1. Algorithms and Complexity
Lecture 8
Introduction to Treewidth

2. Treewidth Definitely today: Possibly today (or next week, or not):
A measure of how `treelike’ a graph is
On a high level, very useful for two reasons:
very many NP-hard graph problems can be solved fast on graphs of small treewidth,
many graphs of interest have small treewidth.
Definitely today:
Defn of treewidth, how to use it, one application
Possibly today (or next week, or not):
Planar graphs, apx scheme in planar graphs
FPT via graph minors.

3. Weighted Independent Set in Trees
Let 𝑇 be a rooted tree, 𝑇[𝑣] the subtree rooted at v∈𝑉.
Define 𝐴 𝑣 as the max weight of an ind. set of 𝑇[𝑣], then v
=max , 𝜔 𝑣 +

4. Weighted Independent Set in Trees
Let 𝑇 be a rooted tree, 𝑇[𝑣] the subtree rooted at v∈𝑉.
Define 𝐴 𝑣 as the max weight of an ind. set of 𝑇[𝑣], then
For each bag 𝑣 in 𝑇 and subsets 𝑋 of the vertices in it, store the optimal IS of 𝑇 𝑣 containing that set in the bag T
𝐴[𝑣, 𝑎,𝑑 ] v a b c d

5. Treewidth
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇.

6. Treewidth
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇.

7. Treewidth Definition a b c d f g h e
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. a b c d f g h e

8. Treewidth Definition d b a a b c d e f g h
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. d b a a b c d e f g h

9. Treewidth Definition d b a a b c d g b d e f g h
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. d b a a b c d g b d e f g h

10. Treewidth Definition d b a a b c d g b d e f g d f g h
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. d b a a b c d g b d e f g d f g h

11. Treewidth Definition d b a a b c d g b b e g d e f g d f g h
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. d b a a b c d g b b e g d e f g d f g h

12. Treewidth Definition c e b d b a a b c d g b b e g d e f g d f g h
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇. c e b d b a a b c d g b b e g d e f g d f g h

13. Treewidth Definition c e b d b a a b c d g b b e g d e f g d g h e f g
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇.
Separates a, f and ceh. c e b d b a a b c d g b b e g d e f g d g h e f g h

14. Treewidth Definition Definition
Refer to        as a bag.
Definition
A tree decomposition of graph 𝐺= 𝑉,𝐸 is a pair (𝑋,𝑇) where 𝑋={ 𝑋 1 ,…, 𝑋 𝑙 } with 𝑋 𝑖 ⊆𝑉 and 𝑇 a tree with vertex set 𝑋 such that
𝑖=1 𝑙 𝑋 𝑖 =𝑉 ,
𝐸⊆ 𝑖=1 𝑙 𝑋 𝑖 × 𝑋 𝑖 ,
∀𝑣∈𝑉: all 𝑋 𝑖 containing 𝑣 induce connected subtree if 𝑇.
Definition
The width of a treedecomposition is                                . The treewidth of a graph is the minimum width among all possible tree decompositions of G.

15. Cops and Robber Interpretation
𝑤 cops try to capture a robber
Everybody sees everybody
In each turn, the cops can move from vertex to vertex arbitrarily with helicopters.
The robber moves infinitely fast on the edges, and sees where the cops will land.
𝑤+1 cops can eventually win iff graph has treewidth ≤𝑤.
Proof is not easy and we skip it
has treewidth 2

16. Cops and Robber Interpretation
𝑤 cops try to capture a robber
Everybody sees everybody
In each turn, the cops can move from vertex to vertex arbitrarily with helicopters.
The robber moves infinitely fast on the edges, and sees where the cops will land.
𝑤+1 cops can eventually win iff graph has treewidth ≤𝑤.
Proof is not easy and we skip it l
has treewidth ≤l
(exercise) r b a
has treewidth 2 c b

17. Finding Good Tree Decompositions
Treewidth problem: Given 𝐺 and integer w, does 𝐺 have treewidth w?
Treewidth param’d by w is NP-hard, but also FPT:
Idea is to use iterative compression:
A given tree decomposition of width w+1 can be used to determine the treewidth
Full algorithm is technical and beyond the scope of the course
There are algorithms that given graph of TW w,
return TD of width w in 𝑤 𝑂(𝑤 3 ) 𝑛 time.
return TD of width 4𝑤+1 in 𝑂( 3 3𝑤 𝑤 𝑛 2 ) time
return TD of width O(𝑤 lg 𝑤 ) in polynomial time

