Tree decomposition: Difference between revisions

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{{Short description|Mapping of a graph into a tree}}

{{about|tree structure of graphs|decomposition of graphs into trees|Graph theory#Decomposition problems|decomposition of trees in nature|Nurse log}}

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In [[graph theory]], a '''tree decomposition''' is a mapping of a [[Graph (discrete mathematics)|graph]] into a [[tree (graph theory)|tree]] that can be used to define the [[treewidth]] of the graph and speed up solving certain computational problems on the graph.

Tree decompositions are also called '''junction trees''', '''clique trees''', or '''join trees''';. theyThey play an important role in problems like [[belief propagation|probabilistic inference]], [[constraint satisfaction]], [[query optimization]],{{citation neededsfnp|reason=It is not apparent from the linked Wikipedia article how tree decompositions are used there. Further neither this nor the linked Wikipedia article references any scientific paper on the subject. It is impossible for someone not knowledgeable in that ___domain to verify this. Gottlob|date=June 2016Lee|Valiant|Valiant|2012}} and [[matrix decomposition]].

The concept of tree decompositionsdecomposition was originally introduced by {{harvs|last=Halin|first=Rudolf|authorlink=Rudolf Halin|year=1976|txt}}. Later it was rediscovered by {{harvs|first1=Neil|last1=Robertson|author1-link=Neil Robertson (mathematician)|first2=Paul|last2=Seymour|author2-link=Paul Seymour (mathematician)|year=1984|txt}} and has since been studied by many other authors.<ref>{{harvtxt|Diestel|2005}} pp.354–355</ref>

==Definition==

Intuitively, a tree decomposition represents the vertices of a given graph ''{{mvar|G''}} as subtrees of a tree, in such a way that vertices in the given graph{{mvar|G}} are adjacent only when the corresponding subtrees intersect. Thus, ''{{mvar|G''}} forms a [[Glossary of graph theory#Subgraphs|subgraph]] of the [[intersection graph]] of the subtrees. The full intersection graph is a [[chordal graph]].

Each subtree associates a graph vertex with a set of tree nodes. To define this formally, we represent each tree node as the set of vertices associated with it.

Thus, given a graph {{math|1=''G'' = (''V'', ''E'')}}, a tree decomposition is a pair {{math|(''X'', ''T'')}}, where {{math|1=''X'' = {''X''<{{sub>|1</sub>}}, ...…, ''X''<{{sub>''|n}}''</sub>} }} is a family of subsets (sometimes called ''bags'') of ''{{mvar|V''}}, and ''{{mvar|T''}} is a tree whose nodes are the subsets ''{{mvar|X''<{{sub>''|i''</sub>}}}}, satisfying the following properties:<ref>{{harvtxt|Diestel|2005}} section 12.3</ref>

# The union of all sets ''{{mvar|X''<{{sub>''|i''</sub>}}}} equals ''{{mvar|V''}}. That is, each graph vertex is associated with at least one tree node.

# For every edge {{math|(''v'', ''w'')}} in the graph, there is a subset ''{{mvar|X''<{{sub>''|i''</sub>}}}} that contains both ''{{mvar|v''}} and ''{{mvar|w''}}. That is, vertices are adjacent in the graph only when the corresponding subtrees have a node in common.

# If ''{{mvar|X''<{{sub>''|i''</sub>}}}} and ''{{mvar|X''<{{sub>''|j''</sub>}}}} both contain a vertex ''{{mvar|v''}}, then all nodes ''{{mvar|X''<{{sub>''|k''</sub>}}}} of the tree in the (unique) path between ''{{mvar|X''<{{sub>''|i''</sub>}}}} and ''{{mvar|X''<{{sub>''|j''</sub>}}}} contain ''{{mvar|v''}} as well. That is, the nodes associated with vertex ''{{mvar|v''}} form a connected subset of ''{{mvar|T''}}. This is also known as coherence, or the ''running intersection property''. It can be stated equivalently that if <math>X_i</math>{{mvar|X{{sub|i}}}}, <math>X_j</math>{{mvar|X{{sub|j}}}} and <math>X_k</math>{{mvar|X{{sub|k}}}} are nodes, and <math>X_k</math>{{mvar|X{{sub|k}}}} is on the path from <math>X_i</math>{{mvar|X{{sub|i}}}} to <math>X_j</math>{{mvar|X{{sub|j}}}}, then <math>X_i \cap X_j \subseteq X_k</math>.

The tree decomposition of a graph is far from unique; for example, a trivial tree decomposition contains all vertices of the graph in its single root node.

