Treewidth: a complexity measure for graphs

1
Treewidth a complexity measure for graphs

• Ioan Todinca
• LIFO - Université dOrléans
• France

2
Outline

• Optimization problems on graphs
• Motivation
• General approaches for hard problems
• Treewidth and tree decomposition
• Definitions
• Applications
• Computing treewidth

3
Graphs

• G(V,E)
• n vertices, m edges
• Complexity of the algorithms O(f(n))
• Decision problems
• yes/no
• Optimization problems
• Minimize/maximize some quantity

4
Shortest path

• Non-negative weights
• Dijkstras algorithm
• O(mn log n)
• General case
• Floyds algorithm - all pairs shortest path
• O(n3)

5
Minimum weight spanning tree

• A tree touching all the vertices, of minimum
  weight
• Kruskals algorithm
• O(m log n)
• Prims algorithm
• O(n²)

6
Chinese postman

• Find a cycle passing at least once through each
  edge
• Minimize the length of the cycle
• O(n3)
• Eulerian cycle passing exactly once through each
  edge
• O(m)

7
Polynomial and NP-hard problems

• Efficient algorithms polynomial worst-case
  runing time O(Poly (n))
• Maximum flow, connectivity,
• NP-Hard problems no polynomial algorithms,
  unless PNP
• Exponential running time
• Coloring, max independent set, TSP,

8
The coloring problem

• Assign a color to each vertex, such that adjacent
  vertices receive different colors.
• Use a minimum number of colors.
• Even the 3-coloring problem is NP-hard.

9
Maximum independent set

• Independent set set of pairwise non-adjacent
  vertices
• Find an independent set of maximum size.

10
Traveling salesman

• Find a cycle passing at least once through each
  vertex, of minimum length
• Hamiltonicity find a cycle passing exactly once
  through each vertex

11
Optimization problems the output

• Optimization problem
• Find an object satisfying a property P
• Minimize/maximize the cost of the object
• Solutions and optimal solutions
• Solution any object satisfying property P
• Optimal solution solution of minimum/maximum
  cost
• OPT the cost of the optimal solution

12
How to cope with NP-hard optimization problems?

• What we can not do
• Obtain efficient algorithms, giving the optimal
  solution for any input
• What we can do
• Exponential time algorithms
• Algorithms giving non-optimal solutions
• Optimal algorithms for particular cases

13
Exponential algorithms

• Too slow for big data
• May be interesting for small instances
• O(2n) and O(1.01n) are quite different
  complexities!
• Maximum independent set O(1.1844n)
• Pruning the search tree
• Branch and bound

14
Heuristics

• No guarantee on the solution
• Validated on benchmarks
• More or less efficient
• Sometimes very simple and efficient
• Approaches
• Ad-hoc heuristics
• Genetic algorithms, simulated annealing,
• Randomized algorithms

15
Approximation algorithms

• Guarantee on the solution!
• c-approximation algorithm (cgt1)
• minimization problem gives a solution of cost at
  most c OPT
• Maximization problem at least OPT/c
• Polynomial algorithms!

16
Approximation algorithm for TSP

• 2-approximation
• Compute a minimum weight spanning tree T
• w(T) lt OPT
• Walk around the tree
• Also a 1.5-approximation for TSP
• Better approximation?

17
Non-metric TSP bad news

• Complete graph G
• A cycle passing through each vertex exactly once
• Of minimum length.
• For any constant c, there is no c-approximation
  for this problem, unless PNP

2
5
6
2
1
8
8
2
1
1
18
If non-metric TSP has a c-approximation
algorithm, then Hamiltonicity is polynomial
1
cn1

• If G hamiltonian length(output) cn
• If length(output) cn G is hamiltonian

19
Bad news for Coloring, MIS,

• For any cgt1, no c-approximation algorithm (unless
  PNP or similar)
• No 1000-approximation!
• For any egt0, no n1-e approximation
• No coloring algorithms with at most OPTn0.99
  colors!
• What can we do?
• n/log n approximations

20
Solving the problems for particular graph classes

• The input may satisfy particular properties
• Planar graphs the four colors theorem
• Interval graphs Coloring, MIS, Hamiltonicity
  become polynomial
• But
• Solve one problem at a time
• The classes are often very restricted

21
Graph decomposition techniques

• General principle
• Split the graph into several pieces that glue
  according some simple rules
• Recursively split the pieces
• Solve the problems bottom-up, by dynamic
  programming

