Introduction to Treewidth

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Introduction to Treewidth

Marcin Pilipczuk most slides by Łukasz Kowalik

some slides by Marek Cygan and Michał Pilipczuk

Algorytmika, 11.05.2020r.

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Treewidth

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The book

Contained in Chapter 7 of

http://parameterized-algorithms.mimuw.edu.pl

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Motivation of treewidth

Independent Set: given a graph with vertex weights, find the maximum weight of a subset of vertices that are pairwise non-adjacent.

NP-hard in general, tractable in many special cases. Easy exercise: linear-time algorithm on trees.

I Principle of dynamic programming.

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Motivation of treewidth

NP-hard in general, tractable in many special cases.

Easy exercise: linear-time algorithm on trees.

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Motivation of treewidth

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Motivation of treewidth

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Weighted Independent Set on trees

Root the tree in an arbitrary vertex r .

I Tu is the subtree rooted at u. Compute dynamic programming tables:

I A[u]: maximum weight of an IS in T_u.

I B[u]: maximum weight of an IS in Tu that contains u.

I C [u]: maximum weight of an IS in Tu that excludes u.

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ]. ^u

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Weighted Independent Set on trees

I Tu is the subtree rooted at u.

Compute dynamic programming tables:

I B[u]: maximum weight of an IS in Tu that contains u.

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

I B[u]: maximum weight of an IS in T_u that contains u.

I C [u]: maximum weight of an IS in Tu that excludes u. Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ]. Answer: A[r ].

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ].

u

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Weighted Independent Set on trees

Recursive formulas:

I A[u] = max(B[u], C [u]).

I B[u] = w (u) +P

v ∈chld(u)C [v ].

I C [u] =P

v ∈chld(u)A[v ].

Answer: A[r ]. ^u

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Independent Set on trees: Why does the DP work?

Very limited dependence between vertices

Essential information about subtree stored in a compact way.

u

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Generalizations of paths and trees

pathlike path

Pathwidth

Graph of small pathwidth is path-like.

We hope that problems tractable on paths are also tractable on graphs of small pathwidth.

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Generalizations of paths and trees

tree treelike

Treewidth

Graph of small treewidth is tree-like.

We hope that problems tractable on trees are also tractable on graphs of small treewidth.

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Generalizations of paths and trees

treelike tree

Treewidth

Graph of small treewidth is tree-like.

We hope that problems tractable on trees are also tractable on graphs of small treewidth.

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Introduction

Crucial property of small pathwidth/treewidth graphs: separators.

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Introduction

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Introduction

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Introduction

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Introduction

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Introduction

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Introduction

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Introduction

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Introduction

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Definition (mapping to intervals)

Definition

A graph G has pathwidth ≤ pw if there exists a function f : V → intervals such that:

1 if (u, v ) ∈ E then f (u) ∩ f (v ) 6= ∅

2 ∀_{x ∈Z}|{v ∈ V : x ∈ f (v )}| ≤ pw + 1

v₂ v₃ v₄ v₅ v₁

f (v₁) f (v2) f (v4)

f (v₃) f (v₅)

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Definition (mapping to intervals)

Definition

1 if (u, v ) ∈ E then f (u) ∩ f (v ) 6= ∅

2 ∀_{x ∈Z}|{v ∈ V : x ∈ f (v )}| ≤ pw + 1

v₂ v₃ v₄ v₅ v₁

f (v₁) f (v2) f (v4)

f (v₃) f (v₅)

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Definition (mapping to intervals)

Definition

1 if (u, v ) ∈ E then f (u) ∩ f (v ) 6= ∅

2 ∀_{x ∈Z}|{v ∈ V : x ∈ f (v )}| ≤ pw + 1

v₂ v₃ v₄ v₅ v₁

f (v₁) f (v2) f (v4)

f (v3) f (v5)

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Definition (mapping to intervals)

Definition

1 if (u, v ) ∈ E then f (u) ∩ f (v ) 6= ∅

2 ∀_{x ∈Z}|{v ∈ V : x ∈ f (v )}| ≤ pw + 1

v₂ v₃ v₄ v₅ v₁

f (v₁) f (v2) f (v4)

f (v3) f (v5)

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Path decompositions and pathwidth

Graph G = (V , E ) Path decomposition of G

c b

a

f e d

i h g

abdef bcdef defgh efhi

Path decomposition is a path of bags (subsets of V )

such that For every edge uv ∈ E some bag contains u and v

For every vertex v ∈ V bags containing v form nonempty subpath (connected!)

