1. Parameterized complexity Bounded tree width approaches
Algorithms for hard problems
Parameterized complexity
Bounded tree width approaches
Juris Viksna, 2017

2. Tree decomposition of graph
The purpose of the “−1” in the definition of the width of a decomposition is to let trees have tree width 1
[Adapted from R.Downey and M.Fellows]

3. Tree decomposition of graph
This is equivalent with requirement that for all x,zT,
if y belongs to path between x an z then TxTzTy.
This is slightly differently phrased definition of tree decomposition from J.Flum and M.Grohe. However note that also somewhat different notation is used here.
[Adapted from J.Flum and M.Grohe]

4. Tree decomposition - example
[Adapted from J.Flum and M.Grohe]
Tree decompositions are undirected trees. However sometimes in proofs or applications it may be convenient to regard them as rooted trees; the choice of root in such cases can be arbitrary.

5. Tree decomposition - some terminology
Some other extra bits of notation used by J.Flum and M.Grohe.
[Adapted from J.Flum and M.Grohe]

6. Tree decompositions of subgraphs and minors
If H is a minor of G then we also have tw(H) ≤tw(G).
Why?
[Adapted from J.Flum and M.Grohe]

7. Treewidth and k-trees
[Adapted from R.Downey and M.Fellows]

8. k-trees - examples
[Adapted from R.Downey and M.Fellows]

9. k-trees - examples Example of Tree Decomposition of Width 2
[Adapted from R.Downey and M.Fellows]

10. Treewidth – some notes on terminology
In slides we have mainly adapted terminology used by R.Downey and M.Fellows:
Treewidth tw(G) of graph G is defined as the smallest k such that G is a partial k-tree.
The width of tree decomposition k of graph G is maximal size of bag assigned to some vertex of G tree decomposition T minus one:
k = max{|Tx|1| xT }.
Then we proceed to show that tw(G) = k, where k is minimal width of all the possible G tree decompositions.
Note that some inconsistencies might be present regarding this terminology both in slides and textbooks 

11. Treewidth - equivalence of definitions
[Adapted from R.Downey and M.Fellows]

12. Treewidth - equivalence of definitions
[Adapted from R.Downey and M.Fellows]

13. Treewidth - equivalence of definitions
[Adapted from R.Downey and M.Fellows]

14. Treewidth - equivalence of definitions
  
assume we have is a partial k-tree G’
G’ is a subgraph of k-tree G
we are building a tree T describing construction of G
the root of T represents an initial (k+1)-clique K and is labeled with K vertices
each step of “cloning” some (k+1)-clique K’ to (k+1)-clique K’’ is represented by edge from T node labeled with vertices of K’ to
a new tree node labeled with vertices from K’’
- T is a tree decomposition of G with treewidth k
- there is a subtree T’ of T with treewidth k that is a tree decomposition of G’

15. Treewidth - equivalence of definitions
   
By induction of size of T for trees with
fixed treewidth k
if T has single node of width k, the underlying graph G is a subgraph
of (k+1)-clique, thus G is a partial k-tree
assume Tx is a leave of T attached under node Ty and the underlying graph G‘ of TTx is a partial k-tree
if Tx Ty, we can just ignore (remove) it
otherwise Tx = T’x  T’’x, where T’x = Tx  Ty
- T’x is a part of (k+1)-clique K (by induction hypothesis)
we need to add each z T’’x to a (k+1)-clique and to guarantee that
z is connected by edges to all vertices of T’x
we start by cloning of K, then in each step remove from K a vertex not from T’x and replace it with the vertex z generated in previous step
T’x
T’x T’’x
remaining T

16. Computing of treewidth – Bodlaender’s theorem
[Adapted from R.Downey and M.Fellows]

17. Computing of treewidth – Bodlaender’s theorem
Sounds almost too good, however running time is (roughly) O(n232k3) 
[Adapted from R.Downey and M.Fellows]

18. Computing of treewidth – Bodlaender’s theorem
[Adapted from J.Flum and M.Grohe]

19. Computing of treewidth – Bodlaender’s theorem
p(k) = 32k3 :) Although the algorithm is simple and constructive, it is still non-practical.
Note that a simple non-constructive proof follows from the fact that graphs with treewidth bounded by k forms an ideal with respect to minor ordering.
[Adapted from J.Flum and M.Grohe]

