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[[Image:Graph fractional coloring.svg|right|thumb| 5:2-coloring of [[Dodecahedron|Dodecahedral graph]]. A 4:2-coloring of | |||
this graph does not exist.]] | |||
'''Fractional coloring''' is a topic in a young branch of [[graph theory]] known as [[fractional graph theory]]. It is a generalization of ordinary [[graph coloring]]. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. | |||
Fractional graph coloring can be viewed as the [[linear programming relaxation]] of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems. | |||
==Definitions== | |||
[[File:Fractional coloring of C5.png|thumb|Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring.<br/>Below: A 5:2 coloring of the same graph.]] | |||
A '''''b''-fold coloring''' of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets. | |||
An '''''a'':''b''-coloring''' is a ''b''-fold coloring out of ''a'' available colors. | |||
The '''''b''-fold chromatic number''' χ<sub>''b''</sub>(''G'') is the least ''a'' such that an ''a'':''b''-coloring exists. | |||
The '''fractional chromatic number''' χ<sub>f</sub>(''G'') is defined to be | |||
:<math>\chi_{f}(G) = \lim_{b \to \infty}\frac{\chi_{b}(G)}{b} = \inf_{b}\frac{\chi_{b}(G)}{b}</math> | |||
Note that the limit exists because χ<sub>''b''</sub>(''G'') is ''[[subadditive function|subadditive]]'', meaning χ<sub>''a''+''b''</sub>(''G'') ≤ χ<sub>''a''</sub>(''G'') + χ<sub>''b''</sub>(''G''). | |||
The fractional chromatic number can equivalently be defined in probabilistic terms. χ<sub>f</sub>(''G'') is the smallest ''k'' for which there exists a probability distribution over the [[Independent_set_(graph_theory)|independent sets]] of ''G'' such that for each vertex ''v'', given an independent set ''S'' drawn from the distribution, | |||
:<math>\Pr(v\in S) \geq \frac{1}{k}</math>. | |||
Some properties of χ<sub>''f''</sub>(''G''): | |||
:<math>\chi_f(G)\ge n(G)/\alpha(G)</math> | |||
and | |||
:<math>\omega(G) \le \chi_f(G) \le \chi(G)</math>. | |||
Here n(''G'') is the [[glossary of graph theory#Basics|order]] of ''G'', α(''G'') is the [[independence number]], ω(''G'') is the [[clique number]], and χ(''G'') is the [[chromatic number]]. | |||
==Linear Programming (LP) Formulation== | |||
The fractional chromatic number χ<sub>f</sub>(''G'') of a graph ''G'' can be obtained as a solution to a [[linear programming|linear program]]. Let <math>\mathcal{I}</math>(''G'') be the set of all independent sets of ''G'', and let <math>\mathcal{I}</math>(''G'',''x'') be the set of all those independent sets which include vertex ''x''. For each independent set ''I'', define a nonnegative real variable ''x<sub>I</sub>''. Then χ<sub>f</sub>(''G'') is the minimum value of | |||
: <math>\sum_{I\in\mathcal{I}(G)} x_I\,</math>, | |||
: subject to <math>\sum_{I\in\mathcal{I}(G,x)} x_I \ge 1</math> for each vertex <math>x</math>. | |||
The [[dual problem|dual]] of this linear program computes the "fractional clique number", a relaxation to the rationals of the integer concept of [[clique number]]. That is, a weighting of the vertices of ''G'' such that the total weight assigned to any independent set is at most ''1''. The [[strong duality]] theorem of linear programming guarantees that the optimal solutions to both linear programs have the same value. Note however that each linear program may have size that is exponential in the number of vertices of ''G'', and that computing the fractional chromatic number of a graph is [[NP-hard]].<ref>[[Carsten Lund]] and [[Mihalis Yannakakis]]: "[http://dx.doi.org/10.1145/185675.306789 On the hardness of approximating minimization problems]", J. ACM 41:5(1994), p. 960-981.</ref> This stands in contrast to the problem of fractionally coloring the edges of a graph, which can be solved in polynomial time. This is a straightforward consequence of Edmonds' matching polytope theorem.<ref>Bruce Hajek and Galen Sasaki: "[http://dx.doi.org/10.1109/18.21215 Link scheduling in polynomial time]", IEEE Transactions on Information Theory 34:5(1988), p. 910-917.</ref><ref name=schrijver>{{cite book|last=Schrijver|first=Alexander|title=Combinatorial Optimization: Polyhedra and Efficiency|year=2003|publisher=Springer-Verlag|location=Berlin ; Heidelberg ; New York, N.Y.|isbn=3540443894|pages=474}}</ref> | |||
==Applications== | |||
Applications of fractional graph coloring include ''activity scheduling''. In this case, the graph ''G'' is a ''conflict graph'': an edge in ''G'' between the nodes ''u'' and ''v'' denotes that ''u'' and ''v'' cannot be active simultaneously. Put otherwise, the set of nodes that are active simultaneously must be an independent set in graph ''G''. | |||
An optimal fractional graph coloring in ''G'' then provides a shortest possible schedule, such that each node is active for (at least) 1 time unit in total, and at any point in time the set of active nodes is an independent set. If we have a solution ''x'' to the above linear program, we simply traverse all independent sets ''I'' in an arbitrary order. For each ''I'', we let the nodes in ''I'' be active for <math>x_I</math> time units; meanwhile, each node not in ''I'' is inactive. | |||
In more concrete terms, each node of ''G'' might represent a ''radio transmission'' in a wireless communication network; the edges of ''G'' represent ''interference'' between radio transmissions. Each radio transmission needs to be active for 1 time unit in total; an optimal fractional graph coloring provides a minimum-length schedule (or, equivalently, a maximum-bandwidth schedule) that is conflict-free. | |||
===Comparison with traditional graph coloring=== | |||
