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

On-line chain partitions of an order can be described as a two-person game between Algorithm and Spoiler. The game is played in rounds. Spoiler presents an on-line order, one point at a time. Algorithm responds by making an irrevocable assignment of the new point to one of the chains of the chain partition. The performance of Algorithm's strategy is measured by comparing the number of chains used with the number of chains of an optimal chain partition. By Dilworth's Theorem the size of an optimal chain partition equals the width of the order. The value of the game for orders of width w, denoted by val(w), is the least integer n for which some Algorithm has a strategy using at most n chains for every on-line order of width w. Alternatively, it is the largest integer n for which Spoiler has a strategy that forces any Algorithm to use n chains on order of width w.

The study of chain partition games goes back to the early 80’s when Kierstead [4] (upper bound) and Szemerédi (lower bound published in [5]) proved the estimates for on-line orders of width w: w+1 2 val(w) 5 w -1 4 . It took almost 30 years until these bounds had been slightly improved. The story can be found in the survey [2].

The study of on-line chain partition on restricted classes of orders began in 1981 when Kierstead and Trotter [6] proved the following result: when Spoiler is restricted to presenting interval orders of width w, the value of the game is 3w -2. Among other classes of orders that have been studied thereafter are (k + k)-free orders and semi-orders. Again we refer to [2] for details.

Up-growing on-line orders have been introduced by Felsner [3]. In this variant Spoiler’s power is restricted by the condition that the new element has to be a maximal element of the order presented so far. Felsner [3] showed that the value of the chain partition game on up-growing orders is w+1 exists a mapping I of points of the order into unit length intervals on a real line so that x < y in P iff interval I(x) is entirely to the left of I(y). Alternatively semi-orders are characterized as the (2 + 2) and (3 + 1)-free orders (see Fig. 2).

Considering on-line chain partitions of semi-orders note that the general (not upgrowing) case is easy to analyze: First, observe that the number of chains used by Algorithm can be bounded by 2w -1. Let x be the new point and consider the set Inc(x) of points incomparable with x. Clearly, the only chains forbidden for x are those used in Inc(x). Now width(Inc(x)) w -1 since the width of the whole order does not exceed w. Moreover, height(Inc(x)) 2 as the presented order is (3 + 1)-free. Therefore, | Inc(x)| 2(w -1) = 2w -2, proving that x can be assigned to at least one of 2w -1 legal chains.

It turns out that there is no better strategy for Algorithm. In other words, Spoiler may force Algorithm to use 2w -1 chains on semi-orders of width w. A strategy for Spoiler looks as follows:

(1) Present two antichains A and B, both consisting of w points in such a way that A < B, i.e., all points from A are below all points from B. If Algorithm uses 2w -1 or more chains, the construction is finished. Otherwise, suppose that k chains (2 k w) contain elements from A and B, namely let a i ∈ A i , b i ∈ B i for 1 i k lie in the same chain. (2) Present k-1 points x 1 , . . . , x k-1 in such a way that {a 1 , . . . , a i } x i {b i+1 , . . . , b k } and x i is incomparable to all the rest (the interval representation of the whole order looks as in Fig. 1). It is easy to verify that in such setting Algorithm is forced to use 2w -1 chains. The contribution of this paper is the following theorem.

Theorem 1.1. The value of the on-line chain partition game for up-growing semi-orders of width w is

• w⌋.

2.1. Outline. In this section we prove that the value of the on-line chain partition game for up-growing semi-orders equals ⌊ϕ • w⌋, where ϕ = 1+ √ 5 2

is the golden number. First, in Sect. 2.2 we collect some facts about semi-orders. Section 2.3 describes a strategy for Spoiler which forces Algorithm to use at least ⌊ϕ • w⌋ chains on a semi-order of width w. This sets the lower bound for the value of the game. In Sect. 2.4 we propose a strategy for Algorithm using at most ⌊ϕ • w⌋ chains on semi-orders of width at most w.

The presence of the golden number ϕ in the result of a chain partition game may seem surprising. In fact, it is the Fibonacci sequence (F 0 = 0, F 1 = 1 and F i+2 = F i + F i+1 ) which appears in the counting argument of the upper bound and serves as a discrete counterpart of ϕ.

2.2. Basic Facts. For x, y ∈ P by x P y we mean that x and y are incomparable in P. Let x↓ P = {y ∈ P : y < x}, called a down set of x in P, denote the set of predecessors of x in P. Dually, let x↑ P = {y ∈ P : y > x}, called an up set of x, denote the set of successors of x in P. If the order P is unambiguous from the context we also write x↑ instead of x↑ P and x↓ instead of x↓ P . By X↓ we mean x∈X x↓.

The maximum and the minimu

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