Paper introduction to Combinatorial Optimization on Graphs of Bounded Treewidth

Combinatorial Optimization on Graphs of Bounded Treewidth

Combinatorial Optimization on Graphs of BoundedTreewidth

HANS L. BODLAENDER AND ARIE M. C. A. KOSTERThe Computer Journal Volume 51 Issue 3, May 2008

Yu LIU @ IPL Camp

Aug 10th, 2014

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Main Topic

This Paper:

Introduce the concepts of treewidth and tree decompositions

Introduce a useful approach for obtaining fixed-parametertractable algorithms

surveys some of the latest (till 2007) developments

Applicability

Algorithms that exploit tree decompositions

Algorithms that determine or approximate treewidth and findoptimal tree decompositions

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Outline

1 Background

2 Efficient DP Algorithms on Graphs of Small TreewidthFor Series-Parallel GraphsGeneralization of DP Algorithms Using Tree Decompositions

3 Designing Algorithms To Solve problems on Graphs Given withA Tree Decomposition with Small Treewidth

Weighted Independent SetTreewidth and Fixed-Parameter Tractability

4 Determining The Treewidth of A Given GraphExact AlgorithmsApproximation AlgorithmsAlgorithmic Results for Planar Graphs

5 Remarks and Conclusions

6 Appendix


Background

Turn NP-hard Problems to Be More Tractable

Many combinatorial optimization problems defined on graphsbelong to the class of NP-hard problems in general. However, inmany cases, if the graphs are

trees (connected graphs without cycles), or

can construct some special trees form them

the problem becomes polynomial time solvable.

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Background

Weighted Independent Set Problem (WIS)

Definition ((Maximum) Weighted Independent Set)

Input is a graph G = (V ,E ) with vertex weightsc(v) ∈ Z+, v ∈ V .

Output is a subset S ⊆ V such that ∀v ∈ S are pairwisenon-adjacent so that the sum of the weightsc(S) =

∑v∈S c(v) is maximized.

NP-hard for general cases

Linear solvable on trees

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Background

Linear Algorithm for WIS on Trees

Root the tree at an arbitrary vertex r

let T (v) denote the subtree with v as root:

A(v) denotes the maximum weight of an independent set inT (v)B(v) denotes the maximum weight of an independent set inT (v) not containing vFor a non-leaf vertex v its children are x1, ..., xr

Algorithm

leaf: A(v) := c(v) and B(v) := 0

non-leaf:A(v) := c(v) + B(x1) + ...+ B(xr ) ↑ A(x1) + ...+ A(xr )

bottom-to-top compute A(v) for every v , until A(r)

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Outline

1 Background






6 Appendix


Efficient DP Algorithms on Graphs of Small Treewidth

For Series-Parallel Graphs

Series-Parallel Graphs

A two-terminal labeled graph (G , s, t) consists of a graph G with amarked source s ∈ V , and sink t ∈ V . New graphs can becomposed from two two-terminal labeled graphs in two ways: inseries or in parallel.

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SP-Tree

For every series-parallel graph, we can construct a so-calledSP-tree:

The leafs of the SP-tree T (G ) correspond to the edges e ∈ Ethe internal nodes are either labelled S or P for series andparallel composition of the series-parallel graphs associated bythe child-subtrees

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Polynomial-Time Algorithm for WIS on SP-Trees

G (i) denotes the series-parallel graph that is associated with nodei of the SP-tree.

AA(i): maximum weight of independent set containing both sand t

AB(i): maximum weight of independent set containing both sbut not t,

BA(i): maximum weight of independent set containing both tbut not s

BB(i): maximum weight of independent set containingneither s nor t,

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For leaves,

AA(i) := −∞AB(i) := c(s)

BA(i) := c(t)

BB(i) := 0

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For non-leaves (S-node),

AA(i) := AA(i1) + AA(i2) + c(s ′) ↑AB(i1) + BA(i2)

AB(i) := AA(i1) +AB(i2) + c(s ′) ↑AB(i1) + BB(i2)

BA(i) := BA(i1) + AA(i2) + c(s ′) ↑BB(i1) + BA(i2)

BB(i) := BA(i1) +AB(i2) + c(s ′) ↑BB(i1) + BB(i2)

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For non-leaves (P-node),

AA(i) := AA(i1) + AA(i2)− c(s)− c(t)

AB(i) := AB(i1) + AB(i2)− c(s)

BA(i) := BA(i1) + BA(i2)− c(t)

BB(i) := BB(i1) + BB(i2)

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Graphs of Bounded Treewidth

Similar approach can be found for more general graphs, if theyhave bounded treewidth.

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Tree Decompositions (TD) of Graphs

A Tree Decomposition [Robertson and Seymour ’86] of a graphG (V ,E ) is a pair: ({Xi |i ∈ I},T (I ,F )).

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Tree Decompositions of Graphs

Tree Decomposition

Related concepts:

Path-decomposition [Robertson and Seymour ’83]

Branch-decomposition [Robertson and Seymour ’91]

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Treewidth of Graph

Treewidth is a decisive parameter on the computation complexity.

The width of a tree decomposition (T , {Xi |i ∈ I}):max i∈I ||Xi | − 1|The treewidth of G is the minimum width over all treedecompositions of G .

Computing treewidth of general graphs is a NP-hard problem[Arnborg’87]

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Nice Tree Decomposition

A special type of tree decomposition that is very useful fordescribing dynamic programming algorithms.

Definition (Nice Tree Decomposition)

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Nice Tree Decomposition

A special type of tree decomposition that is very useful fordescribing dynamic programming algorithms.

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Generalization of DP Algorithms Using Tree Decompositions

Dynamic Programming on Tree Decompositions

Constructing DP tables inbottom-up manner [Arn-borg+’87,Bodlaender+’88].

The computation finishedwhen achieve the root node.

The size of table is decidedby the treewidth.

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Outline

1 Background






6 Appendix


Designing Algorithms To Solve problems on Graphs Given with A Tree Decomposition with Small Treewidth

Weighted Independent Set

Polynomial-Time Algorithm for WIS on Nice TD

Suppose input is a graph G (V ,E ) and a (nice) tree decompositionof G of width k , say ({Xi |i ∈ I},T (I ,F ), an O(2k n)-timealgorithm exists.

For each node i ∈ I , we compute a table, which we term Ci

that contains an integer value for each subset S ⊆ Xi . Eachtable Ci contains at most 2k+1 values.

For a node i , we use Gi (Vi ,Ei ) to denote the subgraphinduced by Xi and all its descendant (corresponding to thesubtree rooted by i).

Each of these values Ci (S), for S ⊆ Xi , equals the maximumweight of an independent set W ⊆ Vi in Gi such thatXi ∩W = S .

In case no independent set exists, we set Ci (S) = −∞

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Computing Leaf-Node Table

If i is a leaf of T , then |Xi | = 1, say Xi = {v}. The table Ci hasonly two entries, and we can compute these trivially: Ci (∅) = 0and Ci ({v}) = c(v).

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Computing Introduce-Node Table

Suppose i is an introduce node with child j . SupposeXi = Xj ∪ {v}.

Each of the at most 2k+1 entries can be computed in O(k) time.

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Computing Forget-Node Table

Suppose i is an forget node with child j . Suppose Xi = Xj − {v}.

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Computing Join-Node Table

Suppose i is a join node with children j1 and j2.

Any two vertices picked form Gj1 and Gj2 are not adjacent if theyare not in Xi .

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Computing Join-Node Table

Suppose i is a join node with children j1 and j2.

x y z

x y z

x y z

...

...

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Putting It All Together

When we have computed the table for root node we can find theWIS.

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Treewidth and Fixed-Parameter Tractability

On Fixed-Parameter Tractab (FPT) Problems

Many problems that can be solved in O(f (k).n) time when thegraph is given with a (nice) tree decomposition of width k (withO(n) nodes), for some function f .

Hamiltonian Circuit

Chromatic Number (vertex colouring),

Vertex Cover

Steiner Tree

Feedback Vertex Set ...

Similar algorithms (as to WIS) exist [Telle and Proskurowski],[Koster et al.], [Arnborg and Proskurowski], [Bern et al.] and[Wimer et al.].

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Establishing Fixed-Parameter Tractability

Treewidth can be used in several cases to quickly establish that aproblem is FPT. For example,

Longest Cycle problem: given an undirected graph G (V ,E ),and an integer k , and ask if G has a cycle of at least k edges.

Feedback Vertex Set problem: given an undirected graphG (V ,E ) and an integer k , and ask for a set of vertices W ofsize at most k that is a feedback vertex set, i.e. G [V −W ] isa forest.

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Other Results

If we have a bounded width tree decomposition of G .

For each graph property that can be formulated in MonadicSecond-Order Logic (MSOL), there is a linear time algorithmthat verifies if the property holds for the given graph G[Courcelle]

in [Arnborg et al.] and [Borie et al.], it is shown that theabove result can be extended to optimisation problems

A more extensive overview of MSOL and its applications canbe found in [Hlineny et al].

The use of tree decompositions for solving problems can alsobe found in the area of probabilistic networks [Cooper],[Lauritzen and Spiegelhalter].

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The Theoretical and Actual Table Sizes

The picture shows partial constraint satisfaction graph and actualtable sizes versus theoretical table sizes during dynamicprogramming algorithm.

With pre-processing and reduction techniques the actual table sizescould be kept within the main memory size [Koster et al] .

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Outline

1 Background






6 Appendix


Determining The Treewidth of A Given Graph

Exact Algorithms

FPT and Exact Algorithms

Treewidth belongs to FPT. [Bodlaender’96] gives a linearalgorithm.

