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In [[computer science]] and [[operations research]], '''approximation algorithms''' are [[algorithm]]s used to find approximate solutions to [[optimization problem]]s. Approximation algorithms are often associated with [[NP-hard]] problems; since it is unlikely that there can ever be efficient [[polynomial time|polynomial-time]] exact algorithms solving NP-hard problems, one settles for polynomial-time sub-optimal solutions. Unlike [[heuristic (computer science)|heuristics]], which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run-time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size.
A typical example for an approximation algorithm is the one for [[Vertex cover problem|vertex cover]] in [[Graph (mathematics)|graph]]s: find an uncovered edge and add ''both'' endpoints to the vertex cover, until none remain.  It is clear that the resulting cover is at most twice as large as the optimal one. This is a [[constant factor approximation algorithm]] with a factor of 2.


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NP-hard problems vary greatly in their approximability; some, such as the [[bin packing problem]], can be approximated within any factor greater than 1 (such a family of approximation algorithms is often called a [[polynomial time approximation scheme]] or ''PTAS''). Others are impossible to approximate within any constant, or even polynomial factor unless [[P = NP]], such as the [[maximum clique problem]].
 
NP-hard problems can often be expressed as [[integer programs]] (IP) and solved exactly in [[exponential time]]. Many approximation algorithms emerge from the [[linear programming relaxation]] of the integer program.
 
Not all approximation algorithms are suitable for all practical applications. They often use IP/LP/[[semidefinite programming|Semidefinite]] solvers, complex data structures or sophisticated algorithmic techniques which lead to difficult implementation problems. Also, some approximation algorithms have impractical running times even though they are polynomial time, for example O(''n''<sup>2000</sup>). Yet the study of even very expensive algorithms is not a completely theoretical pursuit as they can yield valuable insights. A classic example is the initial PTAS for [[Euclidean traveling salesman problem|Euclidean TSP]] due to [[Sanjeev Arora]] which had prohibitive running time, yet within a year, Arora refined the ideas into a linear time algorithm. Such algorithms are also worthwhile in some applications where the running times and cost can be justified e.g. [[computational biology]], [[financial engineering]], [[transportation planning]], and [[inventory management]]. In such scenarios, they must compete with the corresponding direct IP formulations.
 
Another limitation of the approach is that it applies only to optimization problems and not to "pure" [[decision problem]]s like [[boolean satisfiability problem|satisfiability]], although it is often possible to conceive optimization versions of such problems, such as the [[maximum satisfiability problem]] (Max SAT).
 
Inapproximability has been a fruitful area of research in computational complexity theory since the 1990 result of Feige, Goldwasser, Lovasz, Safra and Szegedy on the inapproximability of [[Independent set (graph theory)|Independent Set]]. After Arora et al. proved the [[PCP theorem]] a year later, it has now been shown that Johnson's 1974 approximation algorithms for Max SAT, Set Cover, Independent Set and Coloring all achieve the optimal approximation ratio, assuming P != NP.
 
== Performance guarantees ==
For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, a '''''ρ''-approximation algorithm''' ''A'' is defined to be an algorithm for which it been proven that the value/cost, ''f''(''x''), of the approximate solution ''A''(''x'') to an instance ''x'' will not be more (or less, depending on the situation) than a factor ''ρ'' times the value, OPT, of an optimum solution.
 
:<math>\begin{cases}\mathrm{OPT} \leq f(x) \leq \rho \mathrm{OPT},\qquad\mbox{if } \rho > 1; \\ \rho \mathrm{OPT} \leq f(x) \leq \mathrm{OPT},\qquad\mbox{if } \rho < 1.\end{cases}</math>
 
The factor ''ρ'' is called the ''relative performance guarantee''. An approximation algorithm has an ''absolute performance guarantee'' or ''bounded error'' ''c'', if it has been proven for every instance ''x'' that
 
:<math> (\mathrm{OPT} - c) \leq f(x) \leq (\mathrm{OPT} + c).</math>
 
Similarly, the ''performance guarantee'', ''R''(''x,y''), of a solution ''y'' to an instance ''x'' is defined as
 
:R(x,y) = <math> \max \left ( \frac{OPT}{f(y)}, \frac{f(y)}{OPT} \right ),</math>
 
where ''f''(''y'') is the value/cost of the solution ''y'' for the instance ''x''. Clearly, the performance guarantee is greater than or equal to 1 and equal to 1 if and only if ''y'' is an optimal solution. If an algorithm ''A'' guarantees to return solutions with a performance guarantee of at most ''r''(''n''), then ''A'' is said to be an ''r''(''n'')-approximation algorithm and has an ''approximation ratio'' of ''r''(''n''). Likewise, a problem with an ''r''(''n'')-approximation algorithm is said to be r''(''n'')''-''approximable'' or have an approximation ratio of ''r''(''n'').<ref name=ausiello99complexity>{{cite book|title=Complexity and Approximation: Combinatorial Optimization Problems and their Approximability Properties|year=1999|author=G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi}}</ref><ref name="kann92onthe">{{cite book|title=On the Approximability of NP-complete Optimization Problems|author=Viggo Kann|year=1992|url=http://www.csc.kth.se/~viggo/papers/phdthesis.pdf}}</ref>
 
One may note that for minimization problems, the two different guarantees provide the same result and that for maximization problems, a relative performance guarantee of ρ is equivalent to a performance guarantee of <math>r = \rho^{-1}</math>. In the literature, both definitions are common but it is clear which definition is used since, for maximization problems, as ρ ≤ 1 while r ≥ 1.
 
