AN INTRODUCTION TO
TREEWIDTH
Daniel Lokshtanov, UCSD
WHAT IS TREEWIDTH?
A graph parameter that measures how close a
graph is to being a tree.
 Graphs with constant treewidth share many
properties with trees, this can be exploited
algorithmically






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For polynomial time algorithms (ex; minor testing)
Faster Exponential Time Algorithms for hard
problems
Space-Bounded Computation
Parameterized Algorithms
Approximation Algorithms
Preprocessing Heuristics (Kernelization)
ORIGINS



Discovered independently in several fields; graph
grammars, graph minors, logic.
Several different names in the 80’s. The ones i
remember are Treewidth and partial k-trees.
Generalized to digraphs, hypergraphs, matroids
... here; only undirected graphs.
EVERYTHING IS EASY ON TREES
Independent Set (ISET):
IN: Graph G, integer k.
Q: Does G have a vertex set S of size k such that
every pair of vertices in S are non-adjacent?
NP-Complete in general, but linear time for trees.
Why?
DYNAMIC PROGRAMMING FOR ISET ON
TREES
Root the tree T at r
Let Tv be the subtree of T rooted at r
In[v] = Largest iset S in Tv such that v ∈ S.
Out[v] = Largest iset S in Tv such that v ∉ S.
Best[v] = Largest iset in Tv.
In[v] = Σu Out[u]
Out[v] = Σu Best[u]
Best[v]=max{In[v], Out[v]}
GENERALIZE TREES?
Dynamic Programming works because the
information must transfer through a single
vertex.
 Generalize to information transfer through a
constant number of vertices.

Width = 3
Tree-Partition
MORE GENERAL DYNAMIC PROGRAMMING
G
= graph
 T = decomposition tree. For each node v of T
there is a bag Bv that contains vertices of G.
 The bags of T partition the vertices of G.
 Root the tree T, Tv is the subtree rooted at T.
 Gv is G[∪u∈ Tv Bu]
ISET ON TREE-PARTITIONS
Define Best[v,S] where v is a node in T and S ⊆ Bv
to be the size of the largest independent set X of
Gv such that X ∩ Bv = S.
Test[X] = 1 if G[X] is independent, -∞ if
not.
Best[v,S] = |S| + Σu MaxS’⊆Bu × Test[S ∪ S’]
ARE TREE-PARTITIONS THE RIGHT
GENERALIZATION?
u
v1
v2
v3
vn-2
vn-1
vn
What is the tree-partition-width?
Decomposition tree must be a star with the bag B
containing u in the center.
At least one component of size ≥ n/|B|
tree-partition-width ≥ n1/2
TOWARDS TREE-WIDTH


Need to make tree-partitions more robust; adding
a new vertex should increase the ”width” by at
most one, retaining nice algorithmic properties.
What if every vertex could appear several times,
but every edge only needed to appear once?
v

y
Problem: Can decompose anything as a path!
u
v
x
y
u
u
v
v
x
x
x
u
y
y
TREE-DECOMPOSITION DEFINITION
A tree-decomposition of a graph G=(V,E) is a tree T
and a collection X of bags, with a bag Ba ∈ X for
every node a ∈ T, such that:
1.
2.
For every uv ∈ E, some bag contains both u and v
For every vertex v ∈ V, v appears in a non-empty,
connected subtree of T.
Treewidth is NP-complete (Yannakakis)
The width of the decomposition is max |Bv| - 1
The treewidth of G is the smallest width over any
decomposition.
EXAMPLE

Treewidth of trees is 1
{2}
2
{3}
3
{8,a}
8
1
4
7
b
9
5
6
c
a
CONNECTIVITY PROPERTIES:
Lemma: Let (T,X) be a tree-decomposition of G,
and let e=ab be an edge in T, splitting T into T1
and T2. Set V1 = ∪v ∈ T1 Bv and V2 = ∪v ∈ T2 Bv.
Then,
V1 ∩ V2 = Ba ∩ Bb and there are no edges between
V1 \ (Ba ∩ Bb) and V2 \ (Ba ∩ Bb ).
V1
ua
ub
u
u
v
T1
T2 uv
V2
MORAL OF PREVIOUS SLIDE
 For
a node v of T, G \ Bv splits into
components corresponding to the
components of T \ v.
EXERCISE


