On the Maximum Cardinality Search Lower Bound for Treewidth

1
On the Maximum Cardinality Search Lower Bound
for Treewidth

• Hans Bodlaender
• Utrecht University
• Arie Koster
• ZIB Berlin

2
Maximum Cardinality Search is fun!
3
Contents

• Introduction
• Treewidth
• Maximum Cardinality Search
• Lucenas result
• Complexity
• Bounds on planar graphs
• Comparison with degeneracy lower bound
• Heuristics and experiments
• Conclusions

4
Treewidth

• Treewidth of graph smallest width of tree
  decomposition (no definition today!)
• Many problems have treewidth-based algorithm with
  running time exponential only in treewidth
• E.g. Probabilistic inference, frequency
  assignment
• Method to solve problems
• Make tree decomposition of G with small width
• Run dynamic programming-like algorithm on tree
  decomposition

5
Computing treewidth andtree decomposition

• We need algorithms to find tree decompositions
  and determine treewidth
• Theoretical solutions often not useable ?
• Experimental work
• Preprocessing
• Upper bound heuristics
• Branch and bound
• ILP methods
• Lower bounds

6
Lower bounds

• Inform about quality of upper bounds
• Help to speed up Branch and Bound and ILP methods
• Degeneracy
• Ramachandramurthi bound
• Clautiaux et. al technique
• Contraction technique
• Lucena (2003) MCS-lower bound

Can be combined with other lowerbound methods
7
Recognizing chordal graphs

• Rose, Tarjan, Lueker, 1976 Lexicographic
  Breadth First Search
• Tarjan, Yannakakis, 1984 Maximum Cardinality
  Search
• Used later also as triangulation heuristic /
  upper bound heuristic for treewidth
• M-variant by Berry, Blair, Heggernes (2002)

8
Maximum Cardinality Search

• Simple mechanism to make a permutation of the
  vertices of an undirected graph
• Let the visited degree of a vertex be its number
  of visited neighbors
• the maximum visited degree of a graph is the
  maximum over all MCS-orderings of the maximum
  visited degree of a vertex in the ordering.

• Pseudocode for MCS
• First, all vertices are unvisited
• Repeat
• Visit an unvisited vertex that has the largest
  visited degree
• Until all vertices are visited

9
Example a 4-clique with one subdivision
a
b
c
d
e
10
We can start at any vertex each vertex has 0
visited neighbors
a
0
a,
0
b
c
0
0
d
0
e
11
Say, we start at a
a
0
The next vertexmust be b, c, ord. It can not be
e.
a,
1
b
c
1
1
d
0
e
12
After b, we must visit c.
a
0
a, b,
2
b
c
1
1
d
1
e
13
And then d
a
0
a, b, c,
2
b
c
1
2
d
1
e
14
And finally e
a
0
a, b, c, d,
2
b
c
1
2
d
1
e
15
We made an MCS-ordering of the graph
a
0
a, b, c, d, e
2
b
c
1
2
d
2
e
16
Lucenas theorem

• (SIAM J. Disc. Math, 2003)
• The treewidth of a graph is at least its visited
  degree.
• Task find an MCS-ordering such that the largest
  visited degree of a vertex is as large as
  possible.

17
Different orderings can give different maximum
visited degree
a
0
e,
0
b
c
1
1
d
0
e
18
a
1
e, b,
1
b
c
1
1
d
0
19
a
2
e, b, d,
2
b
c
1
1
d
0
20
a
3
e, b, d, c,
2
b
c
1
1
d
0
21
a has visited degree 3, so the treewidth is at
least 3
a
3
e, b, d, c,
2
b
c
1
1
d
0
22
Some easy cases

• Clique maximum visited degree n 1.
• Grid maximum visited degree 2.

23
Complexity

• Instance graph G, integer k
• Question is the maximum visited degree of G at
  least k?
• Is an NP-complete problem
• Can not be approximated in polynomial time with
  constant ratio unless PNP
• Construction a graph that has maximum visited
  degree at least k when a 3-sat formula can be
  satisfied, and maximum visited degree at most 6
  otherwise.

24
Planar graphs

• Example of planar graph with O(k!) vertices and
  with maximum visited degree k.
• Upper bound O(log n) on maximum visited degree of
  planar graphs with n vertices

25
Example planar graphs can have large maximum
visited degree
0
26
Start in bottom row, and visit vertices on bottom
row first
0
1
27
0
1
1
28
0
1
1
1
29
0
1
1
1
1
30
0
1
1
1
1
1
31
0
1
1
1
1
1
1
32
0
1
1
1
1
1
1
1
33
0
1
1
1
1
1
1
1
1
34
0
1
1
1
1
1
1
1
1
1
35
0
1
1
1
1
1
1
1
1
1
1
36
0
1
1
1
1
1
1
1
1
1
1
1
37
Next row
1
0
1
1
1
1
1
1
1
1
1
1
1
38
Visited degree 2
1
2
0
1
1
1
1
1
1
1
1
1
1
1
39
1
2
2
0
1
1
1
1
1
1
1
1
1
1
1
40
1
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
41
1
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
42
1
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
43
1
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
44
1
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
45
1
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
46
1
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
47
1
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
48
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
49
Third row
2
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
50
Visited degree 3
2
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
51

2
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
52

2
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
53

2
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
54

2
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
55

2
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
56

2
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
57

3
2
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
58

3
3
2
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
59
End of 3rd row
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
60

4th row
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
61

Visited degree 4
4
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
62

4
4
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
63

4
4
4
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
64

4
4
4
4
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
65

And reaching a node with visited degree 5
5
4
4
4
4
3
3
3
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
1
1
1
Generalizes to larger values
66
Upper bound for planar graphs

• Planar graphs have visited degree O(log n).
• Let G be a planar graph with an MCS with visited
  degree k, of minimum size.
• At each point
• The set of visited vertices is connected
• By minimality the set of unvisited vertices is
  connected.

