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On the Maximum Cardinality

Search Lower Bound for Treewidth

Hans BodlaenderUtrecht University

Arie KosterZIB Berlin

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Maximum Cardinality Search is

fun!

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Contents

• Introduction– Treewidth – Maximum Cardinality Search– Lucena’s result

• Complexity• Bounds on planar graphs• Comparison with degeneracy lower bound• Heuristics and experiments• Conclusions

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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

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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

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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 combinedwith other lowerbound methods

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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)

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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

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Example: a 4-clique with one subdivision

a

bc

d

e

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We can start at any vertex: each vertex has 0

visited neighbors0

0

0

0

0

a

bc

d

e

a, …

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Say, we start at a

0

1

1

0

1

a

bc

d

e

a, …The next vertexmust be b, c, ord. It can not be e.

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After b, we must visit c.

0

1

2

1

1

a

bc

d

e

a, b, …

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And then d

0

1

2

1

2

a

bc

d

e

a, b, c, …

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And finally e

0

1

2

1

2

a

bc

d

e

a, b, c, d, …

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We made an MCS-ordering of the graph

0

1

2

2

2

a

bc

d

e

a, b, c, d, e …

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Lucena’s 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.

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Different orderings can give different maximum

visited degree0

1

0

0

1

a

bc

d

e

e, …

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1

1

1

0

1

a

bc

d

e, b, …

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2

1

2

0

1

a

bc

d

e, b, d, …

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3

1

2

0

1

a

bc

d

e, b, d, c, …

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a has visited degree 3, so the treewidth is at least 3

3

1

2

0

1

a

bc

d

e, b, d, c, …

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Some easy cases

• Clique: maximum visited degree n – 1.

• Grid: maximum visited degree 2.

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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 P=NP– 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.

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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

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0

Example: planar graphs can havelarge maximum visited degree

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0 1

Start in bottom row, and visitvertices on bottom row first

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0 1 1

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0 1 1 1

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0 1 1 1 1

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0 1 1 1 1 1

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0 1 1 1 1 1 1

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0 1 1 1 1 1 1 1

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0 1 1 1 1 1 1 1 1

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0 1 1 1 1 1 1 1 1 1

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0 1 1 1 1 1 1 1 1 1 1

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0 1 1 1 1 1 1 1 1 1 1 1

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Next row

1

0 1 1 1 1 1 1 1 1 1 1 1

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Visited degree 2

1 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1 1 1

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1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2

1 1

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1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

Third row

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2 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

Visited degree 3

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2 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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32 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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332 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

2 2

1 1

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332 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

End of 3rd row

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33

3

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

4th row

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33

43

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

Visited degree 4

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33

443

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

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3

4

3

443

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

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4

3

4

3

443

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

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4

3

4

3

443

2 3 3 3 3 3 3 3

1 2 2 2 2 2 2 2 2 2

5

0 1 1 1 1 1 1 1 1 1

3

2 2

1 1

And reaching a node with visited degree 5

Generalizes to larger values

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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.

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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.

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Lemma

• If v has visited degree l 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.

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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

After z is visited, we must visitv (v.d. 2) instead of vertex inunvisited part with v.d. 1.

z

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Proof of upper bound

• Look at forest of edges to last successor.

• Each node of vd l > 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).

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MCS lower bound is bad for planar graphs

• Treewidth of planar graphs can be ( n1/2).

• Can be approximated with ratio 1.5 (Seymour & Thomas, Hicks)

• MCS gives O( log n)

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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(|V|+|E|) time

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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

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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 onmaximum visited degree,hence on treewidth

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Graphs alsoused in otherexperiments.

Three casesgive matchingbounds.

Default isneveroutperformed!

Improvements

are incremental.

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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

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Thank you!Questions?

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Results

Barley-pp 5 0.00 6 0.00 5 0.00 6 0.00 7Link 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 27Munin1-pp 4 0.00 5 0.02 5 0.04 5 0.05 17Munin2-pp 4 0.00 5 0.15 5 0.30 5 0.29 8Munin3-pp 4 0.00 5 0.05 4 0.10 5 0.10 17Oesoca+-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 6Pignet2-pp 5 0.01 6 7.59 6 17.42 6 18.02 239

network MMD UPBMCS-def MCS-max MCS-min

Water-pp 6 0.00 8 0.00 7 0.00 8 0.01 10

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Upper bound heuristic

• Each vertex v has upper bound u(v) variable– Invariant: u(v) 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 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.

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.

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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

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Bounds of upper bound heuristics

MCSLB ub1 cpu ub2 cpulink 5 11 0.02 10 0.01link-pp 6 10 0 9 0.02munin1 4 7 0 6 0.01munin1-pp 5 8 0 7 0pignet2 5 9 0.41 8 0.16pignet2-pp 6 11 0.06 9 0.06celar06pp 11 16 0.01 16 0graph11 8 13 0.01 13 0.01graph11-pp 9 14 0 13 0.01graph13 8 14 0.01 13 0.01graph13-pp 8 14 0.01 12 0.02

InstanceMCSLB-max

ub3 cpu10 0.059 0.026 0.017 08 0.169 0.06

16 013 0.0113 0.0113 0.0112 0.02