First-Fit is Linear on ( r + s )-free Posets Kevin G. Milans ( - - PowerPoint PPT Presentation

First-Fit is Linear on (r + s)-free Posets

Kevin G. Milans (milans@math.sc.edu) University of South Carolina Joint with Gwena¨ el Joret Universit´ e Libre de Bruxelles 24th Cumberland Conference University of Louisville 2011 May 14

The Online Chain Partition Problem

◮ A game between Spoiler and Algorithm.

The Online Chain Partition Problem

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P.

The Online Chain Partition Problem

1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2 3

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2 3

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ It is not clear that val(w) is finite, even if

w = 2.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ It is not clear that val(w) is finite, even if

w = 2.

◮ Kierstead (1981); Felsner: val(2) = 5.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ It is not clear that val(w) is finite, even if

w = 2.

◮ Kierstead (1981); Felsner: val(2) = 5. ◮ Kierstead (1981): val(w) ≤ 5w−1 4

.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ It is not clear that val(w) is finite, even if

w = 2.

◮ Kierstead (1981); Felsner: val(2) = 5. ◮ Kierstead (1981): val(w) ≤ 5w−1 4

.

◮ Szemer´

edi; Saks: val(w) ≥ w+1

2

• .

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ Bosek–Krawczyk (2009):

val(w) ≤ w16 lg w.

The Online Chain Partition Problem

1 2 2 3 1

◮ A game between Spoiler and Algorithm. ◮ Spoiler presents an element x of P. ◮ Algorithm assigns x to a chain.

Definition

The least k such that Algorithm has a strategy to partition posets of width w into at most k chains is val(w).

◮ Bosek–Krawczyk (2009):

val(w) ≤ w16 lg w.

◮ Bosek et al. (2010):

val(w) ≥ (2 − o(1)) w+1

2

• .

The First-Fit Algorithm

◮ One simple strategy for Alg: First-Fit.

The First-Fit Algorithm

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

The First-Fit Algorithm

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3 2

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

The First-Fit Algorithm

1 2 1 3 2 1 4 3 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

◮ When P has additional structure, First-Fit

does much better.

The First-Fit Algorithm

1 2 1 3 2 1 4 3 2 1

◮ One simple strategy for Alg: First-Fit. ◮ First-Fit puts x in the first possible chain.

Example (Kierstead)

First-Fit uses arbitrarily many chains on posets

• f width 2.

◮ When P has additional structure, First-Fit

does much better.

◮ First-Fit uses only the comparability

graph of P.

Interval Orders

Definition

An interval order is a poset whose elements are closed intervals on the real line such that [a, b] < [c, d] if and only if b < c.

Interval Orders

Definition

An interval order is a poset whose elements are closed intervals on the real line such that [a, b] < [c, d] if and only if b < c.

Example

An Interval Order P Hasse Diagram of P

First-Fit on Interval Orders

Definition

The least k such that First-Fit partitions interval orders of width w into at most k chains is FF(w).

First-Fit on Interval Orders

Definition

The least k such that First-Fit partitions interval orders of width w into at most k chains is FF(w).

Upper Bounds

◮ (Woodall (1976)): FF(w) = O(w log w) ◮ (Kierstead (1988)): FF(w) ≤ 40w ◮ (Kierstead–Qin (1995)): FF(w) ≤ 25.8w ◮ (Pemmaraju–Raman–Varadarajan (2003)): FF(w) ≤ 10w ◮ (Brightwell–Kierstead–Trotter (2003; unpub)):

FF(w) ≤ 8w

◮ (Narayansamy–Babu (2004)): FF(w) ≤ 8w − 3 ◮ (Howard (2010+)): FF(w) ≤ 8w − 4

First-Fit on Interval Orders

Definition

The least k such that First-Fit partitions interval orders of width w into at most k chains is FF(w).

Lower Bounds

◮ (Kierstead–Trotter (1981)): There is a positive ε such that

FF(w) ≥ (3 + ε)w when w is sufficiently large.

◮ (Chrobak–´

Slusarek (1990)): FF(w) ≥ 4w − 9 when w ≥ 4.

◮ (Kierstead–Trotter (2004)): FF(w) ≥ 4.99w − O(1). ◮ (D. Smith (2009)): If ε > 0, then FF(w) ≥ (5 − ε)w when

w is sufficiently large.

