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{{about|tree structure of graphs|decomposition of graphs into trees|Graph_theory#Decomposition_problems}}
 
[[Image:Tree decomposition.svg|thumb|240px|A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two.]]
In [[graph theory]], a '''tree decomposition''' is a mapping of a [[graph (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.
 
In [[machine learning]], tree decompositions are also called '''junction trees''', '''clique trees''', or '''join trees'''; they
play an important role in problems like [[belief propagation|probabilistic inference]], [[constraint satisfaction]], [[query optimization]], and [[matrix decomposition]].
 
The concept of tree decompositions 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 G as subtrees of a tree, in such a way that vertices in the given graph are adjacent only when the corresponding subtrees intersect. Thus, 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 ''G'' = (''V'', ''E''), a tree decomposition is a pair (''X'', ''T''), where ''X'' = {''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>} is a family of subsets of ''V'', and ''T'' is a tree whose nodes are the subsets ''X''<sub>''i''</sub>, satisfying the following properties:<ref>{{harvtxt|Diestel|2005}} section 12.3</ref>
 
# The union of all sets ''X''<sub>''i''</sub> equals ''V''. That is, each graph vertex is associated with at least one tree node.
# For every edge (''v'', ''w'') in the graph, there is a subset ''X''<sub>''i''</sub> that contains both ''v'' and ''w''. That is, vertices are adjacent in the graph only when the corresponding subtrees have a node in common.
# If ''X''<sub>''i''</sub> and ''X''<sub>''j''</sub> both contain a vertex ''v'', then all nodes ''X''<sub>''k''</sub> of the tree in the (unique) path between ''X''<sub>''i''</sub> and ''X''<sub>''j''</sub> contain ''v'' as well. That is, the nodes associated with vertex ''v'' form a connected subset of ''T''. This is also known as coherence, or the ''running intersection property''. It can be stated equivalently that if <math>X_i</math>, <math>X_j</math> and <math>X_k</math> are nodes, and <math>X_k</math> is on the path from <math>X_i</math> to <math>X_j</math>, 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.
 
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]].
 
==Treewidth==
{{main|Treewidth}}
 
The ''width'' of a tree decomposition is the size of its largest set ''X''<sub>''i''</sub> 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]].
 
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.
 
==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 ''k''. To solve this problem, first choose one of the nodes of the tree decomposition to be the root, arbitrarily. For a node ''X<sub>i</sub>'' of the tree decomposition, let ''D<sub>i</sub>'' be the union of the sets ''X<sub>j</sub>'' descending from ''X<sub>i</sub>''. For an independent set ''S''&nbsp;⊂&nbsp;''X<sub>i</sub>'', let ''A''(''S'',''i'') denote the size of the largest independent subset ''I'' of ''D<sub>i</sub>'' such that ''I''&nbsp;∩&nbsp;''X<sub>i</sub>''&nbsp;=&nbsp;''S''. Similarly, for an adjacent pair of nodes ''X<sub>i</sub>'' and ''X<sub>j</sub>'', with ''X<sub>i</sub>'' farther from the root of the tree than ''X<sub>j</sub>'', and an independent set ''S''&nbsp;⊂&nbsp;''X<sub>i</sub>''&nbsp;∩&nbsp;''X<sub>j</sub>'', let ''B''(''S'',''i'',''j'') denote the size of the largest independent subset ''I'' of ''D<sub>i</sub>'' such that ''I''&nbsp;∩&nbsp;''X<sub>i</sub>''&nbsp;∩&nbsp;''X<sub>j</sub>''&nbsp;=&nbsp;''S''. We may calculate these ''A'' and ''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>.
 
At each node or edge, there are at most 2<sup>''k''</sup> sets ''S'' for which we need to calculate these values, so if ''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"/>
 
==See also==
*[[Bramble (graph theory)|Brambles]] and [[Haven (graph theory)|havens]], two kinds of structures that can be used as an alternative to tree decomposition in defining the treewidth of a graph
*[[Branch-decomposition]], a closely related structure whose width is within a constant factor of treewidth
 