22
Graph decompositions

• Only work if the input graph admits a good
  decomposition
• Allow to solve classes of problems by the same
  algorithm
• Decompositions
• Modular decompositions (cographs, )
• Tree decompositions (trees, cycles, outerplanar
  graphs)
• Clique decompositions (combines both, but no
  results about computing the decompositions)

23
Techniques for hard optimization problems

• Approximations
• Guaranteed, efficient
• Restricted input
• Graph classes
• Decompositions
• Heuristics
• Exponential algorithms

24
Outline

• Optimization problems on graphs
• Motivation
• General approaches for hard problems
• Treewidth and tree decomposition
• Definitions
• Applications
• Computing treewidth

25
Tree decompositions and treewidth

• Every graph can be seen as a generalized tree
• treewidth the smaller it is, the more the graph
  behaves like a tree
• Many problems can be solved in O(2tw(G)n) time

26
Tree decompositions for G(V,E)

• Tree T, each node i has a bag Xi
• Each vertex x of G is in some bag
• For each edge xy, there is a bag containing both
  x and y
• For each vertex x, the set of nodes i s.t. Xi
  contains x forms a subtree Tx of T

27
Treewidth Robertson, Seymour 84

• Width of a decomposition (T,X)
• width max Xi -1
• Treewidth of G
• tw(G) min width(T,X)
• over all possible decompositions

28
Treewidth examples

• Trees treewidth one
• Cycles treewidth two
• Complete graph Kn treewidth n-1
• tw(G)n-1

29
Treewidth of the grid

• Grid ab
• tw(grid ab) min(a,b)
• Hard to prove that min(a,b) is a lower bound!

30
Graph minors

• H is a minor of G if it can be obtained by
  repeatedly applying
• Edge deletion
• Vertex deletion
• Edge contraction

a
b
c
a
b
c
x
y
xy
d
d
31
The grid minor theorem

• Conjecture. A graph G that does not contain the
  rr grid as a minor has treewidth at most r2 log
  r
• Theorem. If no rr grid minor then tw29r5
• For any G, either G has small tw or it contains a
  big grid.

32
Treewidth applications the graph minors theorem

• The class of planar graphs minor closed
• Kuratowskis theorem a graph G is planar iff K5
  and K3,3 are not minors of G

33
Graph minors theorem

• G a minor closed class of graphs. There is a
  finite set of graphs obstr(G) s.t.G in G iff G
  has no minors in obstr(G) .
• Consequence O(n²) recognition algorithm for the
  class...
• if you know the obstruction set!

34
Treewidth algorithmic applications

• Input G and (T,X) of width k
• Most of the classical problems can be solved in
  O(exp(k) Poly(n)) time
• Example maximum independent set
• Coloring, TSP,

35
Decomposition
yc
xa

• Xi
• x, y not in Xi
• x, y in different subtrees of T i
• Xi separates x and y in G
• Any x,y path passes through Xi

Xi
36
Maximum independent set
root

• Gi
• restrict to the vertices in the subtree rooted
  in I
• Bottom-up Gi increasing
• Groot G

gh
beg
i
bcg
bgd
j
abd
cgf
37
MIS(Z,i)
root

• Independent in Gi
• Intersects Xi exactly in Z
• Of maximum size

gh
beg
i
bcg
bgd
abd
cgf
38
MIS(Z,i) Z U MIS(Z,j) U MIS(Z,p)

• Consider all sets Z, Z and compute MIS(Z,I)
• MIS(G) in O(n23k) time
• Can be done in O(n2k)

root
gh
beg
i
bcg
bgd
j
p
abd
cgf
39
Most problems can be solved in O(exp(k)Poly(n))
time

• Dynamic programming
• Logic Courcelle et al.
• every problem expressible in ExtMSOL2 can be
  solved in O(exp(k)n) time
• MIS
• 3Col

40
Surprising treewidth application

• The disjoint paths problem
• s1, s2,,sp
• t1, t2,,tp
• Find disjoint paths from si to ti
• Small treewidth gt previous techniques
• Big treewidth gt use the grid minor!

41
Treewidth applications

• Mathematics
• Graph minors theorem, Wagners conjecture
• Algorithms
• Most hard problems can be solved efficiently on
  graphs of bounded treewidth
• Treewidth techniques may help even on arbitrary
  graphs!