67
Definition Last successor

• The last successor of a vertex v is its last
  visited neighbor, if it is visited later than v.
• If all neighbors of v are visited before v, v has
  no last successor.
• Edges from vertices to last successors form a
  forest.

68
Lemma

• If v has visited degree l ³ 6, then there are two
  vertices w and x with v the last successor of w
  and of x, and the visited degree of w and x at
  least l 4.

69
Proof

• Look at set S of last four neighbors of v,
  visited before v.
• Each has visited degree at least l 4, otherwise
  v is visited earlier.
• Look at subgraph of visited vertices. W.l.o.g.,
  suppose unvisited vertices are in exterior face.

• If vertex from S is not at exterior face, v is
  its last successor.
• We cannot have three vertices from S at exterior
  face.

v
z
After z is visited, we must visit v (v.d. ³ 2)
instead of vertex inunvisited part with v.d. 1.
70
Proof of upper bound

• Look at forest of edges to last successor.
• Each node of vd l gt 5 has two children with vd at
  least l 4.
• There is a node with vd k.
• Forest, and hence G must have more than 2k/4-1
  vertices, so k O(log n).

71
MCS lower bound is bad for planar graphs

• Treewidth of planar graphs can be Q( n1/2).
• Can be approximated with ratio 1.5 (Seymour
  Thomas, Hicks)
• MCS gives O( log n)

72
Degeneracy

• Degeneracy or MMD
• Maximum over all subgraphs H of G of minimum
  degree of vertex in H
• Lower bound on treewidth
• Easily computable in O(VE) time

73
Degeneracy versus maximum visited degree

• Maximum visited degree never smaller than
  degeneracy
• Consider last visited vertex of subgraph with
  large minimum degree
• Maximum visited degree sometimes larger than
  degeneracy
• Planar graphs have degeneracy at most 5.
• Example shows planar graphs with MCS with large
  maximum visited degree

74
Heuristics

• Three heuristics for tie-breaking MCS
• First
• Largest degree
• Smallest degree
• Heuristics that gives upper bound on maximum
  visited degree (omitted from talk)

Give lower bound on maximum visited degree,hence
on treewidth
75
Graphs also used in other experiments.
Three casesgive matchingbounds.
Default isneveroutperformed!
Improvementsare incremental.
76
Conclusions

• MCS gives incremental improvement to existing
  lower bounds for treewidth
• Interesting combinatorial questions
• Additional techniques (contraction,
  Clautiaux-et.al technique) can give considerable
  improvements
• Sometimes still big gaps, e.g., (close to) planar
  graphs

77
Thank you! Questions?
78
Results
UPB
network
MMD
MCS-def
MCS-max
MCS-min
Barley-pp
5
0.00
6
0.00
5
0.00
6
0.00
7
Link
4
0.00
5
3.65
4
7.79
5
7.89
25
Link-pp
6
0.01
6
0.60
6
1.27
6
1.20
27
Munin1-pp
4
0.00
5
0.02
5
0.04
5
0.05
17
Munin2-pp
4
0.00
5
0.15
5
0.30
5
0.29
8
Munin3-pp
4
0.00
5
0.05
4
0.10
5
0.10
17
Oesoca-pp
9
0.00
10
0.00
9
0.00
10
0.00
11
Pathfinder-pp
5
0.00
6
0.00
5
0.00
6
0.00
6
Pignet2-pp
5
0.01
6
7.59
6
17.42
6
18.02
239
Water-pp
6
0.00
8
0.00
7
0.00
8
0.01
10
79
Upper bound heuristic

• Each vertex v has upper bound u(v) variable
• Invariant u(v) ³ maximum visited degree of v
• Try to improve upper bounds until no longer
  possible
• Compute set UN(v) of u(w) of neighbors w of v
• Sort UN(v), say d1, , dr. l 0.
• For i 1 to r do if di ³ l then l
• Set u(v) min u(v), l

If v is visited with visited degree k, then v
has neighbors w1, , wk with wi visited before v
with visited degree at least i 1.
80
More on upper bound heuristic

• Improvement looking at maximum visited degree
  when v has a neighbor that is visited later
• Improvement 2 looking at maximum visited degree
  when v is visited before neighbor w a bound per
  edge

81
Bounds of upper bound heuristics
MCSLB-max
MCSLB
Instance
ub1
cpu
ub2
cpu
ub3
cpu
link
5
0.02
11
10
10
0.01
0.05
link-pp
6
0
10
9
9
0.02
0.02
munin1
4
0
6
6
7
0.01
0.01
munin1-pp
5
0
7
7
8
0
0
pignet2
5
9
0.41
8
8
0.16
0.16
pignet2-pp
6
0.06
11
9
9
0.06
0.06
celar06pp
11
0.01
16
16
16
0
0
graph11
8
0.01
13
13
13
0.01
0.01
graph11-pp
9
0
14
13
13
0.01
0.01
graph13
8
0.01
13
13
14
0.01
0.01
graph13-pp
8
0.01
12
12
14
0.02
0.02