Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w into w14lgw 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 1 ≤ ⋯ ≤ x m, y 1 ≤ ⋯ ≤ y m and the set of comparabilities Zx 1 ≤ y 1, x m ≤ 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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics