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# Computing Branchwidth via Efficient Triangulations and Blocks - PowerPoint PPT Presentation

Computing Branchwidth via Efficient Triangulations and Blocks. Authors: F.V. Fomin, F. Mazoit, I. Todinca Presented by: Elif Kolotoglu, ISE, Texas A&M University. Outline. Introduction Definitions Theorems Algorithm Conclusion. Introduction.

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### Computing Branchwidth via Efficient Triangulations and Blocks

Authors:

F.V. Fomin, F. Mazoit, I. Todinca

Presented by:

Elif Kolotoglu, ISE, Texas A&M University

Outline Blocks

• Introduction

• Definitions

• Theorems

• Algorithm

• Conclusion

Introduction Blocks

• 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 + √3)n · nO(1) is presented

• The best known was 4n · nO(1)

Introduction Blocks

• 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)

Introduction Blocks

• Branchwidth and branch decompositions are strongly related with treewidth and tree decompositions

• For any graph G, bw(G) ≤ tw(G) + 1 ≤ Floor(3/2bw(G))

• But the algorithmic behaviors are not quite same

Introduction Blocks

• 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

Introduction Blocks

• 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

Outline Blocks

• Introduction

• Definitions

• Theorems

• Algorithm

• Conclusion

Definitions Blocks

• Let G = (V, E) be an undirected and simple graph with |V|=n, |E|=m and let T be a ternary tree with m leaves

• Letηbe a bijection from the edges of G to the leaves of T

• Then the pair(T, η)is called abranch decompositionof G.

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Definitions Blocks

• 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).

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Definitions Blocks

• The maximum size over all lab(e) is the width of the branch decomposition (T, η)

• Thebranchwidthof G, denoted byβ(G),is the minimum width over all branch decompositions of G

• A branch decomposition of G with width equal to the branchwidth is anoptimal branch decomposition

Example Blocks

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Definitions Blocks

• 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

Definitions Blocks

• 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, η) of G.

• Lab(e) induces a clique in H

Outline Blocks

• Introduction

• Definitions

• Theorems

• Algorithm

• Conclusion

Theorems Blocks

• The basic result states that, for any graph G, there is an optimal branch decomposition (T, η) such that H(T, η) 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

Theorems Blocks

• 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

Theorems Blocks

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

Theorems Blocks

• A set of vertices B in V of G is called a block if, for each connected component Ci of G-B,

• its neighborhood Si=N(Ci) is a minimal separator

• B\Si is non empty and contained in a connected component of G-Si

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Theorems Blocks

• 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

Theorems Blocks

• Lemma 1&2: 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

Theorems Blocks

• Let B be a block of G and K(B) be the complete graph with vertex set B. A branch decomposition (TB,ηB) 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

Theorems Blocks

• 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(nm|B(G)|) time

Theorems Blocks

• 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

Theorems Blocks

• 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*(√3n) time

Thorems Blocks

• Theorem 8: The branchwidth of a graph can be computed in O*((2+√3)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*(√3n) time is needed. And the branchwidth is computed using Theorem 4 in O*(2n) time.

Outline Blocks

• Introduction

• Definitions

• Theorems

• Algorithm

• Conclusion

Algorithm Blocks

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

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Algorithm Blocks

• 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 S=N(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 Ω with

• compute the components Ci of G-Ω contained in C and let Si=N(Ci)

• let ∆*G be the set of inclusion minimal separators of G

• bw(G)=

Outline Blocks

• Introduction

• Definitions

• Theorems

• Algorithm

• Conclusion

Conclusion Blocks

• Enumerating the blocks in a graph and finding the block-branchwidth of the blocks leads to a O*((2+√3)n) time algorithm for the branchwidth problem

• This is the best algorithm for branchwidth problem

Conclusion Blocks

• 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, η), for some optimal branch decompositions of the graph?