20. Small tree decompositions
[Adapted from J.Flum and M.Grohe]

21. Small tree decompositions
[Adapted from J.Flum and M.Grohe]

22. Separability Lemma
[Adapted from J.Flum and M.Grohe]

23. Separability Lemma
In principle a useful result, unfortunately with a bit confusing statement. So, what does it actually mean?
G – a graph.
T – a tree decomposition of G.
x,y – adjacent vertices of T, removal of edge {x,y} splits T into T1 and T2.
u – a vertex of some bag from T1, v – a vertex of some bag from T2.
Then every path in G connecting u and v (possibly with exception of the edge {u,v} itself) intersects with set TxTy.
[Adapted from J.Flum and M.Grohe]

24. Treewidth examples – trees and cycles
[Adapted from J.Flum and M.Grohe]

25. Treewidth examples - cliques
[Adapted from J.Flum and M.Grohe]

26. Treewidth examples – k-connected graphs
[Adapted from J.Flum and M.Grohe]

27. Treewidth examples - grids
[Adapted from J.Flum and M.Grohe]

28. Balanced seperators
[Adapted from J.Flum and M.Grohe]

29. Treewidth examples - grids
Thus, planar graphs do not have bounded treewidth...
[Adapted from J.Flum and M.Grohe]

30. Treewidth examples - grids
[Adapted from J.Flum and M.Grohe]

31. Treewidth of planar graphs
[Adapted from R.Niedermeier]

32. Treewidth for some types of graphs
[Adapted from
R.Downey and
M.Fellows]

33. Treewidth for some types of graphs
[Adapted from H.Bodlaender]

34. Treewidth - yet another property
[Adapted from J.Flum and M.Grohe]

35. Algorithms on structures of bounded tree width
[Adapted from J.Flum and M.Grohe]

36. Algorithms on structures of bounded tree width
Independent set – a set of pairwise non-adjacent vertices.
When parameterized by the size of independent set k the problem is known to be in W[1] – so very unlikely to be FPT. However the problem is FPT when parameterized by tw(G).
[Adapted from J.Flum and M.Grohe]

37. tw-INDEPENDENT SET G – graph with treewidth k.
Find maximal independent set of G in time poly(n)f(k).

38. tw-INDEPENDENT SET
[Adapted from R.Downey and M.Fellows]

39. tw-INDEPENDENT SET
Dynamic programming of tree decomposition for tw-IDEPENDENT SET problem:
G – graph with treewidth k.
Chose an arbitrary rooted tree decomposition T of G with width k and with branching factor 2 or less.
For each xT and each subset STx assign label l(S)=|S| if vertices of S form and independent set and l(S)=1 otherwise. (Number of such subsets S is 2k or less, for each subset the labeling can be done in time k2 or less).

40. tw-INDEPENDENT SET
Starting from leaves of T recursively for each parent vertex x, child vertices y and z of x, and for each set STx with l(S)1 find two subsets S1Ty and S2Ty such that:
l(S1)1 and l(S2)1;
S, S1 and S2 are mutually disjoint and there are no edges between them;
L= l(S)+l(S1)+l(S2) is maximal;
and reassign l(S)=L. (This can be done in time 23k or less.)
Output the largest label l(S) in root of T as a size of maximal independent set.
Note. If x has only a single child y, assume S2= and l(S2)=0. If no sets S1 and S2 with required properties can be found then leave l(S) its initial value.
Running time: O(n23k).

41. 3-COLOURABILITY
[Adapted from J.Flum and M.Grohe]

42. 3-COLOURABILITY
[Adapted from J.Flum and M.Grohe]

43. tw-HAMILTONICITY
[Adapted from J.Flum and M.Grohe]

44. Three ‘similarly sounding’ graph problems
[Adapted from R.Downey and M.Fellows]

45. Three ‘similarly sounding’ graph problems
[Adapted from R.Niedermeier]

46. VERTEX COVER parameterized by treewidth
[Adapted from R.Niedermeier]

47. VERTEX COVER parameterized by treewidth
[Adapted from R.Niedermeier]

48. VERTEX COVER parameterized by treewidth
[Adapted from R.Niedermeier]

49. VERTEX COVER parameterized by treewidth
[Adapted from R.Niedermeier]

50. VERTEX COVER parameterized by treewidth
[Adapted from R.Niedermeier]

51. DOMINATING SET parameterized by tw
[Adapted from R.Niedermeier]

52. DOMINATING SET parameterized by tw
[Adapted from R.Niedermeier]

53. Colour coding
[Adapted from R.Niedermeier]

54. Colour coding
[Adapted from R.Niedermeier]

55. Colour coding
[Adapted from R.Niedermeier]

56. Colour coding
[Adapted from R.Niedermeier]

57. Some treewidth related notions - cutdwidth
[Adapted from H.Bodlaender]

58. Some treewidth related notions - bandwidth
[Adapted from H.Bodlaender and J.Bottcher]