Bartłomiej Bosek

On-line Chain Partitioning of Up-growing Interval Orders

On-line chain partitioning problem of on-line posets has been open for the past 20 years. The best known on-line algorithm uses

Shortest Augmenting Paths for Online Matchings on Trees

\frac{5^w-1}{4}

A subexponential upper bound for the on-line chain partitioning problem

chains to cover poset of width w. Felsner (Theor. Comput. Sci., 175(2):283–292, 1997) introduced a variant of this problem considering only up-growing posets, i.e. on-line posets in which each new point is maximal at the moment of its arrival. He presented an algorithm using

First-Fit Coloring of Incomparability Graphs

{\left( {\begin{array}{*{20}c} {{w + 1}} \\ {2} \\ \end{array} } \right)}

Variants of Online Chain Partition Problem of Posets

chains for width w posets and proved that his solution is optimal. In this paper, we study on-line chain partitioning of up-growing interval orders. We prove lower bound and upper bound to be 2w−1 for width w posets.

News about Semiantichains and Unichain Coverings

The shortest augmenting path (\(\textsc {Sap}\)) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp in 1972 have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although is has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph \(G=(W\uplus B, E)\) is being revealed online, i.e., in each round one vertex from \(B\) with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by \(\textsc {Sap}\) is \(O(n \log n)\). However, no better bound than \(O(n^2)\) is known even for trees. In this paper we prove an \(O(n \log ^2n)\) upper bound for the total length of augmenting paths for trees.

Forbidden structures for efficient First-Fit chain partitioning (extended abstract)

AbstractThe main question in the on-line chain partitioning problem is to decide whether there exists an on-line algorithm that partitions posets of width at most w into polynomial number of chains — see Trotter’s chapter Partially ordered sets in the Handbook of Combinatorics. So far the best known on-line algorithm of Kierstead used at most (5w − 1)/4 chains; on the other hand Szemerédi proved that any on-line algorithm requires at least

Shortest augmenting paths for online matchings on trees

\left( {\begin{array}{*{20}c} {w + 1} \\ 2 \\ \end{array} } \right)