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Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Online Chain Partitioning of Upgrowing Interval Orders

Bartlomiej Bosek

Algorithmics Research Group Jagiellonian University, Cracow, Poland

Chambery-Krakow-Lyon Workshop on Computational Logic and Applications, 21 June 2005

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Poset

Definition We say that P = (X, ) is a partially

• rdered set(poset) if for each x, y, z

from X

x x (reflexive) x y, y x imply x = y (antisymmetric) x y, y z imply x z (transitive)

Example

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Chain and Antichain

Definition The subposet C is a chain if for each x, y from C

x y or y x. (complete)

The subposet A is an antichain if for each x, y from A

x y and y x. (incomparability)

Example

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Chain and Antichain

Definition The subposet C is a chain if for each x, y from C

x y or y x. (complete)

The subposet A is an antichain if for each x, y from A

x y and y x. (incomparability)

Example

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Width and Chain Partition

Definition The width of poset P = (X, ) is a maximal size of antichain in P. The chain partition of poset P = (X, ) is a family of chains C1, . . . , Ck so that

C1 ∪ . . . ∪ Ck = X.

Example Question Is there a connection between the minimal chain covering of and the width of he poset?

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Width and Chain Partition

Definition The width of poset P = (X, ) is a maximal size of antichain in P. The chain partition of poset P = (X, ) is a family of chains C1, . . . , Ck so that

C1 ∪ . . . ∪ Ck = X.

Example Question Is there a connection between the minimal chain covering of and the width of he poset?

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Dilworth’s Theorem

Theorem (Dilworth) Minimal chain partition of poset is equal to poset’s width. Example

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Online Chain Partitioning of the Posets

Online chain partitioning of the posets can be viewed as two-person game. We call the players Algorithm and Spoiler. During each round: Spoiler introduces a new point with a comparability status of previously presented points to the new one. Algorithm covers this new point by some chain.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 d c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 d 1 c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 d 1 e c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 d 1 e 3 c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 1. Spoiler puts e. Algorithm covers e by 3.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 d 1 e 3 c 1 result of the game a 1 b 2 d 2 e 1 c 1

• ptimal off-line solution

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Value of Online Chain Partitioning Game

Definition The value of the chain partitioning game for width w posets is the largest integer CP(w) so that there is a strategy of presenting points that forces any algorithm to use CP(w) chains.

Note that we may as well define CP(w) as the least integer so that there is an algorithm that never uses more chains.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Theorem (Kierstead, Szemeredi) w + 1 2

• CP(w) 5w − 1

4 This is already a complicated result, and no progress has been made for the last 20 years.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Upgrowing version

We suppose that Spoiler introduces points which are always maximal at the moment of their arrival. Online posets with this property are called upgrowing posets. We define also the value of this game for width w posets and we denote it CPU(w).

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d 3 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d 3 e Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d 3 e 3 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d 3 e 3 f Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Example a 1 b 2 c 1 d 3 e 3 f 2 Spoiler puts a. Algorithm covers a by 1. Spoiler puts b. Algorithm covers b by 2. Spoiler puts c. Algorithm covers c by 1. Spoiler puts d. Algorithm covers d by 3. Spoiler puts e. Algorithm covers e by 3. Spoiler puts f. Algorithm covers e by 2.

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Theorem (Felsner) CPU(w) = w + 1 2

• Bartlomiej Bosek

Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Interval Orders

Definition A poset P = (X, ) is an interval order if there is a function F assigning to each point x from X an interval F(x) = [ax, bx] of the real line R, so that x < y iff bx < ay. Example a b c d e f g a b c d e f g interval representation

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Fishburn’s Theorem

Theorem (Fishburns) The poset P = (X, ) is an interval order iff P = (X, ) is a (2 + 2)-free poset

i.e. X does not contain elements a, b, c, d so that a < b, c < d, a is incomparable with d and c is incomparable with b .

a b c d poset 2 + 2

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Online Chain Partitioning of Upgrowing Interval Orders

Consider a two-person game analogous to previous ones. Now Spoiler has to present an interval order. Let CPUI(w) be the value of this version for width w postes.

A bound of not-interval variant gives us w CPUI(w) w + 1 2

• .

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Our result

Theorem (Bosek, Micek) CPUI(w) = 2w − 1

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders

Values of the game in all variants

upgr. inter. value

• w+1

2

• . . . 5w−1

4

Kierstead, Szemeredi

• +

3w − 2 Kierstead, Trotter +

• w+1

2

• Felsner

+ + 2w − 1 Bosek, Micek

Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders