On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width w into w^O(loglogw) chains. This improves over previously best known algorithms using w^O(logw) chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in w^O(√(w))n time, where w is the width and n is the size of a presented poset.