O(nk+2) algorithm [Arnborg et al. ’87]

An O(nn−k) branch and bound algorithm based onvertex-ordering has been proposed by [Gogate and Dechter]

Others: O(2np(n)) [Arnborg et al. ’87], O(1.8899np(n))[Fomin et al.] (p(n) denotes a polynomial in n)

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Approximation Algorithms

Overview

Algorithms with algorithmic guarantee

(log k)-approximation algorithm (O(k log k), O(k√log k))

exit algorithms that run in time polynomial in n butexponential in the treewidth knot known whether there exist constant approximationalgorithms that run in time polynomial in n and k.

Heuristic algorithms (without algorithmic guarantee)

based on Lexicographic Breadth First Search (LBFS)/Maximum Cardinality Search (MCS)fill-in based algorithms ∗

meta-heuristics have been applied to find good treedecompositions (Tabu search [Clautiaux et al. ’04], geneticalgorithms (Larranaga et al. ’97)

∗The Fill-In problem is to determine the minimum number of edges to beadded to a graph G such that the result is chordal.

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Some Definiations on Graph Theory

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Important Lemmers for Approximation Algorithm

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Lower Bound

The lower bound bounds the treewidth from below.

A good estimate of the true treewidth of a graph mightobtained

A high lower bound on the treewidth for a particular class ofgraphs indicates that the applicability of the dynamicprogramming methodology is limited

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Key Points for Approximation

The treewidth of graphs is closed under taking subgraphs andminors

However, easy-to-compute lower bounds for treewidth are notclosed under taking subgraphs and minors

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Some Results of Lower Bound

let the minimum degree of graph G be denoted asδ(G ) := minv∈V |N(v)| [Scheffler ’89] then δ(G ) ≤ tw(G ),and we can also have two bounds:

δD(G ) := maxH⊆Gδ(H) (H is a subset)δC (G ) := maxH�Gδ(H) (H is a minor)δC (G ) can only be approximated (from below)

MCS algorithm can also be used for lower bounding

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Algorithmic Results for Planar Graphs

Some Results

A polynomial time algorithm for branchwidth exits [Seymourand Thomas ’94]

Hicks [’05a, ’05b] has shown that it is practical.

It is an open question whether the treewidth of planar graphscan be computed in polynomial time.

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Some Results

In polynomial time we can obtain a tree decomposition ofwidth at most 1.5 k

It is an open question whether the treewidth of planar graphscan be computed in polynomial time.

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Treewidth of Planar Graphs

For instance, a planar graph of treewidth k has a 1/2-balancedseparator † of size at most k + 1.

†A set S is a 1/2-balanced separator in a graph G(V ,E), if each connectedcomponent of G [V − S ] has at most 1/2n vertices.

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Theorem 14 can be used to obtain faster exponential timealgorithms for NP-hard problems on planar graphs.

O∗(c√n)-time algorithm (for some constant c) for WIS,

Dominating Set, Vertex Cover, etc.

If a problem has an algorithm solving it in O(ckn) time whena tree decomposition of width k is given, then it can be solvedon planar graphs in O∗(c ′

√n) time. [Dorn et al.]

If a planar graph G has a dominating set of size at most k ,then its treewidth is O(

√k).

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Approximation Algorithms on Planar Graphs

Given a plane embedding of a planar graph G (V ,E ), we divide itsvertices into layersL1, L2, ..., LT in the following way.

All vertices that are incident to the exterior face are in layerL1.

For i ≤ 1, suppose we remove from the embedding all verticesin layers L1, ..., Li , and their incident edges.

All vertices that are then incident to the exterior face are inlayer Li+1. LT is thus the last nonempty layer.

A plane graph that has an embedding where the vertices are in klayers is called k-outerplanar.

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Outline

1 Background






6 Appendix


Remarks and Conclusions

This paper surveyed the concept of treewidth in the context ofFPT algorithms.

Treewidth provides a powerful tool for determining thefixed-parameter tractability of general NP-hard combinatorialoptimization problems.

Research about treewidth has exposed the strong potential ofthe concept of bounded treewidth for addressing thechallenges posed by NP-hard combinatorial optimizationproblems

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Outline

1 Background






6 Appendix


Appendix

FPT

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Appendix

MCS/LBFS

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Appendix

Branch-decomposition

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Appendix

Branch-decomposition

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Appendix

Path-decomposition

A path-decomposition is a tree decomposition ({Xi |i ∈ I},T (I ,F ))in which the underlying tree T of the decomposition is a pathgraph.

For each edge of G , there exists an i such that both endpointsof the edge belong to Xi , and

For every three indices i ≤ j ≤ k ,Xi ∩ Xk ⊆ Xj .

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Paper introduction to Combinatorial Optimization on Graphs of Bounded Treewidth

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