If one further required that each node must be active ''continuously'' for 1 time unit (without switching it off and on every now and then), then traditional graph [[vertex coloring]] would provide an optimal schedule: first the nodes of color 1 are active for 1 time unit, then the nodes of color 2 are active for 1 time unit, and so on. Again, at any point in time, the set of active nodes is an independent set. | |||
In general, fractional graph coloring provides a shorter schedule than non-fractional graph coloring; there is an [[integrality gap]]. It may be possible to find a shorter schedule, at the cost of switching devices (such as radio transmitters) on and off more than once. | |||
==Notes== | |||
<references /> | |||
==References== | |||
*{{citation | |||
| last1 = Scheinerman | first1 = Edward R. | author1-link = Ed Scheinerman | |||
| last2 = Ullman | first2 = Daniel H. | |||
| isbn = 0-471-17864-0 | |||
| location = New York | |||
| publisher = Wiley-Interscience | |||
| title = Fractional graph theory | |||
| year = 1997}}. | |||
*{{citation | |||
| last1 = Godsil | first1 = Chris | author1-link = Chris Godsil | |||
| last2 = Royle | first2 = Gordon | author2-link = Gordon Royle | |||
| isbn = 0-387-95241-1 | |||
| location = New York | |||
| publisher = Springer-Verlag | |||
| title = Algebraic Graph Theory | |||
| year = 2001}}. | |||
[[Category:Graph coloring]] | |||
Revision as of 18:35, 24 May 2013
Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common.
Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems.
Definitions
Below: A 5:2 coloring of the same graph.
A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors. The b-fold chromatic number χb(G) is the least a such that an a:b-coloring exists.
The fractional chromatic number χf(G) is defined to be
Note that the limit exists because χb(G) is subadditive, meaning χa+b(G) ≤ χa(G) + χb(G).
The fractional chromatic number can equivalently be defined in probabilistic terms. χf(G) is the smallest k for which there exists a probability distribution over the independent sets of G such that for each vertex v, given an independent set S drawn from the distribution,
- .
Some properties of χf(G):
and
- .
Here n(G) is the order of G, α(G) is the independence number, ω(G) is the clique number, and χ(G) is the chromatic number.
Linear Programming (LP) Formulation
The fractional chromatic number χf(G) of a graph G can be obtained as a solution to a linear program. Let (G) be the set of all independent sets of G, and let (G,x) be the set of all those independent sets which include vertex x. For each independent set I, define a nonnegative real variable xI. Then χf(G) is the minimum value of
- ,
- subject to for each vertex .
The dual of this linear program computes the "fractional clique number", a relaxation to the rationals of the integer concept of clique number. That is, a weighting of the vertices of G such that the total weight assigned to any independent set is at most 1. The strong duality theorem of linear programming guarantees that the optimal solutions to both linear programs have the same value. Note however that each linear program may have size that is exponential in the number of vertices of G, and that computing the fractional chromatic number of a graph is NP-hard.[1] This stands in contrast to the problem of fractionally coloring the edges of a graph, which can be solved in polynomial time. This is a straightforward consequence of Edmonds' matching polytope theorem.[2][3]
Applications
Applications of fractional graph coloring include activity scheduling. In this case, the graph G is a conflict graph: an edge in G between the nodes u and v denotes that u and v cannot be active simultaneously. Put otherwise, the set of nodes that are active simultaneously must be an independent set in graph G.
An optimal fractional graph coloring in G then provides a shortest possible schedule, such that each node is active for (at least) 1 time unit in total, and at any point in time the set of active nodes is an independent set. If we have a solution x to the above linear program, we simply traverse all independent sets I in an arbitrary order. For each I, we let the nodes in I be active for time units; meanwhile, each node not in I is inactive.
In more concrete terms, each node of G might represent a radio transmission in a wireless communication network; the edges of G represent interference between radio transmissions. Each radio transmission needs to be active for 1 time unit in total; an optimal fractional graph coloring provides a minimum-length schedule (or, equivalently, a maximum-bandwidth schedule) that is conflict-free.
Comparison with traditional graph coloring
If one further required that each node must be active continuously for 1 time unit (without switching it off and on every now and then), then traditional graph vertex coloring would provide an optimal schedule: first the nodes of color 1 are active for 1 time unit, then the nodes of color 2 are active for 1 time unit, and so on. Again, at any point in time, the set of active nodes is an independent set.
In general, fractional graph coloring provides a shorter schedule than non-fractional graph coloring; there is an integrality gap. It may be possible to find a shorter schedule, at the cost of switching devices (such as radio transmitters) on and off more than once.
Notes
- ↑ Carsten Lund and Mihalis Yannakakis: "On the hardness of approximating minimization problems", J. ACM 41:5(1994), p. 960-981.
- ↑ Bruce Hajek and Galen Sasaki: "Link scheduling in polynomial time", IEEE Transactions on Information Theory 34:5(1988), p. 910-917.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
References
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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.