TY - JOUR
T1 - On first-fit coloring of ladder-free posets
AU - Kierstead, Henry
AU - Smith, Matt Earl
N1 - Funding Information:
The research of both authors is supported in part by NSF grant DMS-0901520 .
PY - 2013/2
Y1 - 2013/2
N2 - 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.
AB - 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.
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U2 - 10.1016/j.ejc.2012.07.007
DO - 10.1016/j.ejc.2012.07.007
M3 - Article
AN - SCOPUS:84867298612
SN - 0195-6698
VL - 34
SP - 474
EP - 489
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
IS - 2
ER -