42
Outline

• Optimization problems on graphs
• Motivation
• General approaches for hard problems
• Treewidth and tree decomposition
• Definitions
• Applications
• Computing treewidth

43
Computing treewidth the main results

• The Treewidth problem is NP-hard
• For fixed k, deciding if tw(G)k is polynomial,
  even linear Bodlaender 94
• O(24k²n) time
• First output tw(G)gtk or a decomposition of width
  2k
• Use the decomposition to decide if tw(G)k
• O(log OPT) approximation algorithm

44
Treewidth an alternative definition

• G(V,E) (T,X)
• H(T,X)
• Vertex set V
• x,y adjacent iff there exists a bag containing
  them
• H(T,X) contains G
• Bags gt cliques of H

45
Chordal graphs

• G is chordal if each cycle of at least four
  vertices has a chord
• Edge between two non-consecutive vertices of the
  cycle
• H(T,X) is chordal

46
H(T,X) as an intersection graph

• Tx induced by the bags containing x
• xy adjacent in H iff Tx and Ty intersect
• H(T,X) is the intersection graph of the subtrees
  of a tree

47
Chordal graphs as intersection graphs

• If H is the intersection graph of the subtrees of
  a tree then H is chordal
• and conversely!

a
b
c
d
48
From tree decompositions to chordal graphs

• width (T,X) ?(H)-1

49
Treewidth and minimal triangulations

• Triangulations of G chordal supergraphs
• tw(G) min ?(H) -1
• over all triangulations H of G
• Equivalently
• over all minimal triangulations H of G

50
(Minimal) triangulation problems

• Treewidth
• Find a minimal triangulation of the input graph
• Minimize the cliquesize of the triangulation
• Minimum fill-in
• Find a minimal triangulation of the input graph
• Minimize the number of added edges
• How to control the minimal triangulations?

51
Minimal separators

• x,y-separator S
• x and y are in different connected components of
  G-S
• minimal x,y-separator S
• inclusion-minimal x,y-separator
• minimal separator S of G
• exist x,y s.t. S is a x,y-minimal separator

52
Chordal graphs and minimal separators Dirac 61

• A graph is chordal iff all its minimal separators
  are cliques
• If G non chordal there is a separator which is
  not a clique

a
b
c
d
53
Parallel and crossing separators

• S crosses T if S separates two vertices of T.
  Otherwise S is parallel to T
• S T ? T S

54
Minimal triangulations and minimal separators

• Parra, Scheffler 96, Bodlaender, Kloks,
  Kratsch, Müller, Spinrad 90-96
• Theorem. H is a minimal triangulation of G iff H
  is obtained from G by considering a maximal set
  of pairwise parallel separators and completing
  each of them into a clique
• The minimal separators of H are exactly the
  minimal separators chosen in G.
• They form the same components in H and in G.

55
The Parra-Scheffler theorem
56
Treewidth and minimal separators

• Are the minimal separators sufficient for
  computing the treewidth?
• Yes, for many graph classes Bodlaender, Kloks,
  Kratsch, Müller, permutation graphs, circle
  graphs, circular arc graphs,
• Actually, yes ? Bouchitté, T. 00
• Do they help for approximating treewidth?
• Sometimes (AT-free graphs, ) Bouchitté,
  Kratsch, Müller, T. 01

57
Decomposition using minimal separators the
principle

• Blocks G
• While there is a non-complete block B
• Choose a minimal separator S of B
• For each component C of B-S create the block BSS
  U C
• Remove B

C1
C2
S
C2
C1
S
S
58
Decomposition using minimal separators example
59
Decomposition using minimal separators example
b
a
b
c
a
b
c
d
d
f
d
f
e
e
e
e
d
f
d
f
h
g
g
g
g
h
h
60
Decomposition using minimal separators example
b
c
b
f
d
f
e
e
d
f
g
g
g
h
61
Tree decompositions, triangulations, minimal
separators
b
c
b
f
d
f
e
e
d
f
g
g
g
h
62
Blocks
63
Potential maximal cliques

• G(V,E)
• Definition. A set of vertices K is a potential
  maximal clique of G if there is a minimal
  triangulation H of G such that K is a maximal
  clique of H.
• K is a potential maximal clique iff it is a
  terminal block in the decomposition algorithm.