Prove that ISET can be solved in time
O(4wpoly(n)) if a tree-decomposition (T,X) of
width ≤ w is given.
Hint: Root the decoposition tree T at a vertex r.
Define Tv to be the subtree below v, and Vv to be
∪u∈ Tv Bu.
SOLUTION


Define Best[v,S] (where S ⊆ Bv) and G[S] is an
iset, to be the size of the largest independent set
X in G[Vv] such that X ∩ Bv = S.
Best[v,S] = |S|Σ+
u
Children of
v
Max Best[u,S’]-|S∩ Bu|
S ∩ Bu ⊆ S’
S’ ⊆ S ∪ (Bu \ Bv)
S ∪ S’ is an iset.
STRUCTURAL PROPERTIES
ADDING / DELETING VERTICES
Lemma: (a) Adding a vertex v to G increases tw(G)
by at most 1. (b) Deleting a vertex v from G can
not increase tw(G).
Proof (a): Add v to all bags.
Proof (b): Delete v from all bags it appears in.
Corollary: (a) Adding an edge e to G increases
tw(G) by at most 1. (b) Deleting an edge e from G
can not increase tw(G).
EXERCISE

Show that treewidth of graph below is ≤ 2.
u
v1
v2
v3
vn-2
vn-1
vn
Proof: G \ u is a tree, hence of treewidth 1. Removing u
decreased treewidth by at most 1.
CONTRACTING AND SUBDIVIDING EDGES

Contracting an edge uv in G can not increase
tw(G).
Proof: Replace all occurences of u and v by the new
vertex ”uv”. Since some bag contained u and v, all
is well.

Subdividing an edge uv can not increase tw(G).
Proof: Some bag contains uv. Add a new bag to it.
TREEWIDTH OF CLIQUES
Lemma: In a tree-decomposition of a clique C, all
of C must appear in some bag.
Proof: (Board for figure)
Suppose not. Root the decomposition at a node a
that maximizes |Ba ∩ C| (of course, |Ba ∩ C| <
|C|).
Let u ∈ C and u ∉ Ba. Let b be the topmost bag
where u is.
Some v ∈ Ba ∩ C is not in Bb ∩ C.
Now uv can’t be in any bag!
EXERCISE
Prove that a graph G has treewidth ≤ 1 if and only
if it is a forest (contains no cycles)
Soln:
 Trees have treewidth 1 so forests do to.
 If G contains a cycle Cn then tw(G) ≥ tw(Cn)
because deleting vertices does not increase
treewidth. tw(Cn)≥ tw(C3) because contracting
edges does not increase tw. tw(C3)=2.
BALANCED SEPARATORS
Lemma: Let G be a graph of treewidth t, and X ⊆
V. There exists a set S of size at most t+1 such
that for every connected component C of G\S,
|C ∩ X| ≤ |X|/2. (*)
Proof: Root the tree at a node set p initially to be
the root. While there is a child q of p such that
|Vq ∩ X| > |X|/2, set p=q.
BALANCED SEPARATORD CONT’D
When the algorithm terminates, the subtrees of all
children satisfy |C ∩ X| ≤ |X|/2. If p has a
parent then |Vp ∩ X| > |X/2| (since the parent
wasn’t chosen) so the component C = G \ Tv
satisfies |C ∩ X| ≤ |X/2|.
≤ |X|/2
p
> |X|/2
≤ |X|/2
BALANCED SEPARATORS 2: EXERCISE
Lemma: For a graph G=(V,E) of treewidth t and a
vertex set X, there is a partition of V into L ∪ S ∪
R such that:



|S| ≤ t+1
There are no edges from L to R
max{|L ∩ X|, |R ∩ X|} < 3|X|/4
Hint: Use previous Lemma + bin packing
arguments.
SOLN.
Find a set S of size t+1 such that all components C
of G \ S satisfy |C ∩ X| ≤ |X|/2. Let C1 and C2
be the componentsC with
largest
intersection
with
fits into
L if |(L ∪
C) ∩ X| ≤ |X|/2
X. Put C1 into L, C2 intoIf R,
and place the
two components Ca and Cb
remaining componetns don’t
intofit
L then
or Rtheif smallest
they fit.
of them
fits into the smallest of L and R.
At most one component C, does not fit.
wlog:
so:
|L ∩ X| ≤ |X-C|/2
|(L ∪ C) ∩ X| ≤ |X+C|/2 ≤ 2|X|/3
ISET IN POLYNOMIAL SPACE
Theorem: Independent set in graphs of treewidth t
can be solved in time nO(t) and using polynomial
space.
Proof: Find partition of G into L, R and S as
described, using X=V. Guess the intersection of
solution with S. Solve for L and R recursively.
T(n) ≤ 2t × 2T(2n/3) ≤ 2t × log3/2 n ≤ nO(t)
WHAT ABOUT ISET IN F(TW)NO(1) TIME
AND POLYNOMIAL SPACE?
Theorem: Independent Set in graphs of treewidth
t can be solved in time 2O(t2)nO(1) and using
polynomial space.
Proof:
If n ≥ 2t then the 4tnO(1), 2tnO(1) space DP algorithm
runs in polynomial space.
2
If n ≤ 2t then nO(t) ≤ (2t)O(t) ≤ 2O(t )nO(1).
COMPUTING TREEWIDTH
Theorem: [Bodlaender, early 90’s]: The treewidth
of G can be computed in time 2O(t3)n.
WHAT TO OPTIMIZE?


For algorithms on graphs on bounded treewidth
one can try to optimize the dependence on t, or
the dependence on n.
For the dependence on n, Courcelle’s theorem
gives very strong results.
(C)MSOL
Quantifiers: ∃ and ∀ for variables for vertex sets
and edge sets, vertices and edges.
Operators: = and ∊
Operators: inc(v,e) and adj(u,v)
Logical operators: ∧, ∨ and ¬
(C): Size modulo fixed integers operator:
eqmodp,q(S)
EXAMPLE: p(G,S) = “S is an independent set of G”:
p(G,S) = ∀u, v ∊ S, ¬adj(u,v)
CMSOL OPTIMIZATION PROBLEMS
FOR COLORED GRAPHS
Φ-Optimization
Input: G, C1, ... Cx
Max / Min |S|
So that Φ(G, C1, Cx, S)
holds.
CMSOL definable
proposition
MODEL CHECKING CMSOL PROPERTIES
Φ-Model-Checking:
IN: G, C1, ... Cx
Q: Does Φ(G, C1, ... Cx) hold?
The properties ”G has an independent set of size
10” and ”G has an independent set of size 100”
need different formulas!
COURCELLE’S THEOREM(S)
V1.0: For every CMSOL property Φ and integer t, there
is a linear time algorithm for Φ-Model-Checking on
graphs of treewidth t.
V1.1: For every CMSOL property Φ and integer t, there
is a linear time algorithm for Φ-Optimization on
graphs of treewidth t.
Often accredited to Courcelle, actually proved
by Arnborg, Lagergren and Seese.
OPTIMIZING F(TW)

Until recently, fairly naive algorithms were the
best ones known for most problems, and we did
not know why. In 2008, the list to beat was:
2twn for ISET
 4twn for DOMSET
 qtwn for q-Coloring
 tw!n for Hamiltonian Cycle

OPTIMIZING F(TW) – CURRENT PICTURE
2twn for ISET
 3twn for DOMSET [new upper bound, ESA09]
 qtwn for q-Coloring
 4twn for Hamiltonian Cycle
[new upper bound, Arxiv’11]

Matching Lower bounds
under SETH, SODA’11