The ''absolute performance guarantee'' <math>\Rho_A</math> of some approximation algorithm ''A'', where ''x'' refers to an instance of a problem, and where <math>R_A(x)</math> is the performance guarantee of ''A'' on ''x'' (i.e. ρ for problem instance ''x'') is:
 
:<math> \Rho_A = \inf \{ r \geq 1 \mid R_A(x) \leq r, \forall x \}.</math>
 
That is to say that <math>\Rho_A</math> is the largest bound on the approximation ratio, ''r'', that one sees over all possible instances of the problem. Likewise, the ''asymptotic performance ratio'' <math>R_A^\infty</math> is:
 
:<math> R_A^\infty = \inf \{ r \geq 1 \mid \exists n \in \mathbb{Z}^+, R_A(x) \leq r, \forall x, |x| \geq n\}. </math>
 
That is to say that it is the same as the ''absolute performance ratio'', with a lower bound ''n'' on the size of problem instances. These two types of ratios are used because there exist algorithms where the difference between these two is significant.
 
{| class="wikitable"
|+Performance guarantees
|-
!  !! ''r''-approx<ref name="ausiello99complexity"/><ref name="kann92onthe"/> !! ''ρ''-approx !! rel. error<ref name="kann92onthe"/> !! rel. error<ref name="ausiello99complexity"/> !! norm. rel. error<ref name="ausiello99complexity"/><ref name="kann92onthe"/> !! abs. error<ref name="ausiello99complexity"/><ref name="kann92onthe"/>
|-
! max: <math>f(x) \geq</math>
| <math>r^{-1} \mathrm{OPT}</math> || <math>\rho \mathrm{OPT}</math> || <math>(1-c)\mathrm{OPT}</math> || <math>(1-c)\mathrm{OPT}</math> || <math>(1-c)\mathrm{OPT} + c\mathrm{WORST}</math> || <math>\mathrm{OPT} - c</math>
|-
! min: <math>f(x) \leq</math>
| <math>r \mathrm{OPT}</math> || <math>\rho \mathrm{OPT}</math> || <math>(1+c)\mathrm{OPT}</math> || <math>(1-c)^{-1}\mathrm{OPT}</math> || <math>(1-c)^{-1} \mathrm{OPT} + c\mathrm{WORST}</math> || <math>\mathrm{OPT} + c</math>
|-
|}
 
==Algorithm design techniques==
By now there are several standard techniques that one tries to design an approximation algorithm. These include the following ones.
# [[Greedy algorithm]]
# [[Local search (optimization)|Local search]]
# Enumeration and [[dynamic programming]]
# Solving a [[convex programming]] relaxation to get a fractional solution. Then converting this fractional solution into a feasible solution by some appropriate rounding. The popular relaxations include the following.
## [[Linear programming]] relaxation
## [[Semidefinite programming]] relaxation
# Embedding the problem in some simple metric and then solving the problem on the metric. This is also known as metric embedding.
 
== Epsilon terms ==
In the literature, an approximation ratio for a maximization (minimization) problem of ''c'' - ϵ (min: ''c'' + ϵ) means that the algorithm has an approximation ratio of ''c'' ∓ ϵ  for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable [[MAX-3SAT]] instances due to [[Johan Håstad]].<ref name="hastad99someoptimal">{{cite journal|title=Some Optimal Inapproximability Results|journal=Journal of the ACM|year=1999|url=http://www.nada.kth.se/~johanh/optimalinap.ps|author=[[Johan Håstad]]}}</ref> As mentioned previously, when ''c'' = 1, the problem is said to have a [[polynomial-time approximation scheme]].
 
An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size ''n'' goes to infinity as ''n'' does. In this case, the approximation ratio is ''c'' ∓ ''k'' / OPT = ''c'' ∓ o(1) for some constants ''c'' and ''k''. Given arbitrary ϵ > 0, one can choose a large enough ''N'' such that the term ''k'' / OPT < ϵ for every ''n ≥ N''. For every fixed ϵ, instances of size ''n < N'' can be solved by brute force , thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of ''c'' ∓ ϵ for every ϵ > 0.
 
== See also ==
* [[Domination analysis]] considers guarantees in terms of the rank of the computed solution.
 
