An Improved Subexponential Bound for On-line Chain Partitioning

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w into w lgw 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 + 1)-ladder-free posets of width w, where an m-ladder is the transitive closure of the union of two incomparable chains x1 6 . . . 6 xm, y1 6 . . . 6 ym and the set of comparabilities {x1 6 y1, . . . , xm 6 ym}. Here, we improve the subexponential upper bound to w lgw+O(1) with a simplified proof, exploiting the First-Fit algorithm on ladder-free posets.