On-line Chain Partitions of Up-growing Semi-orders

Felsner

Kloch

et al. 2011

Order

Self Cite

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents, point by point, a partially ordered set. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. There are many recent results about variants of the general on-line chain partition problem. With this survey we attempt to give an overview over

“…The result is shown in the next theorem. The proof can be found in an independent paper [10]. We now turn to the variants where the semi-order is presented together with a representation.…”

Section: Proposition 51 the Value Of The On-line Chain Partition Gammentioning

confidence: 99%

“…Felsner et al[10]) The value of the on-line chain partition game for up-growing semi-orders of width w is 1+√ 5 2 w .…”

mentioning

confidence: 99%

On-Line Chain Partitions of Orders: A Survey

Felsner

Kloch

et al. 2011

Order

Self Cite

“…Recently, Baier et al [7] resolved the case of up-growing interval orders, they proved val(w) = 2w − 1. For up-growing semi-orders, Felsner et al [8] proved a matching lower and upper bound of…”

Section: Introductionmentioning

confidence: 99%

On-Line Dimension of Semi-Orders

Kloch

Krawczyk

et al. 2012

Order

Self Cite

We analyze the on-line dimension of partially ordered sets as a value of a two-person game between Algorithm and Spoiler. The game is played in rounds. Spoiler presents an on-line order of width at most w, one point at a time. Algorithm maintains its realizer, i.e., the set of d linear extensions which intersect to the presented order. Algorithm may not change the ordering of the previously introduced elements in the existing linear extensions. The value of the game val(w) is the least d such that Algorithm has a strategy against Spoiler presenting any order of width at most w. For interval orders Hopkins showed that val(w) 4w − 4. We analyze the on-line dimension of semi-orders i.e., interval orders admitting a unit-length representation. For up-growing semi-orders of width w we prove a matching lower and upper bound of w. In the general (not necessarily up-growing) case we provide an upper bound of 2w.

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

Krawczyk

2014

Combinatorica

The 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 (5 w − 1)/4 chains; on the other hand Szemerédi proved that any on-line algorithm requires at least w+1 2 chains. These results were obtained in the early eighties and since then no progress in the general case has been done.We provide an on-line algorithm that partitions posets of width w into at most w 13 log w chains. This yields the first subexponential upper bound for the on-line chain partitioning problem.