18. Max weight IS with given TD in 𝑂 ∗ (2 𝑤 )
Use given TD to compute a nice TD of width k.
For a bag 𝑖, let 𝐺 𝑖 =( 𝑉 𝑖 , 𝐸 𝑖 ) be the subgraph of 𝐺 formed by all vertices in sets 𝑋𝑗, with 𝑗=𝑖 or j a descendant of i in the tree.
𝐴[𝑖,𝑌] = maximum weight of an independent set W of 𝐺 𝑖 with 𝑊∩ 𝑋 𝑖 =𝑌 (which is −∞ if such 𝑊 does not exist)
𝐴[6,{ℎ}]=1, 𝐴 4, 𝑒,𝑔 =−∞,𝐴 1, 𝑏 =3, 𝐴 3,{𝑔} =1
1=root 5 a b c e b c d b a 3 4 d g b b d e e g f g d g h e f g h 2 6

19. Nice tree decompositions
𝑇,𝑋 where 𝑇 is a rooted tree, and four types of bags i:
Leaf: leaf of tree with | 𝑋 𝑖 | = 1
Join: bag with two children 𝑗,𝑗′ with 𝑋 𝑖 = 𝑋 𝑗 = 𝑋 𝑗′ ’
Introduce: bag with one child j with 𝑋 𝑖 = 𝑋 𝑗 ∪𝑣 for some vertex 𝑣
Forget: bag with one child j with 𝑋 𝑖 = 𝑋 𝑗 ∖𝑣 for some vertex v
Given any TD, we can compute a nice TD of same width in polynomial time.

20. Nice tree decompositionsg b a c e h f g d b a c e h f
root
Leaf
Join
Introduce
Forget

21. 𝐺 𝑖 is a graph with one vertex
Leaf bags
Let i be a leaf bag. Say 𝑋 𝑖 ={𝑣}
𝐴 𝑖, 𝑣 =𝜔 𝑣
𝐴 𝑖,∅ =0 v
𝐺 𝑖 is a graph with one vertex

22. 𝜔( ) Join bags - = + Let i be a join bag with children 𝑗1, 𝑗2.
𝐴 𝑖,𝑌 =𝐴 𝑗 1 ,𝑌 +𝐴 𝑗 2 ,𝑌 − 𝑣∈𝑌 𝜔(𝑣) + -
𝜔( ) =

23. Introduce bags = = + 𝜔(𝑣) = −∞
Let i be an introduce bag with child 𝑗, 𝑋 𝑖 = 𝑋 𝑗 ∪{𝑣}
𝐴 𝑖,𝑆 =𝐴[𝑗,𝑆] for 𝑣∉𝑆
𝐴 𝑖,𝑆∪{𝑣} =𝐴 𝑗,𝑆 +𝜔(𝑣), if 𝑁(𝑣)∩𝑆=∅
𝐴 𝑖,𝑆∪{𝑣} =−∞ if 𝑁(𝑣)∩𝑆≠∅ v = v =
+ 𝜔(𝑣) v
= −∞

24. Forget bags = 𝑚𝑎𝑥 , Let i be a forget bag with child 𝑗, 𝑋 𝑖 = 𝑋 𝑗 ∖{𝑣}
𝐴 𝑖,𝑆 = max { 𝐴 𝑗,𝑆 ,𝐴 𝑗,𝑆∪𝑣 }
= 𝑚𝑎𝑥 ,

25. Maximum weighted independent set on graphs with treewidth k
Compute all table entries in bottom up fashion (starting at the leaves of T) in the usual way.
Gives 2 𝑤 𝑤 𝑂(1) 𝑛= 𝑂 ∗ 2 𝑤 -time algorithm for max weight ind. set if TD of width w is given.
FPT algorithms for very many, NP-hard graph problems when parameterized by treewidth.

26. Weighted Dominating Set in Trees
Dominating set: 𝑋⊆𝑉 such that each vertex is in 𝑋 or is a neighbor of some vertex in X.
Can also be solved in 𝑂(𝑛) time on trees, and 3 𝑤 𝑤 𝑂(1) 𝑛 time if a tree decomposition of width w is given ( 9 𝑤 𝑤 𝑂 1 𝑛 in ex).
Has a dominating set of size 2

27. Weighted Dominating Set in Trees
Dominating set: 𝑋⊆𝑉 such that each vertex is in 𝑋 or is a neighbor of some vertex in 𝑥.
Define 𝐴 𝐷 𝑣 as min weight DS of 𝑇 𝑣 ∖𝑣
Define 𝐴 𝐼 𝑣 as min weight DS of 𝑇[𝑣] including 𝑣
Define 𝐴 𝐸 𝑣 as min weight DS of 𝑇[𝑣] excluding 𝑣
If 𝑣 is leaf, 𝐴 𝐸 𝑣 =∞, 𝐴 𝐷 [𝑣]=0, 𝐴 𝐼 𝑣 =𝜔(𝑣), otherwise:
𝐴 𝐷 𝑣 = 𝑐∈𝑐ℎ(𝑣) min { 𝐴 𝐼 𝑐 , 𝐴 𝐸 [𝑐]} v

28. Weighted Dominating Set in Trees
Dominating set: 𝑋⊆𝑉 such that each vertex is in 𝑋 or is a neighbor of some vertex in 𝑥.
Define 𝐴 𝐷 𝑣 as min weight DS of 𝑇 𝑣 ∖𝑣
Define 𝐴 𝐼 𝑣 as min weight DS of 𝑇[𝑣] including 𝑣
Define 𝐴 𝐸 𝑣 as min weight DS of 𝑇[𝑣] excluding 𝑣
If 𝑣 is leaf, 𝐴 𝐸 𝑣 =∞, 𝐴 𝐷 [𝑣]=0, 𝐴 𝐼 𝑣 =𝜔(𝑣), otherwise:
𝐴 𝐷 𝑣 = 𝑐∈𝑐ℎ(𝑣) min { 𝐴 𝐼 𝑐 , 𝐴 𝐸 [𝑐]}
𝐴 𝐼 𝑣 =𝜔 𝑣 + 𝑐∈𝑐ℎ(𝑣) min { 𝐴 𝐷 𝑐 , 𝐴 𝐼 𝑐 } v

29. Weighted Dominating Set in Trees
Dominating set: 𝑋⊆𝑉 such that each vertex is in 𝑋 or is a neighbor of some vertex in 𝑥.
Define 𝐴 𝐷 𝑣 as min weight DS of 𝑇 𝑣 ∖𝑣
Define 𝐴 𝐼 𝑣 as min weight DS of 𝑇[𝑣] including 𝑣
Define 𝐴 𝐸 𝑣 as min weight DS of 𝑇[𝑣] excluding 𝑣
If 𝑣 is leaf, 𝐴 𝐸 𝑣 =∞, 𝐴 𝐷 [𝑣]=0, 𝐴 𝐼 𝑣 =𝜔(𝑣), otherwise:
𝐴 𝐷 𝑣 = 𝑐∈𝑐ℎ(𝑣) min { 𝐴 𝐼 𝑐 , 𝐴 𝐸 [𝑐]}
𝐴 𝐼 𝑣 =𝜔 𝑣 + 𝑐∈𝑐ℎ(𝑣) min { 𝐴 𝐷 𝑐 , 𝐴 𝐼 𝑐 }
𝐴 𝐸 𝑣 = min 𝑐∈𝑐ℎ(𝑣) { 𝐴 𝐼 [𝑐] + 𝑐 ′ ∈𝑐ℎ 𝑣 ∖𝑐 min⁡{ 𝐴 𝐼 𝑐 ′ , 𝐴 𝐸 [𝑐′]} v v
still need to dominate v!

30. Weighted Dominating Set and Treewidth
Similar approach as for ind. set.
Table 𝑇[𝑖,𝐷,𝐸,𝐼] for bag 𝑖 and partition D,E,I of 𝑋 𝑖
We need 3 states now for every vertex!
Excluded and dominated already (D)
Excluded and still needs to be dominated (E)
Included (I)
Gives 9 𝑤 𝑤 𝑂(1) 𝑛 time algo, if given TD of width w ( exercise, 𝑂 ∗ (3 𝑤 ) table entries, additional time used for join bags)

31. One first application
In the 𝑘-path problem we are given an undirected graph 𝐺 and need to determine whether there exists a simple path on 𝑘 vertices.

32. DFS /BFS reminder BFS(G,s) Let S be a queue S.push((𝜀,s))
WHILE S not empty
(u,v)= S.pop()
If v is not labeled `visited’
Label v as visited
Add (u,v) to T
For all neighbors w of v
S.push(w)
DFS(G,s)
Let S be a stack
S.push((𝜀,s))
WHILE S not empty
v= S.pop()
If v is not labeled `visited’
Label v as visited
Add (u,v) to T
For all neighbors w of v
S.push(w)

33. A first application of treewidth
𝑘-path: given an undirected graph 𝐺=(𝑉,𝐸), determine whether there is a simple path on 𝑘 vertices.
NP-complete (𝑘=𝑛 is Hamiltonian Path).
Using similar methods as for IS and DS, there is an algorithm determining whether a 𝑘-path exists in 𝑤 𝑂(𝑤) 𝑛 time, if given TD of width w
Using this fact we solve 𝑘-path in 𝑘 𝑂(𝑘) 𝑛 time:
Pick 𝑠∈𝑉 arbitrarily, perform DFS from 𝑠. If height of DFS-tree ≥𝑘, return YES since path from 𝑠 to some leaf has length at least 𝑘.
Otherwise, if the DFS-tree has height ≤𝑘, we use it to find a TD of width at most 𝑘 and we can apply the above algorithm

34. A first application of treewidth≥𝑘

35. A first application of treewidth
<𝑘
𝑙 1
𝑙 5
𝑙 7
𝑙 2
𝑙 3
𝑙 4
𝑙 6
Let leaves be 𝑙 1 ,…, 𝑙 𝑝 .
Create a tree decomposition as follows:
T is the path on vertices 𝑋 1 ,…, 𝑋 𝑝
𝑋 𝑖 contains all ancestors of 𝑙 𝑖 in the DFS tree.
Covers all vertices. Covers all edges since DFS tree.
Third requirement: 𝑇 visits children of 𝑣 consecutively

36. Planar graphs
Graphs that can be drawn in the plane without edge-crossings.
Given such a drawing, regions are called faces
Euler’s formula: If 𝐺 connected, 𝑛 vertices, 𝑚 edges and a drawing with 𝑓 faces, then:
𝑓+𝑛=𝑒+2
Proof by induction on 𝑒. If e>0, we can either:
Remove degree 1 vertex (𝑛 and 𝑒 decrease), or
Remove edge on a cycle (𝑒 and 𝑓 decrease).
If 𝐺 connected and 𝑒≥3, then 𝑒≤3𝑛−6
𝐾 5 has 10 edges -> not planar

37. Graph Minor
𝐻 is a minor of 𝐺: 𝐻 can be obtained from 𝐺 with edge deletion, vertex deletion and edge contraction
Edge contraction:
is a minor of d e a b c f g h i a b c f g h de i

38. Graph Minor
𝐻 is a minor of 𝐺: 𝐻 can be obtained from 𝐺 with edge deletion, vertex deletion and edge contraction
Edge contraction:
is a minor of a f a b c f g h de b d e g c i h i
is a minor of 𝐺 iff ???
𝐺 has a cycle
Kuratowski’s theorem:
𝐺 planar iff it has no 𝐾 5 or 𝐾 3,3 as minor.

39. Grid Minors Grid Minor Theorem
For every integer l, every planar graph either has a (𝑙×𝑙)-grid as a minor or treewidth at most 9𝑙.
Moreover, there exists a polynomial time algorithm that either finds such a minor model or a TD.
Proof uses max-flow min-cut arguments, but beyond scope of the course...
Corollary: Planar graphs have treewidth at most 𝑛
Planar weighted Independent, Dominating Set in 2 𝑂( 𝑛 ) time
Planar Hamiltonian cycle in 2 𝑂( 𝑛 𝑙𝑜𝑔𝑛 ) time