Width of the decomposition: maximum bag size −1 (here: 4). Pathwidth of G: minimum width of a decomposition of G .

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Path decompositions and pathwidth

c b

a

f e d

i h g

Path decomposition is a path of bags (subsets of V ) such that For every edge uv ∈ E some bag contains u and v

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Path decompositions and pathwidth

c b

a

f e d d

i h g

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Path decompositions and pathwidth

c b

a

f e d

i h g

Width of the decomposition: maximum bag size −1 (here: 4).

Pathwidth of G: minimum width of a decomposition of G .

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Nice path decomposition

c b

a

f e d

i h

g ∅ a ab abd abde abdef bdef

bcdef bdef def defg defgh efgh

efh efhi efh ef e ∅

Nice path decomposition is a path decomposition X1, . . . , X_r such that X₁ = X_r = ∅

For i = 2, . . . , r bag X_i is of one of two types:

1 Introduce bag: Xi= Xi −1∪ {v } for some v ∈ V \ Xi −1,

2 Forget bag: Xi= Xi −1\ {v } for some v ∈ Xi −1.

(Easy) Theorem 1: Graph has a path decomposition of width w iff it has a nice path decomposition of width w (at most 2w times longer).

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Nice path decomposition

c b

a

f e d

i h

g ∅ a ab abd abde abdef bdef

bcdef bdef def defg defgh efgh

efh efhi efh ef e ∅

Nice path decomposition is a path decomposition X₁, . . . , Xr such that X1 = X_r = ∅

For i = 2, . . . , r bag X_i is of one of two types:

1 Introduce bag: Xi= Xi −1∪ {v } for some v ∈ V \ Xi −1,

2 Forget bag: Xi= Xi −1\ {v } for some v ∈ Xi −1.

(Easy) Theorem 2: There is a mapping of G to p intervals iff G has nice path decomposition with bags of size p.

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Dynamic programming for Independent Set

For every node t of a path decomposition of graph G : X_t = the bag at t,

Vt =S

i ≤tX_i

X1 Vt · · · Xt

Theorem

The bag X_i is a separator between V_i \ X_i and V (G ) \ V_i.

If S is an independent set in G and S₀ is an independent set in G [V_i] with S0∩ X_i = S ∩ Xi, then (S \ V_i) ∪ S0 is an independent set as well.

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Dynamic programming for Independent Set

Vt =S

i ≤tX_i

X1 Vt · · · Xt

Theorem

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Dynamic programming for Independent Set

Vt =S

i ≤tX_i

X1 Vt · · · Xt

Theorem

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Dynamic programming for Independent Set

For every node t of a path decomposition of graph G : Xt = the bag at t,

V_t =S

i ≤tX_i

X1 V_t · · · X_t

For every f : X_i → {0, 1} we want to compute the following.

Ψ[i , f ] = max{|S | : S is an independent set in V_i and S ∩ X_i = f⁻¹(1)}.

We now give recursive equations on Ψ[·, ·].

Ψ[i , ∅] = 0 if i = 1. The answer is Ψ[r , ∅].

We follow convention that Ψ[i , f ] = −∞ if f⁻¹(1) is already not independent.

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Dynamic programming for Independent Set

V_t =S

i ≤tX_i

X1 V_t · · · X_t

Ψ[i , ∅] = 0 if i = 1.

The answer is Ψ[r , ∅].

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Dynamic programming for Independent Set

V_t =S

i ≤tX_i

X1 V_t · · · X_t

Ψ[i , ∅] = 0 if i = 1.

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Dynamic programming for Independent Set

V_t =S

i ≤tX_i

X1 V_t · · · X_t

Ψ[i , ∅] = 0 if i = 1.

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DP for Independent Set: introduce node

Introduce node i such that X_i = Xi −1∪ {v }.

Goal: For f : X_i → {0, 1} compute

Ψ[i , f ] = max{|S | : S is an independent set in Vi and S ∩ X_i = f⁻¹(1)}. If f (v ) = 0 then

Ψ[i , f ] = Ψ[i − 1, f |_X_{i −1}].

If f (v ) = 1 and there is v⁰ ∈ X_i s.t. vv⁰∈ E (G ) and f (v⁰) = 1, then Ψ[i , f ] = −∞.

Otherwise, we have

Ψ[i , f ] = 1 + Ψ[i − 1, f |_X_{i −1}].

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DP for Independent Set: introduce node

Ψ[i , f ] = max{|S | : S is an independent set in Vi and S ∩ X_i = f⁻¹(1)}.

