Computing Branchwidth via Efficient Triangulations and Blocks

1
Computing Branchwidth via Efficient
Triangulations and Blocks

• Authors
• F.V. Fomin, F. Mazoit, I. Todinca
• Presented by
• Elif Kolotoglu, ISE, Texas AM University

2
Outline

• Introduction
• Definitions
• Theorems
• Algorithm
• Conclusion

3
Introduction

• Proceedings of the 31st Workshop on Graph
  Theoretic Concepts in Computer Science (WG 2005),
  Springer LNCS vol. 3787, 2005, pp. 374-384
• An algorithm to compute the branchwidth of a
  graph on n vertices in time (2 v3)n nO(1) is
  presented
• The best known was 4n nO(1)

4
Introduction

• Branch decomposition and branchwidth were
  introduced by Robertson and Seymour in 1991
• Useful for solving NP-hard problems when the
  input is restricted to graphs of bounded
  branchwidth
• Testing whether a general graph has branchwidth
  bounded by some integer k is NP-Complete (Seymour
  Thomas 1994)

5
Introduction

• Branchwidth and branch decompositions are
  strongly related with treewidth and tree
  decompositions
• For any graph G, bw(G) tw(G) 1 Floor(3/2
  bw(G))
• But the algorithmic behaviors are not quite same

6
Introduction

• On planar graphs computing branchwidth is
  solvable in polynomial time while computing the
  treewidth in polynomial time is still an open
  problem
• On split graphs computing branchwidth is NP hard,
  although it is linear time solvable to find the
  treewidth

7
Introduction

• Running times of exact algorithms for treewidth
• O(1.9601n) by Fomin, Kratsch, Todinca-2004
• O(1.8899n) by Villanger-2006
• O(4n) with polynomial space by Bodlaender,
  Fomin, Koster, Kratsch, Thilikos-2006
• O(2.9512n) with polynomial space by Bodlaender,
  Fomin, Koster, Kratsch, Thilikos-2006

8
Outline

• Introduction
• Definitions
• Theorems
• Algorithm
• Conclusion

9
Definitions

• Let G (V, E) be an undirected and simple graph
  with Vn, Em and let T be a ternary tree
  with m leaves
• Let ? be a bijection from the edges of G to the
  leaves of T
• Then the pair (T, ?) is called a branch
  decomposition of G.

e
a
a
d
e
d
b
c
c
b
10
Definitions

• The vertices of T will be called nodes, and the
  edges of T will be called branches
• Removing a branch e from T partitions T into two
  subtrees T1(e) and T2(e). lab(e) is the set of
  vertices of G both incident to edges mapped on
  T1(e) and T2(e).

e
a
a
d
e
d
b
c
c
b
11
Definitions

• The maximum size over all lab(e) is the width of
  the branch decomposition (T, ?)
• The branchwidth of G, denoted by ß(G), is the
  minimum width over all branch decompositions of G
• A branch decomposition of G with width equal to
  the branchwidth is an optimal branch decomposition

12
Example
8
6
1
2
6
5
8
0
7
7
4
3
3
4
5
6
3
8
4
2
5
0
2
1
7
1
0
13
Definitions

• A graph G is chordal if every cycle of G with at
  least 4 vertices has a chord(an edge between two
  non-consecutive vertices of a cycle)
• A supergraph H(V,F) of G(V, E) (i.e. E is a
  subset of F) is a triangulation of G if H is
  chordal
• If no strict subgraph of H is a triangulation of
  G, then H is called a minimal triangulation

14
Definitions

• For each x in V we can associate a subtree Tx
  covering all the leaves of the branch
  decomposition T that are corresponding to the
  incident edges of x
• The intersection graph of the subtrees of a tree
  is chordal
• The intersection graph of the subtrees Tx is a
  triangulation H(T, ?) of G.
• Lab(e) induces a clique in H

15
Outline

• Introduction
• Definitions
• Theorems
• Algorithm
• Conclusion

16
Theorems

• The basic result states that, for any graph G,
  there is an optimal branch decomposition (T, ?)
  such that H(T, ?) is an efficient triangulation
  of G
• To compute the branchwidth, this result is
  combined with an exponential time algorithm
  computing the branchwidth of hyper-cliques

17
Theorems

• A triangulation H of G is efficient if
• Each minimal separator of H is also a minimal
  separator of G
• For each minimal separator S of H the connected
  components of H-S are exactly the connected
  components of G-S

18
Theorems

• Theorem 1 There is an optimal branch
  decomposition (T, ?) of G s.t. the chordal graph
  H(T, ?) is an efficient triangulation of G.
  Moreover, each minimal separator of H is the
  label of some branch of T.

19
Theorems

• A set of vertices B in V of G is called a block
  if, for each connected component Ci of G-B,
• its neighborhood SiN(Ci) is a minimal separator
• B\Si is non empty and contained in a connected
  component of G-Si

Si
B
Ci
20
Theorems

• The minimal separators Si border the block B and
  the set of these minimal separators are denoted
  by S(B)
• The set of blocks of G is denoted by BG

21
Theorems

• Lemma 12 If H is an efficient triangulation of
  G, then any maximal clique K of H is a block of
  G, and for any block B of G, there is an
  efficient triangulation H(B) of G s.t. B induces
  a maximal clique in H

22
Theorems

• Let B be a block of G and K(B) be the complete
  graph with vertex set B. A branch decomposition
  (TB,?B) of K(B) respects the block B if, for each
  bordering minimal separator S in S(B), there is a
  branch e of the decomposition s.t. S is a subset
  of lab(e). The block branchwidth bbw(B) of B is
  the minimum width over all the branch
  decompositions of K(B) respecting B

23
Theorems

• Theorem 2
• Theorem 4 Given a graph G and the list BG of all
  its blocks together with their block-branchwidth,
  the branchwidth of G can be computed in
  O(nmB(G)) time

24
Theorems

• n(B) of vertices of block B
• s(B) of minimal separators bordering B
• Theorem 5 The block-branchwidth of any block B
  can be computed in O(3s(B)) time
• Theorem 6 The block-branchwidth of any block B
  can be computed in O(3n(B)) time

25
Theorems

• s(B) is at most the of components of G-B, so
  n(B)s(B)n
• At least one s(B) or n(B) n/2, then
• Theorem 7 For any block B of G, the
  block-branchwidth of B can be computed in O(v3n)
  time

26
Thorems

• Theorem 8 The branchwidth of a graph can be
  computed in O((2v3)n) time
• Pf Every subset B of V is checked (in polynomial
  time) if it is a block or not. The
  block-branchwidth is computed for each block. The
  number of blocks is at most 2n) and for each
  block O(v3n) time is needed. And the branchwidth
  is computed using Theorem 4 in O(2n) time.

27
Outline

• Introduction
• Definitions
• Theorems
• Algorithm
• Conclusion

28
Algorithm

• Given a minimal separator S of G and a connected
  component C of G-S, R(S,C) denotes the hypergraph
  obtained from GSUC by adding the hyperedge S.

S
C
29
Algorithm

• Input G, all its blocks and all its minimal
  separators
• Output bw(G)
• Compute all the pairs S,C where S is a minimal
  separator and C a component of G-S with SN(C)
  sort them by the size of SUC
• for each S,C taken in increasing order
• bw(R(S,C))bbw(SUC)
• for each block ? with
• compute the components Ci of G-? contained in C
  and let SiN(Ci)
• let ?G be the set of inclusion minimal
  separators of G
• bw(G)

30
Outline

• Introduction
• Definitions
• Theorems
• Algorithm
• Conclusion

31
Conclusion

• Enumerating the blocks in a graph and finding the
  block-branchwidth of the blocks leads to a
  O((2v3)n) time algorithm for the branchwidth
  problem
• This is the best algorithm for branchwidth problem

32
Conclusion

• Open problems for future research
• Is there a faster way of computing block
  branchwidth?
• Can we find any smaller class of triangulations
  (compared to efficient triangulations) that
  contains H(T, ?), for some optimal branch
  decompositions of the graph?

33
(No Transcript)