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A tree decomposition in which the underlying tree is a [[path graph]] is called a path decomposition, and the width parameter derived from these special types of tree decompositions is known as [[pathwidth]].

A tree decomposition {{math|1=(''X'', ''T'' = (''I'', ''F''))}} of treewidth ''{{mvar|k''}} is ''smooth'', if for all <math>i \in I : |X_i| = k + 1</math>, and for all <math>(i, j) \in F : |X_i \cap X_j| = k</math>.<ref name="b96">{{harvtxt|Bodlaender|1996}}.</ref>

The minimum number of trees in a tree decomposition is the '''tree number''' of ''G.''

==Treewidth==

{{main|Treewidth}}

[[File:Treedecompsnocolour.JPG|thumb|upright=1.5|Two different tree-decompositions of the same graph]]

The ''width'' of a tree decomposition is the size of its largest set ''X''_''i'' minus one. The [[treewidth]] tw(''G'') of a graph ''G'' is the minimum width among all possible tree decompositions of ''G''. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree equal to one. Treewidth may also be defined from other structures than tree decompositions, including [[chordal graph]]s, [[bramble (graph theory)|brambles]], and [[haven (graph theory)|havens]].

The ''width'' of a tree decomposition is the size of its largest set ''{{mvar|X''<{{sub>''|i''</sub>}}}} minus one. The [[treewidth]] {{math|tw(''G'')}} of a graph ''{{mvar|G''}} is the minimum width among all possible tree decompositions of ''{{mvar|G''}}. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree equal to one. Treewidth may also be defined from other structures than tree decompositions, including [[chordal graph]]s, [[bramble (graph theory)|brambles]], and [[haven (graph theory)|havens]].

It is NP-complete to determine whether a given graph ''G'' has treewidth at most a given variable ''k''.<ref>{{harvtxt|Arnborg|Corneil|Proskurowski|1987}}.</ref>

However, when ''k'' is any fixed constant, the graphs with treewidth ''k'' can be recognized, and a width ''k'' tree decomposition constructed for them, in linear time.<ref name="b96">{{harvtxt|Bodlaender|1996}}.</ref> The time dependence of this algorithm on ''k'' is exponential in ''k''².

It is NP-complete to determine whether a given graph ''{{mvar|G''}} has treewidth at most a given variable ''{{mvar|k''}}.<ref>{{harvtxt|Arnborg|Corneil|Proskurowski|1987}}.</ref>

However, when ''{{mvar|k''}} is any fixed constant, the graphs with treewidth ''{{mvar|k''}} can be recognized, and a width ''{{mvar|k''}} tree decomposition constructed for them, in linear time.<ref name="b96">{{harvtxt|Bodlaender|1996}}.</ref> The time dependence of this algorithm on ''{{mvar|k''}} is an exponential infunction of {{math|''k''<{{sup>2</sup>|3}}}}.

==Dynamic programming==

At the beginning of the 1970s, it was observed that a large class of combinatorial optimization problems defined on graphs could be efficiently solved by non-serial [[dynamic programming]] as long as the graph had a bounded ''dimension'',{{sfnp|Bertelé|Brioschi|1972}} a parameter related to treewidth. Later, several authors independently observed, at the end of the 1980s,<ref>{{harvtxt|Arnborg|Proskurowski|1989}}; {{harvtxt|Bern|Lawler|Wong|1987}}; {{harvtxt|Bodlaender|1988}}.</ref> that many algorithmic problems that are [[NP-completeness|NP-complete]] for arbitrary graphs may be solved efficiently by [[dynamic programming]] for graphs of bounded treewidth, using the tree-decompositions of these graphs.

