fit chain partitioning on m-ladder-free posets, as well as an exact quadratic bound when m=2, and an upper bound linear in m when w=2. Using the Bosek-Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.

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