First-Fit on Interval Orders

Definition

The least k such that First-Fit partitions interval orders of width w into at most k chains is FF(w).

Best Known Bounds

If ε > 0 and w is sufficiently large, then (5 − ε)w ≤ FF(w) ≤ 8w − 4.

Beyond Interval Orders

Theorem (Fishburn (1970))

2 + 2

◮ The poset r + s is the disjoint union of a

chain of size r and a chain of size s.

◮ A poset P is an interval order if and only

if P does not contain 2 + 2 as an induced subposet.

Beyond Interval Orders

Theorem (Fishburn (1970))

2 + 2

◮ The poset r + s is the disjoint union of a

chain of size r and a chain of size s.

◮ A poset P is an interval order if and only

if P does not contain 2 + 2 as an induced subposet.

◮ When r ≥ 2 and s ≥ 2, the family of (r + s)-free posets

contains the interval orders.

The Bosek-Krawczyk Algorithm

◮ Bosek-Krawczyk: val(w) ≤ w16 lg w.

The Bosek-Krawczyk Algorithm

2w − 1+2w − 1

◮ Bosek-Krawczyk: val(w) ≤ w16 lg w. ◮ Bosek-Krawczyk uses First-Fit as a

subroutine on an auxiliary poset of width at most w3 that is (2w − 1 + 2w − 1)-free.

The Bosek-Krawczyk Algorithm

2w − 1+2w − 1

◮ Bosek-Krawczyk: val(w) ≤ w16 lg w. ◮ Bosek-Krawczyk uses First-Fit as a

subroutine on an auxiliary poset of width at most w3 that is (2w − 1 + 2w − 1)-free.

Theorem (Bosek–Krawczyk–Szczypka (2010))

If P is an (r + r)-free poset of width w, then First-Fit partitions P into at most 3rw2 chains.

The Bosek-Krawczyk Algorithm

2w − 1+2w − 1

◮ Bosek-Krawczyk: val(w) ≤ w16 lg w. ◮ Bosek-Krawczyk uses First-Fit as a

subroutine on an auxiliary poset of width at most w3 that is (2w − 1 + 2w − 1)-free.

Theorem (Bosek–Krawczyk–Szczypka (2010))

If P is an (r + r)-free poset of width w, then First-Fit partitions P into at most 3rw2 chains.

◮ They asked: can the bound be improved

from O(w2) to O(w)?

The Bosek-Krawczyk Algorithm

2w − 1+2w − 1

◮ Bosek-Krawczyk: val(w) ≤ w16 lg w. ◮ Bosek-Krawczyk uses First-Fit as a

subroutine on an auxiliary poset of width at most w3 that is (2w − 1 + 2w − 1)-free.

Theorem (Bosek–Krawczyk–Szczypka (2010))

If P is an (r + r)-free poset of width w, then First-Fit partitions P into at most 3rw2 chains.

◮ They asked: can the bound be improved

from O(w2) to O(w)?

◮ A positive answer would improve the

constant 16 in val(w) ≤ w16 lg w.

Our Result

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

Our Result

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Let P be an (r + s)-free poset.

Our Result

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Let P be an (r + s)-free poset. ◮ A group is a set of elements of P inducing a subposet of

height at most r − 1.

Our Result

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Let P be an (r + s)-free poset. ◮ A group is a set of elements of P inducing a subposet of

height at most r − 1.

◮ A society (S, F) consists of a set S of groups and a friendship

function F.

Our Result

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Let P be an (r + s)-free poset. ◮ A group is a set of elements of P inducing a subposet of

height at most r − 1.

◮ A society (S, F) consists of a set S of groups and a friendship

function F.

◮ Each group has up to 2(s − 1) friends.

Evolution of Societies

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit.

Evolution of Societies

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m.

Evolution of Societies

(S0, F0)

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0).

Evolution of Societies

(S0, F0)

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Evolution of Societies

(S0, F0) (S1, F1) C1

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Evolution of Societies

(S0, F0) (S1, F1) C1 (S2, F2) C2

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Evolution of Societies

(S0, F0) (S1, F1) C1 (S2, F2) C2

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Key Properties

◮ S0 ⊇ S1 ⊇ · · · .