64
Potential maximal cliques characterization

• C1,,Cp the components of G-K
• S1,,Sp their neighborhoods
• Theorem. K is a potential maximal clique iff
• Each Si strictly included in K
• GK becomes a clique when every Si is completed

C2
S2
S1
C1
S3
C3
65
Potential maximal cliques examples

• Recognition algorithm O(nm)

66
Treewidth and potential maximal cliques

• Theorem Bouchitté, T 99. The potential maximal
  cliques are sufficient to compute the treewidth.
• Input G, PMCG
• Output tw(G) optimal decomposition
• Complexity O(nmp)

67
Enumerating the potential maximal cliques

• Most potential maximal cliques are formed by
  two crossing separators.
• pmc minsep² n
• PMCG can be obtained in O(Poly(n,minsep)) time.

68
Treewidth and minimal separators the conclusion

• Theorem the minimal separators are sufficient
  for computing treewidth.
• Input G
• Output tw(G) optimal decomposition
• Complexity O(nmr²)
• Enumerate the minimal separators
• Use the min seps to enumerate pmcs
• Use pmcs to compute tw

69
Treewidth and potential maximal cliques side
effects

• The same technique allows to compute the minimum
  fill-in
• Exact algorithm for treewidth and minimum
  fill-in O(nm 1.961n) Kratsch, Fomin, T.,
  submitted
• min seps 1.73n
• potential maximal cliques 1.961n

70
Treewidth heuristics and approximations

• Decomposition algorithm each minimal separator
  we chose is completed
• What if we always chose the minimum size
  separator?
• MinCardSep
• Existing heuristic MinDegree
• Not even minimal triangulations, but good for
  minimum fill-in!

71
Blocks and MinCardSep

• Border/Interior
• Suppose
• tw(G) k
• Intgt Brd
• Si 4k, for all i
• Then
• Exists x in Int with 4k neighbors
• So a separator 4k

72
Blocks and MinCardSep

• tw(G) k, Int gt Brd, Si 4k, for all i
• For all x in Int, x has gt4k neighbors
• edges GB gt4k Int/2 kB
• Contradicts tw(GB) k

73
Blocks and asteroidal number

• AT-free graphs
• at most two (inclusion maximal) separators on the
  border
• Asteroidal number
• the maximum number of inclusion maximal
  separators on the border

74
MinCardSep in graphs of asteroidal number a

• If
• B gt 8ak
• the separators on the border are 4k
• Then
• there is a min sep in B with 4k vertices
• split again!
• MCardSep is a 8a approximation algorithm

75
Treewidth approximations and heuristics

• Constant factor approximations
• AT-free graphs (2-approx)
• Bounded asteroidal number (8a-approx)
• O(log OPT) approximations
• 1000 log k only of theoretical interest
• MinDegree
• E. Amir proposes a 2-approx in exp(k) time
• and says that MinDegree always beats it!

76
Treewidth computations and approximations

• Polynomial for fixed k (FPT-problem)
• Polynomial for graphs with few minimal
  separators
• Exact algorithm O(cn Poly(n)) time with clt2
• O(log OPT)-approximation
• Nice heuristic
• Open constant factor approximations

77
Treewidth conclusion

• tw(G) the distance from G to the class of
  trees
• Small treewidth gt efficient algorithms
• Dynamic programming, logic expressions, reduction
  rules
• Not so hard to compute
• NP-hard, but polynomial for fixed k
• Anything left for future research?

78
Treewidth extensions

• G planar for any x,r
• tw(Nr(x)) 3r
• Bounded local treewidth
• Function f
• for any G, any x,r
• tw(Nr(x)) f(r)
• Efficient or fixed parameter algorithms for
  graphs of bounded local treewidth

79
Cliquewidth

• Treewidth the complete graphs are
  undecomposable.
• Cographs very simple graphs!
• Single vertex parallel composition series
  composition

G1
G2
G1
G2
Parallel G1G2
Series G1G2
80
Cliquewidth

• Graph grammars defining a class of k-simple
  graphs
• cwd(G) k
• Quite many NP-hard problems become polynomial
  MIS, Hamiltonicity, Coloring,
• Dynamic programming, logic (ExtMSOL1)
• Almost no results on cliquewidth
  computation/approximations
• cwd 3 polynomial

81
Treewidth approximations?

• Constant factor seems hard
• Treewidth obstructions?
• Find a min-max theorem
• For any G, either something(G) gt r or tw(G)
  r2
• Certifying algorithms?
• When I claim tw(G) k you can check the
  decomposition.
• Why should you trust me when I say tw(G) gt k?

82
Treewidth and tree decompositions a very active
research area!