40. TW of planar graphs with shallow spanning tree
Theorem Let 𝐺 be a planar graph and 𝑆 be a spanning tree of 𝐺 of height ℎ+1. Then given 𝐺 and 𝑆, a TD of 𝐺 of width at most 3ℎ can be found in polynomial time.
Proof idea:
Triangulate
Bag for each face
Edges in 𝑇 for faces sharing edge not in S
Bag contains all verts in face and ancestors in S
Covers all edges and faces
𝑇 has no cycle since 𝑆 is spanning. Bags containing vertex 𝑣 are connected since we can walk around faces adjacent to 𝑆[𝑣]
𝑇 connected by previous property since all bags contain root.
Each bag ≤3(ℎ+1) since all 3 vertices of face ≤ℎ ancestors. 1 5 3 4 6 7 8 9 10 11 14 12 13 15 2 16 a b e d f c i h g j 1 5 3 4 6 7 8 9 10 11 14 12 13 15 2 16 a b e d f c i h g j
X1={abc}
X2={abd}
X3={ade}
X4={aeg}
X5={abcd}
X6={acdef}
X7={acefh}
X8={acegh}
X9 ={acdf}
X10 ={acfh}
X11 ={acghj}
X12 ={achi}
X13 ={achij}
X14 ={agcj}
X15 ={acij}
X16 ={acg} 1 5 3 4 6 7 8 9 10 11 14 12 13 15 2 16 a b e d f c i h g j a b e d f c i h g j a b e d f c i h g j a b e d f c i h g j
root
To see that the constant works out, the root is accounted for thrice in 3(h+1) so subtract 2 and also subtract one because of the definition of treewidth.

41. Baker’s approach for approximation in planar graphs
Given planar graph 𝐺, 𝜔:𝑉→ℕ and 𝜖>0. Find IS of weight at least 1−𝜖 𝑂𝑃𝑇 in 𝑂 ∗ (2 𝑂(1/𝜖) ) time
Pick vertex 𝑠 arbitrarily. Do BFS from 𝑠
Let 𝐿 𝑖 be all vertices at distance 𝑖
Let 𝑘=1/𝜖
Let 𝑉 𝑖 = 𝑗≠𝑖 (𝑚𝑜𝑑 𝑘) 𝐿 𝑗
For 𝑖=1,…,𝑘
Find max weight IS of 𝐺 𝑉 𝑖 s
𝐿 1
𝐿 2
𝐿 3
𝐿 4
𝐿 𝑑

42. Baker’s approach for approximation in planar graphs
Given planar graph 𝐺, 𝜔:𝑉→ℕ and 𝜖>0. Find IS of weight at least 1−𝜖 𝑂𝑃𝑇 in 𝑂 ∗ (2 𝑂(1/𝜖) ) time
Pick vertex 𝑠 arbitrarily. Do BFS from 𝑠
Let 𝐿 𝑖 be all vertices at distance 𝑖
Let 𝑘=1/𝜖
Let 𝑉 𝑖 = 𝑗≠𝑖 (𝑚𝑜𝑑 𝑘) 𝐿 𝑗
For 𝑖=1,…,𝑘
Find max weight IS of 𝐺 𝑉 𝑖 :
Return max weight found
Let 𝑉 𝑖 =𝑉∖ 𝑉 𝑖 . 𝑉 𝑖 partition 𝑉.
If 𝐼 is MWIS, ∃𝑖: 𝜔 𝐼∖ 𝑉 𝑖 ≤𝑂𝑃𝑇
Then MWIS of 𝐺 𝑉 𝑖 ≥𝑂𝑃𝑇−𝑂𝑃𝑇/𝑘=𝑂𝑃𝑇 1−𝜖 s
𝐿 1
𝐿 2
𝐿 3
𝐿 4
𝐿 𝑑

43. Baker’s approach for approximation in planar graphs
For a<b<c, 𝐿 𝑏 separates 𝐿 𝑎 from 𝐿 𝑐 by distance property.
𝐺 𝑉 𝑖 splits in cc’s 𝐿 1 … 𝑙 𝑗−1 , 𝐿 𝑗+1 … 𝐿 𝑗+𝑘−1 , 𝐿 𝑗+𝑘+1 … 𝐿 𝑗+2𝑘−1 ,…
Find TD of each cc (can later obtain TD of 𝐺[ 𝑉 𝑖 ] form this by adding empty bag to connected the tree)
To find TD of each cc:
Let 𝑎 be the min. int such that L 𝑎 is in cc
Add a vertex 𝑧 adjacent to every vertex of 𝐿 𝑎
This new graph is still planar since we can put 𝑧 at 𝑠’s place at the embedding and the BFS tree from 𝑠 has paths from 𝑠 to 𝐿 𝑎 avoiding vertices from 𝐿 𝑏 with 𝑏>𝑎
Perform BFS from 𝑧 in this graph.
Gives tree of height at most 𝑘 by definition of layers. Apply previous thm.
Use the 𝑂 ∗ 2 𝑤 time algorithm for MWIS.
𝐿 5
𝐿 4 z