Evolution of Societies

(S0, F0) (S1, F1) C1 (S2, F2) C2

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Key Properties

◮ S0 ⊇ S1 ⊇ · · · . ◮ If X and Y are friends in Sj−1 and both survive to Sj, then X

and Y are friends in Sj.

Evolution of Societies

(S0, F0) (S1, F1) C1 (S2, F2) C2

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Key Properties

◮ S0 ⊇ S1 ⊇ · · · . ◮ If X and Y are friends in Sj−1 and both survive to Sj, then X

and Y are friends in Sj.

◮ If X and Y are friends in Sj−1 and only one survives to Sj, the

• ther selects a new friend according to a replacement scheme.

Evolution of Societies

(S0, F0) (S1, F1) C1 (S2, F2) C2 · · · (Sn, Fn)

◮ Let C1, . . . , Cm be a chain partition produced by First-Fit. ◮ Extend this by defining Cj = ∅ for j > m. ◮ Construct the initial society (S0, F0). ◮ For j ≥ 1, use Cj to obtain (Sj, Fj) from (Sj−1, Fj−1).

Key Properties

◮ S0 ⊇ S1 ⊇ · · · . ◮ If X and Y are friends in Sj−1 and both survive to Sj, then X

and Y are friends in Sj.

◮ If X and Y are friends in Sj−1 and only one survives to Sj, the

• ther selects a new friend according to a replacement scheme.

◮ The process ends when (Sn, Fn) is generated with Sn = ∅.

Transition Rules

S0 Sj−1 Cj Sj

Transition Rules

S0 Sj−1 Cj Sj

◮ Consider a group X ∈ Sj−1.

Transition Rules

S0 Sj−1 Cj Sj

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

Transition Rules

S0 Sj−1 Cj Sj

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj α

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

• 2. Otherwise, if some friend of X in Sj−1 has nonempty

intersection with Cj, then X makes a β-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj β

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

• 2. Otherwise, if some friend of X in Sj−1 has nonempty

intersection with Cj, then X makes a β-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

• 2. Otherwise, if some friend of X in Sj−1 has nonempty

intersection with Cj, then X makes a β-transition to Sj.

• 3. Otherwise, if the number of α-transitions that X makes from

Si to Sj−1 exceeds (j − i)/2t for some i, then X makes a γ-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj Si α α α

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

• 2. Otherwise, if some friend of X in Sj−1 has nonempty

intersection with Cj, then X makes a β-transition to Sj.

• 3. Otherwise, if the number of α-transitions that X makes from

Si to Sj−1 exceeds (j − i)/2t for some i, then X makes a γ-transition to Sj.

Transition Rules

S0 Sj−1 Cj Sj Si α α α γ

◮ Consider a group X ∈ Sj−1. ◮ There are 3 ways that X can transition from Sj−1 to Sj.

Transition Rules

• 1. If X has nonempty intersection with Cj, then X makes an

α-transition to Sj.

• 2. Otherwise, if some friend of X in Sj−1 has nonempty

intersection with Cj, then X makes a β-transition to Sj.

• 3. Otherwise, if the number of α-transitions that X makes from

Si to Sj−1 exceeds (j − i)/2t for some i, then X makes a γ-transition to Sj.

The Groups in the Initial Society

◮ Let q be the height of P.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y.

◮ Partition P by height.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y

◮ Partition P by height. ◮ Consider y ∈ P.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y)

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y)

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

◮ Add y to sets above ...

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

◮ Add y to sets above ...

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

◮ Add y to sets above ...

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y

◮ Partition P by height. ◮ Consider y ∈ P. ◮ Let B(y) be the set of elements z

such that there is a chain C with

◮ |C| ≥ r and ◮ (min C, max C) = (y, z).

◮ Add y to sets above ... ◮ ... and stop just before y would

enter a set that intersects B(y).

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y

◮ Do this for each y ∈ P.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y X1 X2 X3 X4 X5 X6 X7

◮ Do this for each y ∈ P. ◮ Let X1, X2, . . . , Xq be the resulting

sets.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y X1 X2 X3 X4 X5 X6 X7

◮ Do this for each y ∈ P. ◮ Let X1, X2, . . . , Xq be the resulting

sets.

◮ Each Xj has height at most r − 1.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y X1 X2 X3 X4 X5 X6 X7

◮ Do this for each y ∈ P. ◮ Let X1, X2, . . . , Xq be the resulting

sets.

