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

The First-Fit Algorithm 1 ◮ One simple strategy for Alg: First-Fit. 2 ◮ First-Fit puts x in the first possible chain. 3 Example (Kierstead) 4 First-Fit uses arbitrarily many chains on posets 1 of width 2. 2 3 1 2 1

The First-Fit Algorithm 1 ◮ One simple strategy for Alg: First-Fit. 2 ◮ First-Fit puts x in the first possible chain. 3 Example (Kierstead) 4 First-Fit uses arbitrarily many chains on posets 1 of width 2. 2 ◮ When P has additional structure, First-Fit 3 does much better. 1 2 1

The First-Fit Algorithm 1 ◮ One simple strategy for Alg: First-Fit. 2 ◮ First-Fit puts x in the first possible chain. 3 Example (Kierstead) 4 First-Fit uses arbitrarily many chains on posets 1 of width 2. 2 ◮ When P has additional structure, First-Fit 3 does much better. 1 ◮ First-Fit uses only the comparability 2 graph of P . 1

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 ) ≤ 40 w ◮ (Kierstead–Qin (1995)): FF ( w ) ≤ 25 . 8 w ◮ (Pemmaraju–Raman–Varadarajan (2003)): FF ( w ) ≤ 10 w ◮ (Brightwell–Kierstead–Trotter (2003; unpub)): FF ( w ) ≤ 8 w ◮ (Narayansamy–Babu (2004)): FF ( w ) ≤ 8 w − 3 ◮ (Howard (2010+)): FF ( w ) ≤ 8 w − 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 ) ≥ 4 w − 9 when w ≥ 4 . ◮ (Kierstead–Trotter (2004)): FF ( w ) ≥ 4 . 99 w − 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 ) ≤ 8 w − 4 .

Beyond Interval Orders Theorem (Fishburn (1970)) ◮ 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 2 + 2 if P does not contain 2 + 2 as an induced subposet.

Beyond Interval Orders Theorem (Fishburn (1970)) ◮ 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 2 + 2 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 ) ≤ w 16 lg w .

The Bosek-Krawczyk Algorithm ◮ Bosek-Krawczyk: val ( w ) ≤ w 16 lg w . ◮ Bosek-Krawczyk uses First-Fit as a subroutine on an auxiliary poset of width at most w 3 that is (2 w − 1 + 2 w − 1)-free. 2 w − 1+2 w − 1

The Bosek-Krawczyk Algorithm ◮ Bosek-Krawczyk: val ( w ) ≤ w 16 lg w . ◮ Bosek-Krawczyk uses First-Fit as a subroutine on an auxiliary poset of width at most w 3 that is (2 w − 1 + 2 w − 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 3 rw 2 chains. 2 w − 1+2 w − 1

The Bosek-Krawczyk Algorithm ◮ Bosek-Krawczyk: val ( w ) ≤ w 16 lg w . ◮ Bosek-Krawczyk uses First-Fit as a subroutine on an auxiliary poset of width at most w 3 that is (2 w − 1 + 2 w − 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 3 rw 2 chains. ◮ They asked: can the bound be improved from O ( w 2 ) to O ( w )? 2 w − 1+2 w − 1

The Bosek-Krawczyk Algorithm ◮ Bosek-Krawczyk: val ( w ) ≤ w 16 lg w . ◮ Bosek-Krawczyk uses First-Fit as a subroutine on an auxiliary poset of width at most w 3 that is (2 w − 1 + 2 w − 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 3 rw 2 chains. ◮ They asked: can the bound be improved from O ( w 2 ) to O ( w )? ◮ A positive answer would improve the constant 16 in val ( w ) ≤ w 16 lg w . 2 w − 1+2 w − 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.

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 C 1 , . . . , C m be a chain partition produced by First-Fit.

Evolution of Societies ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m .

Evolution of Societies ( S 0 , F 0 ) ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ).

Evolution of Societies ( S 0 , F 0 ) ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ).

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) C 1 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ).

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) ( S 2 , F 2 ) C 1 C 2 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ).

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) ( S 2 , F 2 ) C 1 C 2 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ). Key Properties ◮ S 0 ⊇ S 1 ⊇ · · · .

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) ( S 2 , F 2 ) C 1 C 2 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ). Key Properties ◮ S 0 ⊇ S 1 ⊇ · · · . ◮ If X and Y are friends in S j − 1 and both survive to S j , then X and Y are friends in S j .

