Exact exponential algorithms for treewidth

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Exact (exponential) algorithms for treewidth

• Fedor Fomin, Bergen, Norway
• Dieter Kratsch, Metz, France
• Ioan Todinca, Orléans, France
• Yngve Villanger, Bergen, Norway

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Tree decompositions and treewidth

• (T,Xi)
• every edge and vertex is covered by some bag
• For each vertex x the set of bags containing it
  forms a (connected) subtree
• width(T,Xi) max Xi -1
• tw(G) min width(T,Xi)

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Why treewidth?

• Major problem, challenging from the point of view
  of exponential algorithms
• O(2n) algorithms
• Elimination orderings TSP-style approach
• Search games Fomin, Fraigniaud, Nisse, 05
• Arnborg, Proskurowski, 87 try all possible
  bags. Works in O(nk2) if tw k.
• How to do better? O(cn) with clt2.

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The main ingredients

• "usefull bags" the potential maximal cliques
• Efficiently compute the treewidth from these bags
• Count and enumerate the potential maximal
  cliques O(1.89n)

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Input k, set of bags BOutput tree dec of width
k using only bags from B
for each bag B 2 B and each component C of
G-B we need a bag B 2 B intersecting C s.t.
N(C) µ B µ B C (otherwise remove B from B)
repeat the loop until B stays unchanged if B is
empty then output FAILED else extract a tree
decomposition
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Running time

• Poly(n) ( of bags)3
• Must be brought down to
• Poly(n) of bags
• Doable because in our case B, B intersect
  exactly on N(C)
• each bag has O(n) glueing areas
• when a bag is deleted, it warns its glueing areas

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Treewidth and 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

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(Minimal) triangulation problems

• Find a minimal triangulation of the input graph
• Treewidth Minimize the cliquesize of the
  triangulation
• Minimum fill-in Minimize the number of added
  edges
• Both problems can be solved in O(1.89n) time.
• How to control the minimal triangulations?
• Minimal separators O(1.71n)
• Potential maximal cliques O(1.89n)

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Minimal separators

• x,y-minimal separator
• inclusion-minimal x,y-separator
• minimal separator S
• exist x,y s.t. S is a minimal x,y separator

Chordal graphs all the minimal separators are
cliques Dirac 61
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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 non-crossing 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.

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The Parra-Scheffler theorem
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Potential maximal cliques

• Definition. A set of vertices K is a potential
  maximal clique of G if there exists a minimal
  triangulation H of G such that K induces a
  maximal clique in H.

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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
K
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Potential maximal cliques examples

• Recognition algorithm O(nm)

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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)

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Counting the minimal separators

• At least two components of G-S are full
  N(C)N(D)S
• S, C or D has n/3 vertices

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Better counting of the minimal separators

• Small ones ßn
• Bad set (S) (x,X)
• Different bad sets represent different separators
• Many bad sets gt few separators
• O(1.7087n)

X
x
S
C the smallest component
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Counting the potential maximal cliques

• Nice potential maximal cliques are formed by
  two crossing separators.
• pmc O(1.89n)

a
b
c
d
e
f
g
h
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Counting hint

• K is the neighborhood of (any) two of the three
  regions.
• Either K 2n/5,
• or K is the neighborhood of a set 2n/5.

K
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Conclusion

• Theorem The treewidth can be computed in
  O(1.89n) time.
• Enumerate the potential maximal cliques.
• Compute the treewidth.
• Similar algorithm for the minimum fill-in.

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Improvements for trewidth?

• Counting and listing the potential maximal
  cliques
• counting with bad sets ! O(1.82n) Villanger,
  05
• listing with polynomial delay / pmc???
• The "worst" number of potential maximal cliques
  3n/3 ¼1.44n
• Difficult cases when treewidth is big

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Related work branchwidth (with F. Fomin and F.
Mazoit)

• bw(G) defined by partitions of the edge set
• Theorem Mazoit optimal branch decompositions
  correspond to efficient triangulations
• 2n potential branch cliques
• 3n/2 time for the branch-width of each

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Open questions

• Lower bounds for treewidth (under some reasonable
  hypothesis)?
• Pathwidth in O(cn) with clt2?
• Polynomial space exact algorithms?

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Thank you!
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