==Notes==
{{reflist|colwidth=30em}}
 
==References==
{{refbegin|colwidth=30em}}
*{{citation
| last1 = Arnborg | first1 = S.
| last2 = Corneil | first2 = D. | author2-link = Derek Corneil
| last3 = Proskurowski | first3 = A.
| title = Complexity of finding embeddings in a ''k''-tree
| journal = SIAM Journal on Matrix Analysis and Applications
| volume = 8 | issue = 2 | year = 1987 | pages = 277–284 | doi = 10.1137/0608024}}.
*{{citation
| last1 = Arnborg | first1 = S.
| last2 = Proskurowski | first2 = A.
| title = Linear time algorithms for NP-hard problems restricted to partial ''k''-trees
| journal = Discrete Applied Mathematics
| volume = 23 | issue = 1 | year = 1989 | pages = 11–24 | doi = 10.1016/0166-218X(89)90031-0}}.
*{{citation
| last1 = Bern | first1 = M. W.
| last2 = Lawler | first2 = E. L. | author2-link = Eugene Lawler
| last3 = Wong | first3 = A. L.
| title = Linear-time computation of optimal subgraphs of decomposable graphs
| journal = Journal of Algorithms
| volume = 8 | issue = 2 | year = 1987 | pages = 216–235 | doi = 10.1016/0196-6774(87)90039-3}}.
*{{citation
| last1 = Bertelé | first1 = Umberto
| last2 = Brioschi | first2 = Francesco
| title = Nonserial Dynamic Programming
| year = 1972
| publisher = Academic Press
| isbn = 0-12-093450-7}}.
*{{citation
| last = Bodlaender | first = Hans L. | authorlink = Hans L. Bodlaender
| contribution = Dynamic programming on graphs with bounded treewidth
| title = Proc. 15th International Colloquium on Automata, Languages and Programming
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| volume = 317 | year = 1988 | pages = 105–118
| doi = 10.1007/3-540-19488-6_110}}.
*{{citation
| last = Bodlaender | first = Hans L. | authorlink = Hans L. Bodlaender
| title = A linear time algorithm for finding tree-decompositions of small treewidth
| journal = SIAM Journal on Computing
| volume = 25 | issue = 6 | year = 1996 | pages = 1305–1317 | doi = 10.1137/S0097539793251219}}.
*{{Citation
| last=Diestel | first=Reinhard
| title=Graph Theory
| publisher=[[Springer Science+Business Media|Springer]]
| year=2005
| edition=3rd
| isbn=3-540-26182-6
| url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
}}.
*{{Citation
| title = ''S''-functions for graphs
| year = 1976
| last = Halin | first = Rudolf | authorlink = Rudolf Halin
| journal = Journal of Geometry
| pages = 171–186
| volume = 8
| doi=10.1007/BF01917434
}}.
*{{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 B
| volume = 36
| issue = 1 | year = 1984 | pages = 49–64
| doi = 10.1016/0095-8956(84)90013-3}}.
{{refend}}
 
[[Category:Trees (graph theory)]]
[[Category:Graph minor theory]]
[[Category:Graph theory objects]]

Revision as of 17:45, 3 September 2013

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A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two.

In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph.

In machine learning, tree decompositions are also called junction trees, clique trees, or join trees; they play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition.

The concept of tree decompositions was originally introduced by Template:Harvs. Later it was rediscovered by Template:Harvs and has since been studied by many other authors.[1]

Definition

Intuitively, a tree decomposition represents the vertices of a given graph G as subtrees of a tree, in such a way that vertices in the given graph are adjacent only when the corresponding subtrees intersect. Thus, G forms a 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 G = (V, E), a tree decomposition is a pair (X, T), where X = {X1, ..., Xn} is a family of subsets of V, and T is a tree whose nodes are the subsets Xi, satisfying the following properties:[2]

  1. The union of all sets Xi equals V. That is, each graph vertex is associated with at least one tree node.
  2. For every edge (v, w) in the graph, there is a subset Xi that contains both v and w. That is, vertices are adjacent in the graph only when the corresponding subtrees have a node in common.
  3. If Xi and Xj both contain a vertex v, then all nodes Xk of the tree in the (unique) path between Xi and Xj contain v as well. That is, the nodes associated with vertex v form a connected subset of T. This is also known as coherence, or the running intersection property. It can be stated equivalently that if , and are nodes, and is on the path from to , then .

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.

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.

Treewidth

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The width of a tree decomposition is the size of its largest set Xi 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 graphs, brambles, and havens.

It is NP-complete to determine whether a given graph G has treewidth at most a given variable k.[3] 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.[4] The time dependence of this algorithm on k is exponential.

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,Template:Sfnp a parameter related to treewidth. Later, several authors independently observed at the end of the 1980s[5] that many algorithmic problems that are 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 k. To solve this problem, first choose one of the nodes of the tree decomposition to be the root, arbitrarily. For a node Xi of the tree decomposition, let Di be the union of the sets Xj descending from Xi. For an independent set S ⊂ Xi, let A(S,i) denote the size of the largest independent subset I of Di such that I ∩ Xi = S. Similarly, for an adjacent pair of nodes Xi and Xj, with Xi farther from the root of the tree than Xj, and an independent set S ⊂ Xi ∩ Xj, let B(S,i,j) denote the size of the largest independent subset I of Di such that I ∩ Xi ∩ Xj = S. We may calculate these A and B values by a bottom-up traversal of the tree:

where the sum in the calculation of is over the children of node .

At each node or edge, there are at most 2k sets S for which we need to calculate these values, so if 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 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.[4]

See also

  • Brambles and havens, two kinds of structures that can be used as an alternative to tree decomposition in defining the treewidth of a graph
  • Branch-decomposition, a closely related structure whose width is within a constant factor of treewidth

Notes

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References

Template:Refbegin

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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

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