Paper introduction to Combinatorial Optimization on Graphs of Bounded Treewidth

Combinatorial Optimization on Graphs of Bounded Treewidth
Combinatorial Optimization on Graphs of Bounded
Treewidth
HANS L. BODLAENDER AND ARIE M. C. A. KOSTER
The 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-parameter
tractable algorithms
surveys some of the latest (till 2007) developments
Applicability
Algorithms that exploit tree decompositions
Algorithms that determine or approximate treewidth and ﬁnd
optimal tree decompositions
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Outline
1 Background
2 Eﬃcient DP Algorithms on Graphs of Small Treewidth
For Series-Parallel Graphs
Generalization of DP Algorithms Using Tree Decompositions
3 Designing Algorithms To Solve problems on Graphs Given with
A Tree Decomposition with Small Treewidth
Weighted Independent Set
Treewidth and Fixed-Parameter Tractability
4 Determining The Treewidth of A Given Graph
Exact Algorithms
Approximation Algorithms
Algorithmic 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 graphs
belong to the class of NP-hard problems in general. However, in
many 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 weights
c(v) ∈ Z+, v ∈ V .
Output is a subset S ⊆ V such that ∀v ∈ S are pairwise
non-adjacent so that the sum of the weights
c(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 in
T(v)
B(v) denotes the maximum weight of an independent set in
T(v) not containing v
For 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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Eﬃcient DP Algorithms on Graphs of Small Treewidth
Series-Parallel Graphs
A two-terminal labeled graph (G, s, t) consists of a graph G with a
marked source s ∈ V , and sink t ∈ V . New graphs can be
composed from two two-terminal labeled graphs in two ways: in
series or in parallel.
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SP-Tree
For every series-parallel graph, we can construct a so-called
SP-tree:
The leafs of the SP-tree T(G) correspond to the edges e ∈ E
the internal nodes are either labelled S or P for series and
parallel composition of the series-parallel graphs associated by
the 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 node
i of the SP-tree.
AA(i): maximum weight of independent set containing both s
and t
AB(i): maximum weight of independent set containing both s
but not t,
BA(i): maximum weight of independent set containing both t
but not s
BB(i): maximum weight of independent set containing
neither 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 they
have bounded treewidth.
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Tree Decompositions (TD) of Graphs
A Tree Decomposition [Robertson and Seymour ’86] of a graph
G(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}):
maxi∈I ||Xi | − 1|
The treewidth of G is the minimum width over all tree
decompositions 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 for
describing dynamic programming algorithms.
Deﬁnition (Nice Tree Decomposition)
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Nice Tree Decomposition
A special type of tree decomposition that is very useful for
describing dynamic programming algorithms.
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Generalization of DP Algorithms Using Tree Decompositions
Dynamic Programming on Tree Decompositions
Constructing DP tables in
bottom-up manner [Arn-
borg+’87,Bodlaender+’88].
The computation ﬁnished
when achieve the root node.
The size of table is decided
by the treewidth.
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Designing Algorithms To Solve problems on Graphs Given with A Tree Decomposition with Small Treewidth
Polynomial-Time Algorithm for WIS on Nice TD
Suppose input is a graph G(V , E) and a (nice) tree decomposition
of G of width k, say ({Xi |i ∈ I}, T(I, F), an O(2k n)-time
algorithm exists.
For each node i ∈ I, we compute a table, which we term Ci
that contains an integer value for each subset S ⊆ Xi . Each
table Ci contains at most 2k+1 values.
For a node i, we use Gi (Vi , Ei ) to denote the subgraph
induced by Xi and all its descendant (corresponding to the
subtree rooted by i).
Each of these values Ci (S), for S ⊆ Xi , equals the maximum
weight of an independent set W ⊆ Vi in Gi such that
Xi ∩ 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 has
only two entries, and we can compute these trivially: Ci (∅) = 0
and Ci ({v}) = c(v).
