Counting Linear Extensions of Small Posets

Fine-Grained and Beyond?

Mikko Koivisto

University of Helsinki

The #LE Problem

Input: A partially ordered set, representedby the cover graph (V, )

Output:# linear extensions L

f

d

g

b

a

c

e

A linear extension: c e a b d g f

≺

The #LE Problem

Input: A partially ordered set, representedby the cover graph (V, )

Output:# linear extensions L

Parameters:# elements nmax degree Δtreewidth tw

f

d

g

b

a

c

e

A linear extension: c e a b d g f

≺

The #LE Problem

Input: A partially ordered set, representedby the cover graph (V, )

Output:# linear extensions L

Parameters:# elements nmax degree Δtreewidth tw# downsets |D|

f

d

g

b

a

c

e

≺

The #LE Problem

Input: A partially ordered set, representedby the cover graph (V, )

Output:# linear extensions L

Parameters:# elements nmax degree Δtreewidth tw# downsets |D|# connected sets |C|

f

d

g

b

a

c

e

≺

HistoryEXACT FPRAS

1985 - 1991 Poly time for trees, series-parallelgraphs, and bounded width posets

Folklore n|D|, worst case n2n

HistoryEXACT FPRAS

1985 - 1991 Poly time for trees, series-parallelgraphs, and bounded width posets

1989 Deyer, Frieze, KannanThe more general case of the volume of convex bodies

1991 Karzanov, Khachiyan ~ n9

1999 Bubley, Deyer ~ n6

Folklore n|D|, worst case n2n

HistoryEXACT FPRAS

1985 - 1991 Poly time for trees, series-parallelgraphs, and bounded width posets

1989 Deyer, Frieze, KannanThe more general case of the volume of convex bodies

1991 Karzanov, Khachiyan ~ n9 1991 Brightwell, Winkler

#P-complete for posets of height > 2 1999 Bubley, Deyer ~ n6

Folklore n|D|, worst case n2n

HistoryEXACT FPRAS

1985 - 1991 Poly time for trees, series-parallelgraphs, and bounded width posets

1989 Deyer, Frieze, KannanThe more general case of the volume of convex bodies

1991 Karzanov, Khachiyan ~ n9 1991 Brightwell, Winkler

#P-complete for posets of height > 2 1999 Bubley, Deyer ~ n6 2006 Huber ~ n6 via exact sampling

2015 Huber ~ n2Δ2 for bipartite graphs

Folklore n|D|, worst case n2n

2004 Peczarski Faster for some posets

HistoryEXACT FPRAS

1985 - 1991 Poly time for trees, series-parallelgraphs, and bounded width posets

1989 Deyer, Frieze, KannanThe more general case of the volume of convex bodies

1991 Karzanov, Khachiyan ~ n9 1991 Brightwell, Winkler

#P-complete for posets of height > 2 1999 Bubley, Deyer ~ n6 2006 Huber ~ n6 via exact sampling

2015 Huber ~ n2Δ2 for bipartite graphs

Folklore n|D|, worst case n2n

2004 Peczarski Faster for some posets

2016 Kangas, Hankala, Niinimäki, K. Yet faster, n |C ∩ D|; also ntw+4

2016 Eiben, Ganian, Kangas, Ordyniak W[1]-hard w.r.t. treewidth

The Issue of Small n

20 40 60 80

104

108

1012

2n/2

n

n6

2n

The Issue of Small n

20 40 60 80

104

108

1012

2n/2

n

n6

2n

THE TRUTH IS OUT THERE

?

The Issue of Small n

20 40 60 80

104

108

1012

deadline

2n/2

n

n6

2n

Result 1: Combining Recurrence Relations

L(V ) = ∑x∈maxV

L(V−x )

L(V ) = ( ∣V∣∣V 1∣,… ,∣V k∣) L(V 1)⋯L(V k )

Recurrence 1:

Recurrence 2: If V1, …, Vk partition V into connected components,

Thm: #LE is in time n |C ∩ D|

Efficient when D is small

Efficient when C is small

[ IJCAI 2016 Kangas, Hankala, Niinimäki, K. ]

Result 2: Inclusion-Exclusion + Tree Decomposition

L = ∑f :V →[n ]

[ f bijection] ∏u≺v

[ f (u)< f (v)]

Sum over permutations that respect the order:

[ IJCAI 2016 Kangas, Hankala, Niinimäki, K. ]

Result 2: Inclusion-Exclusion + Tree Decomposition

L = ∑f :V→[n]

[ f bijection] ∏u≺v

[ f (u)< f (v)]

L = ∑k=1

n

(−1)n−k (nk ) ∑

f :V→[k ]∏u≺v

[ f (u)< f (v)]

Sum over permutations that respect the order:

Remove the bijectivity constraint:

[ IJCAI 2016 Kangas, Hankala, Niinimäki, K. ]

Result 2: Inclusion-Exclusion + Tree Decomposition

L = ∑f :V→[n]

[ f bijection] ∏u≺v

[ f (u)< f (v)]

L = ∑k=1

n

(−1)n−k (nk ) ∑

f :V→[k ]∏u≺v

[ f (u)< f (v)]

Thm: #LE is in time ntw+4

Sum over permutations that respect the order:

Remove the bijectivity constraint: Need ntw+2 additions

[ IJCAI 2016 Kangas, Hankala, Niinimäki, K. ]

Results 1 & 2 in Practice on Random Posets of Treewidth 2 and 3

VEIE = IE + tree decomposition R1 = Recurrence 1 R14-a = Recurrences 1 & 2

Result 3: Parameterizations by Treewidth[ ESA 2016 Eiben, Ganian, Kangas, Ordyniak ]

What is the time complexity of #LE when parameterized by tw ? [ 2013 K. ]

Result 3: Parameterizations by Treewidth[ ESA 2016 Eiben, Ganian, Kangas, Ordyniak ]

Thm: #LE is W[1]-hard w.r.t. the treewidth of the cover graph

What is the time complexity of #LE when parameterized by tw ? [ 2013 K. ]

Proof idea: Reduce from EQUITABLE COLORING parameterized by treewidth

Result 3: Parameterizations by Treewidth[ ESA 2016 Eiben, Ganian, Kangas, Ordyniak ]

Thm: #LE is in FPT w.r.t the treewidth of the incomparability graph

Thm: #LE is W[1]-hard w.r.t. the treewidth of the cover graph

What is the time complexity of #LE when parameterized by tw ? [ 2013 K. ]

Proof idea: Reduce from EQUITABLE COLORING parameterized by treewidth

Result 3: Parameterizations by Treewidth[ ESA 2016 Eiben, Ganian, Kangas, Ordyniak ]

Thm: #LE is in FPT w.r.t the treewidth of the incomparability graph

Thm: #LE is W[1]-hard w.r.t. the treewidth of the cover graph

What is the time complexity of #LE when parameterized by tw ? [ 2013 K. ]

Proof idea: Reduce from EQUITABLE COLORING parameterized by treewidth

Proof idea: 1) tw( combined graph ) < 3 tw( incomparability graph ) + 32) Express #LE in MSO logic and apply the counting extension of Courcelle's thm: “Counting the edge subsets with a desired property is in FPT w.r.t. treewidth”

The Issue of Small n

20 40 60 80

104

108

1012

deadline

2n/2

n

n6

2n

Structure When n Is Small ?

Standard asymptotic paradigm:– Find T : N N such that→ Π is in time O( T(n) )– Possibly consider other parameters as well

Consider a (function) problem Π : Input Output →

Structure When n Is Small ?

Standard asymptotic paradigm:– Find T : N N such that→ Π is in time O( T(n) )– Possibly consider other parameters as well

Consider a (function) problem Π : Input Output →

E.g. lower exponent byadding log factors

Structure When n Is Small ?

Standard asymptotic paradigm:– Find T : N N such that→ Π is in time O( T(n) )– Possibly consider other parameters as well

Consider a (function) problem Π : Input Output →

Inverted non-asymptotic paradigm:– Given t find a subset St Ò Input solvable in time t

– Possibly characterize St using multiple parameters

E.g. lower exponent byadding log factors

Structure When n Is Small ?

Standard asymptotic paradigm:– Find T : N N such that→ Π is in time O( T(n) )– Possibly consider other parameters as well

Consider a (function) problem Π : Input Output →

Inverted non-asymptotic paradigm:– Given t find a subset St Ò Input solvable in time t

– Possibly characterize St using multiple parameters

E.g. lower exponent byadding log factors

What are good algorithmicideas for small n ?