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) ( S 2 , F 2 ) C 1 C 2 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ). Key Properties ◮ S 0 ⊇ S 1 ⊇ · · · . ◮ If X and Y are friends in S j − 1 and both survive to S j , then X and Y are friends in S j . ◮ If X and Y are friends in S j − 1 and only one survives to S j , the other selects a new friend according to a replacement scheme.

Evolution of Societies ( S 0 , F 0 ) ( S 1 , F 1 ) ( S 2 , F 2 ) ( S n , F n ) · · · C 1 C 2 ◮ Let C 1 , . . . , C m be a chain partition produced by First-Fit. ◮ Extend this by defining C j = ∅ for j > m . ◮ Construct the initial society ( S 0 , F 0 ). ◮ For j ≥ 1, use C j to obtain ( S j , F j ) from ( S j − 1 , F j − 1 ). Key Properties ◮ S 0 ⊇ S 1 ⊇ · · · . ◮ If X and Y are friends in S j − 1 and both survive to S j , then X and Y are friends in S j . ◮ If X and Y are friends in S j − 1 and only one survives to S j , the other selects a new friend according to a replacement scheme. ◮ The process ends when ( S n , F n ) is generated with S n = ∅ .

Transition Rules S 0 S j − 1 C j S j

Transition Rules S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 .

Transition Rules S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules

Transition Rules S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j .

Transition Rules α S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j .

Transition Rules S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j . 2. Otherwise, if some friend of X in S j − 1 has nonempty intersection with C j , then X makes a β -transition to S j .

Transition Rules β S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j . 2. Otherwise, if some friend of X in S j − 1 has nonempty intersection with C j , then X makes a β -transition to S j .

Transition Rules S 0 S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j . 2. Otherwise, if some friend of X in S j − 1 has nonempty intersection with C j , then X makes a β -transition to S j . 3. Otherwise, if the number of α -transitions that X makes from S i to S j − 1 exceeds ( j − i ) / 2 t for some i , then X makes a γ -transition to S j .

Transition Rules α α α S 0 S i S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j . 2. Otherwise, if some friend of X in S j − 1 has nonempty intersection with C j , then X makes a β -transition to S j . 3. Otherwise, if the number of α -transitions that X makes from S i to S j − 1 exceeds ( j − i ) / 2 t for some i , then X makes a γ -transition to S j .

Transition Rules γ α α α S 0 S i S j − 1 C j S j ◮ Consider a group X ∈ S j − 1 . ◮ There are 3 ways that X can transition from S j − 1 to S j . Transition Rules 1. If X has nonempty intersection with C j , then X makes an α -transition to S j . 2. Otherwise, if some friend of X in S j − 1 has nonempty intersection with C j , then X makes a β -transition to S j . 3. Otherwise, if the number of α -transitions that X makes from S i to S j − 1 exceeds ( j − i ) / 2 t for some i , then X makes a γ -transition to S j .

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 . ◮ Partition P by height. ◮ Consider y ∈ P . 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. ◮ 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 ). 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 . 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 ). 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 . 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 ... 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 . 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 ). y ◮ Add y to sets above ... 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 . 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 y ◮ | C | ≥ r and ◮ (min C , max C ) = ( y , z ). y ◮ Add y to sets above ... 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 . 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 y ◮ | C | ≥ r and ◮ (min C , max C ) = ( y , z ). y ◮ Add y to sets above ... y ◮ ... 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 . B ( y ) ◮ Do this for each y ∈ P . y y 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 . B ( y ) X 7 ◮ Do this for each y ∈ P . X 6 ◮ Let X 1 , X 2 , . . . , X q be the resulting sets. X 5 y X 4 y X 3 y X 2 X 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 . B ( y ) X 7 ◮ Do this for each y ∈ P . X 6 ◮ Let X 1 , X 2 , . . . , X q be the resulting sets. X 5 ◮ Each X j has height at most r − 1. y X 4 y X 3 y X 2 X 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 . B ( y ) X 7 ◮ Do this for each y ∈ P . X 6 ◮ Let X 1 , X 2 , . . . , X q be the resulting sets. X 5 ◮ Each X j has height at most r − 1. y X 4 ◮ Let S 0 = { X 1 , . . . , X q } . y X 3 y X 2 X 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 . B ( y ) X 7 ◮ Do this for each y ∈ P . X 6 ◮ Let X 1 , X 2 , . . . , X q be the resulting sets. X 5 ◮ Each X j has height at most r − 1. y X 4 ◮ Let S 0 = { X 1 , . . . , X q } . y X 3 ◮ Let I ( y ) = { j : y ∈ X j } . y X 2 X 1

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