As an example, consider the problem of finding the [[maximum independent set]] in a graph of treewidth ''{{mvar|k''}}. To solve this problem, first choose one of the nodes of the tree decomposition to be the root, arbitrarily. For a node ''{{mvar|X<{{sub>|i</sub>''}}}} of the tree decomposition, let ''{{mvar|D<{{sub>|i</sub>''}}}} be the union of the sets ''{{mvar|X<{{sub>|j</sub>''}}}} descending from ''{{mvar|X<{{sub>|i</sub>''}}}}. For an independent set ''S''&nbsp;⊂&nbsp;''X<submath>iS \subset X_i,</submath>'', let {{math|''A''(''S'',''i'')}} denote the size of the largest independent subset ''{{mvar|I''}} of ''{{mvar|D<{{sub>|i</sub>''}}}} such that ''I''&nbsp;∩&nbsp;''X<submath>iI \cap X_i = S.</submath>''&nbsp;=&nbsp;''S''. Similarly, for an adjacent pair of nodes ''{{mvar|X<{{sub>|i</sub>''}}}} and ''{{mvar|X<{{sub>|j</sub>''}}}}, with ''{{mvar|X<{{sub>|i</sub>''}}}} farther from the root of the tree than ''{{mvar|X<{{sub>|j</sub>''}}}}, and an independent set ''S''&nbsp;⊂&nbsp;''X<submath>iS \subset X_i \cap X_j,</submath>''&nbsp;∩&nbsp;''X_j'', let {{math|''B''(''S'',''i'',''j'')}} denote the size of the largest independent subset ''{{mvar|I''}} of ''{{mvar|D<{{sub>|i</sub>''}}}} such that ''I''&nbsp;∩&nbsp;''X<submath>iI \cap X_i \cap X_j = S.</submath>''&nbsp;∩&nbsp;''X_j''&nbsp;=&nbsp;''S''. We may calculate these ''{{mvar|A''}} and ''{{mvar|B''}} values by a bottom-up traversal of the tree:

:<math>A(S,i)=|S| + \sum_{j} \left(B(S\cap X_j, j,i) - |S\cap X_j|\right)</math>

:<math>B(S,i,j)=\max_{S'\subset X_i\atop S=S'\cap X_j} A(S',i)</math>

where the sum in the calculation of <math>A(S,i)</math> is over the children of node <math>X_i</math>{{mvar|X{{sub|i}}}}.

At each node or edge, there are at most {{math|2<{{sup>|''k''</sup>}}}} sets ''{{mvar|S''}} for which we need to calculate these values, so if ''{{mvar|k''}} is a constant then the whole calculation takes constant time per edge or node. The size of the maximum independent set is the largest value stored at the root node, and the maximum independent set itself can be found (as is standard in dynamic programming algorithms) by backtracking through these stored values starting from this largest value. Thus, in graphs of bounded treewidth, the maximum independent set problem may be solved in linear time. Similar algorithms apply to many other graph problems.

This dynamic programming approach is used in [[machine learning]] via the [[junction tree algorithm]] for [[belief propagation]] in graphs of bounded treewidth. It also plays a key role in algorithms for computing the treewidth and constructing tree decompositions: typically, such algorithms have a first step that [[approximation algorithm|approximates]] the treewidth, constructing a tree decomposition with this approximate width, and then a second step that performs dynamic programming in the approximate tree decomposition to compute the exact value of the treewidth.<ref name="b96"/>

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*[[Bramble (graph theory)|Brambles]] and [[Haven (graph theory)|havens]]{{snd}}Two kinds of structures that can be used as an alternative to tree decomposition in defining the treewidth of a graph.

*[[Branch-decomposition]]{{snd}}A closely related structure whose width is within a constant factor of treewidth.

*[[Decomposition_method_Decomposition method (constraint_satisfactionconstraint satisfaction)|Decomposition Method]]{{snd}}Tree Decomposition is used in Decomposition Method for solving constraint satisfaction problem.

==Notes==

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| series = Lecture Notes in Computer Science

| volume = 317 | year = 1988 | pages = 105–118

| doi = 10.1007/3-540-19488-6_110| hdl = 1874/16258 | isbn = 978-3-540-19488-0 | hdl-access = free}}.

*{{citation

| last = Bodlaender | first = Hans L. | authorlink = Hans L. Bodlaender

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| url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/

}}.

*{{citation

| last1 = Gottlob | first1 = Georg

| last2 = Lee | first2 = Stephanie Tien

| last3 = Valiant | first3 = Gregory

| last4 = Valiant | first4 = Paul

| doi = 10.1145/2220357.2220363

| issue = 3

| journal = [[Journal of the ACM]]

| mr = 2946220

| page = A16:1–A16:35

| title = Size and treewidth bounds for conjunctive queries

| volume = 59

| year = 2012}}

*{{Citation

| title = ''S''-functions for graphs

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| pages = 171–186

| volume = 8

| issue = 1–2

| doi=10.1007/BF01917434

| s2cid = 120256194

}}.

*{{citation

| last1 = Robertson | first1 = Neil | authorlink1 = Neil Robertson (mathematician)

| last2 = Seymour | first2 = Paul D. | authorlink2 = Paul Seymour (mathematician)

| title = Graph minors III: Planar tree-width

| journal = [[Journal of Combinatorial Theory,]] | series=Series B

| volume = 36

| issue = 1 | year = 1984 | pages = 49–64