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Computing Introduce-Node Table
Suppose i is an introduce node with child j. Suppose
Xi = 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 they
are 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 the
WIS.
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On Fixed-Parameter Tractab (FPT) Problems
Many problems that can be solved in O(f (k).n) time when the
graph is given with a (nice) tree decomposition of width k (with
O(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 a
problem 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 graph
G(V , E) and an integer k, and ask for a set of vertices W of
size at most k that is a feedback vertex set, i.e. G[V − W ] is
a 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 Monadic
Second-Order Logic (MSOL), there is a linear time algorithm
that veriﬁes if the property holds for the given graph G
[Courcelle]
in [Arnborg et al.] and [Borie et al.], it is shown that the
above result can be extended to optimisation problems
A more extensive overview of MSOL and its applications can
be found in [Hlineny et al].
The use of tree decompositions for solving problems can also
be 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 actual
table sizes versus theoretical table sizes during dynamic
programming algorithm.
With pre-processing and reduction techniques the actual table sizes
could be kept within the main memory size [Koster et al] .
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Determining The Treewidth of A Given Graph
Exact Algorithms
FPT and Exact Algorithms
Treewidth belongs to FPT. [Bodlaender’96] gives a linear
algorithm.
O(nk+2) algorithm [Arnborg et al. ’87]
An O(nn−k) branch and bound algorithm based on
vertex-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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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 but
exponential in the treewidth k
not known whether there exist constant approximation
algorithms 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 tree
decompositions (Tabu search [Clautiaux et al. ’04], genetic
algorithms (Larranaga et al. ’97)
∗
The Fill-In problem is to determine the minimum number of edges to be
added 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 might
obtained
A high lower bound on the treewidth for a particular class of
graphs indicates that the applicability of the dynamic
programming methodology is limited
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Key Points for Approximation
The treewidth of graphs is closed under taking subgraphs and
minors
However, easy-to-compute lower bounds for treewidth are not
closed under taking subgraphs and minors
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Some Results of Lower Bound
let the minimum degree of graph G be denoted as
δ(G) := minv∈V |N(v)| [Scheﬄer ’89] then δ(G) ≤ tw(G),
and we can also have two bounds:
δD(G) := maxH⊆G δ(H) (H is a subset)
δC(G) := maxH G δ(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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Some Results
A polynomial time algorithm for branchwidth exits [Seymour
and Thomas ’94]
Hicks [’05a, ’05b] has shown that it is practical.
It is an open question whether the treewidth of planar graphs
can be computed in polynomial time.
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Some Results
In polynomial time we can obtain a tree decomposition of
width at most 1.5 k
It is an open question whether the treewidth of planar graphs
can 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-balanced
separator † of size at most k + 1.
†
A set S is a 1/2-balanced separator in a graph G(V , E), if each connected
component of G[V − S] has at most 1/2n vertices.
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Theorem 14 can be used to obtain faster exponential time
algorithms 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 when
a tree decomposition of width k is given, then it can be solved
on 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 its
vertices into layersL1, L2, ..., LT in the following way.
All vertices that are incident to the exterior face are in layer
L1.
For i ≤ 1, suppose we remove from the embedding all vertices
in layers L1, ..., Li , and their incident edges.
All vertices that are then incident to the exterior face are in
layer Li+1. LT is thus the last nonempty layer.
A plane graph that has an embedding where the vertices are in k
layers is called k-outerplanar.
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Remarks and Conclusions
This paper surveyed the concept of treewidth in the context of
FPT algorithms.
Treewidth provides a powerful tool for determining the
ﬁxed-parameter tractability of general NP-hard combinatorial
optimization problems.
Research about treewidth has exposed the strong potential of
the concept of bounded treewidth for addressing the
challenges posed by NP-hard combinatorial optimization
problems
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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 path
graph.
For each edge of G, there exists an i such that both endpoints
of 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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