◮ Each Xj has height at most r − 1. ◮ Let S0 = {X1, . . . , Xq}.

The Groups in the Initial Society

◮ Let q be the height of P. ◮ The height of y, denoted h(y), is the size of a longest chain

with top element y. y B(y) y y X1 X2 X3 X4 X5 X6 X7

◮ Do this for each y ∈ P. ◮ Let X1, X2, . . . , Xq be the resulting

sets.

◮ Each Xj has height at most r − 1. ◮ Let S0 = {X1, . . . , Xq}. ◮ Let I(y) = {j : y ∈ Xj}.

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z).

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z

◮ Suppose that h(y) ≤ h(z).

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ...

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ...

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

◮ P has a chain of size r from y to y′.

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

◮ P has a chain of size r from y to y′. ◮ Extend a chain down from z.

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

◮ P has a chain of size r from y to y′. ◮ Extend a chain down from z. ◮ Elements in distinct chains are

incomparable.

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

◮ P has a chain of size r from y to y′. ◮ Extend a chain down from z. ◮ Elements in distinct chains are

incomparable.

Incomparable Elements are in Nearby Groups

Lemma

If y and z are incomparable, then either I(y) ∩ I(z) = ∅, or there are at most s − 2 integers between I(y) and I(z). y z y y y′

◮ Suppose that h(y) ≤ h(z). ◮ We add y to sets above ... ◮ ... and stop only when we encounter

some y′ ∈ B(y).

◮ P has a chain of size r from y to y′. ◮ Extend a chain down from z. ◮ Elements in distinct chains are

incomparable.

◮ The chain at z has size at most s − 1.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

◮ If i < j and x ∈ Cj, then x is incomparable to some y ∈ Ci.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

◮ If i < j and x ∈ Cj, then x is incomparable to some y ∈ Ci. ◮ Some group X with x ∈ X is close to a group Y with y ∈ Y .

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

◮ If i < j and x ∈ Cj, then x is incomparable to some y ∈ Ci. ◮ Some group X with x ∈ X is close to a group Y with y ∈ Y . ◮ X and Y list each other as friends.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

◮ If i < j and x ∈ Cj, then x is incomparable to some y ∈ Ci. ◮ Some group X with x ∈ X is close to a group Y with y ∈ Y . ◮ X and Y list each other as friends. ◮ Y makes an α-transition from Si−1 to Si.

Finding a Large Group

◮ Fact: if X survives to Sn−1, then a constant fraction of the

transitions it makes are α-transitions.

◮ Since the chains are disjoint, the α-transitions correspond to

distinct elements in X.

Lemma

If C1, . . . , Cm is a chain partition produced by First-Fit and (S0, F0), . . . , (Sn, Fn) is the resulting evolution, then n ≥ m + 2.

◮ If i < j and x ∈ Cj, then x is incomparable to some y ∈ Ci. ◮ Some group X with x ∈ X is close to a group Y with y ∈ Y . ◮ X and Y list each other as friends. ◮ Y makes an α-transition from Si−1 to Si. ◮ X is eligible for a β-transition from Si−1 to Si.

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Improve the constant 8 in the upper bound.

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Improve the constant 8 in the upper bound. ◮ Give lower bounds when (r, s) = (2, 2).

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Improve the constant 8 in the upper bound. ◮ Give lower bounds when (r, s) = (2, 2).

Question

For which posets Q is there a polynomial pQ(w) such that First-Fit partitions a Q-free poset of width w into at most pQ(w) chains?

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Improve the constant 8 in the upper bound. ◮ Give lower bounds when (r, s) = (2, 2).

Question

For which posets Q is there a polynomial pQ(w) such that First-Fit partitions a Q-free poset of width w into at most pQ(w) chains?

◮ Note: Kierstead’s example shows that Q must have width 2.

Open Problems

Theorem

If r, s ≥ 2 and P is an (r + s)-free poset of width w, then First-Fit partitions P into at most 8(r − 1)(s − 1)w chains.

◮ Improve the constant 8 in the upper bound. ◮ Give lower bounds when (r, s) = (2, 2).

Question

For which posets Q is there a polynomial pQ(w) such that First-Fit partitions a Q-free poset of width w into at most pQ(w) chains?

◮ Note: Kierstead’s example shows that Q must have width 2.

Thank You.