If f (v ) = 0 then

Ψ[i , f ] = Ψ[i − 1, f |_X_{i −1}].

Otherwise, we have

Ψ[i , f ] = 1 + Ψ[i − 1, f |_X_{i −1}].

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DP for Independent Set: introduce node

Ψ[i , f ] = max{|S | : S is an independent set in Vi and S ∩ X_i = f⁻¹(1)}.

If f (v ) = 0 then

Ψ[i , f ] = Ψ[i − 1, f |Xi −1].

Otherwise, we have

Ψ[i , f ] = 1 + Ψ[i − 1, f |_X_{i −1}].

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DP for Independent Set: forget node

Forget node i such that X_i = Xi −1\ {v }.

Ψ[i , f ] = max(Ψ[i − 1, f [v → 0]], Ψ[i − 1, f [v → 1]]).

There is no need to check edges, since we have already checked them at introduce nodes. That is, even if there is u ∈ X_i such that vu ∈ E (G ) and f (u) = 1, then

Ψ[i − 1, f [v → 1]] = −∞ and the above recursion reduces to

Ψ[i , f ] = Ψ[i − 1, f [v → 0]].

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DP for Independent Set: forget node

Forget node i such that X_i = Xi −1\ {v }.

Ψ[i , f ] = max(Ψ[i − 1, f [v → 0]], Ψ[i − 1, f [v → 1]]).

There is no need to check edges, since we have already checked them at introduce nodes. That is, even if there is u ∈ X_i such that vu ∈ E (G ) and f (u) = 1, then

Ψ[i − 1, f [v → 1]] = −∞

and the above recursion reduces to

Ψ[i , f ] = Ψ[i − 1, f [v → 0]].

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DP for Independent Set: conclusion for pathwidth

Theorem

Given an n-vertex graph and its path decomposition of width p, Independent Set can be solved in time 2^pn^O(1)p^O(1).

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Tree decompositions and treewidth

Graph G = (V , E ) Tree decomposition of G

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi

Tree decomposition is a tree of bags (subsets of V )

such that For every edge uv ∈ E some bag contains u and v

For every vertex v ∈ V bags containing v form nonempty subtree (connected!)

Width of the decomposition: maximum bag size −1 (here: 3). Treewidth of G: minimum width of a decomposition of G .

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Tree decompositions and treewidth

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi

Tree decomposition is a tree of bags (subsets of V ) such that For every edge uv ∈ E some bag contains u and v

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Tree decompositions and treewidth

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi bdef defh

abde dhge

ehfi

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Tree decompositions and treewidth

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi defh

dhge

ehfi

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Tree decompositions and treewidth

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi abde

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Tree decompositions and treewidth

c b

a

f e d

i h g

bdef defh abde

bcf

dhge

ehfi

Width of the decomposition: maximum bag size −1 (here: 3).

Treewidth of G: minimum width of a decomposition of G .

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Tree decompositions and treewidth

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

q, s, u r , u, v q, r , u o, q, r m, n, o

n, p, r

Tree decomposition: Tree of bags

a

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i , s, w g , q, s

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a, b

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f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

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a, b

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v , z b, c, e

c, d

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h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

T1: Every vertex is in some bag

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a, b

e, h, k

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v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

T2: Every edge is in some bag

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a, b

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v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

T3: Bags containing a vertex are connected

a

k b e

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a, b

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v , z b, c, e

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e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

l c f h

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g i

q s w

m o

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

Width of decomposition: Maximum bag size minus 1 Treewidth: Minimum width of a tree decomposition Property: Edge of decomposition induces a separation

a

k b e

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a, b

e, h, k

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c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

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Tree decompositions and treewidth

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

l c f h

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g i

q s w

m o

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

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g i

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m o

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

a

k b e

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g i

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m o

u n

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

a

k b e

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g i

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m o

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

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k b e

l c f h

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g i

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m o

u n

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

l c f h

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g i

q s w

m o

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r v

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

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k b e

l c f h

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g i

q s w

m o

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a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

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Tree decompositions and treewidth

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

a

k b e

l c f h

d

g i

q s w

m o

u n

r v

p

z

a, b

e, h, k

n, o, r

v , z b, c, e

c, d

e, f , h

h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

n, p, r a

k b e

l c f h

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g i

q s w

m o

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a, b

e, h, k

n, o, r

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h, k, l

f , g , h, i g , i , s

i , s, w g , q, s

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f , g , h, i g , i , s

i , s, w g , q, s

n, p, r

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