==Citations==
{{More footnotes|date=April 2009}}
{{reflist}}
 
==References==
* {{cite book
  | last = Vazirani
  | first = Vijay V.
  | authorlink = Vijay Vazirani
  | title = Approximation Algorithms
  | publisher = Springer
  | year = 2003
  | location = Berlin
  | isbn = 3-540-65367-8 }}
* [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 35: Approximation Algorithms, pp.&nbsp;1022&ndash;1056.
* [[Dorit H. Hochbaum]], ed. ''[[Approximation Algorithms for NP-Hard problems]]'', PWS Publishing Company, 1997. ISBN 0-534-94968-1. Chapter 9: Various Notions of Approximations: Good, Better, Best, and More
*{{Citation|last1=Williamson|first1=David P.|last2=Shmoys|first2=David B.|authorlink2=David Shmoys|date=April 26, 2011|title=The Design of Approximation Algorithms|location=|publisher=[[Cambridge University Press]]|isbn=978-0521195270}}
 
==External links==
*Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, [[Marek Karpinski]] and Gerhard Woeginger, [http://www.nada.kth.se/~viggo/wwwcompendium/ ''A compendium of NP optimization problems''].
 
{{optimization algorithms|combinatorial|state=expanded}}
 
[[Category:Computational complexity theory]]
[[Category:Approximation algorithms| ]]

Revision as of 20:27, 29 August 2013

In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial-time exact algorithms solving NP-hard problems, one settles for polynomial-time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run-time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size. A typical example for an approximation algorithm is the one for vertex cover in graphs: find an uncovered edge and add both endpoints to the vertex cover, until none remain. It is clear that the resulting cover is at most twice as large as the optimal one. This is a constant factor approximation algorithm with a factor of 2.

NP-hard problems vary greatly in their approximability; some, such as the bin packing problem, can be approximated within any factor greater than 1 (such a family of approximation algorithms is often called a polynomial time approximation scheme or PTAS). Others are impossible to approximate within any constant, or even polynomial factor unless P = NP, such as the maximum clique problem.

NP-hard problems can often be expressed as integer programs (IP) and solved exactly in exponential time. Many approximation algorithms emerge from the linear programming relaxation of the integer program.

Not all approximation algorithms are suitable for all practical applications. They often use IP/LP/Semidefinite solvers, complex data structures or sophisticated algorithmic techniques which lead to difficult implementation problems. Also, some approximation algorithms have impractical running times even though they are polynomial time, for example O(n2000). Yet the study of even very expensive algorithms is not a completely theoretical pursuit as they can yield valuable insights. A classic example is the initial PTAS for Euclidean TSP due to Sanjeev Arora which had prohibitive running time, yet within a year, Arora refined the ideas into a linear time algorithm. Such algorithms are also worthwhile in some applications where the running times and cost can be justified e.g. computational biology, financial engineering, transportation planning, and inventory management. In such scenarios, they must compete with the corresponding direct IP formulations.

Another limitation of the approach is that it applies only to optimization problems and not to "pure" decision problems like satisfiability, although it is often possible to conceive optimization versions of such problems, such as the maximum satisfiability problem (Max SAT).

Inapproximability has been a fruitful area of research in computational complexity theory since the 1990 result of Feige, Goldwasser, Lovasz, Safra and Szegedy on the inapproximability of Independent Set. After Arora et al. proved the PCP theorem a year later, it has now been shown that Johnson's 1974 approximation algorithms for Max SAT, Set Cover, Independent Set and Coloring all achieve the optimal approximation ratio, assuming P != NP.

Performance guarantees

For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, a ρ-approximation algorithm A is defined to be an algorithm for which it been proven that the value/cost, f(x), of the approximate solution A(x) to an instance x will not be more (or less, depending on the situation) than a factor ρ times the value, OPT, of an optimum solution.

{OPTf(x)ρOPT,if ρ>1;ρOPTf(x)OPT,if ρ<1.

The factor ρ is called the relative performance guarantee. An approximation algorithm has an absolute performance guarantee or bounded error c, if it has been proven for every instance x that

(OPTc)f(x)(OPT+c).

Similarly, the performance guarantee, R(x,y), of a solution y to an instance x is defined as

R(x,y) = max(OPTf(y),f(y)OPT),

where f(y) is the value/cost of the solution y for the instance x. Clearly, the performance guarantee is greater than or equal to 1 and equal to 1 if and only if y is an optimal solution. If an algorithm A guarantees to return solutions with a performance guarantee of at most r(n), then A is said to be an r(n)-approximation algorithm and has an approximation ratio of r(n). Likewise, a problem with an r(n)-approximation algorithm is said to be r(n)-approximable or have an approximation ratio of r(n).[1][2]

One may note that for minimization problems, the two different guarantees provide the same result and that for maximization problems, a relative performance guarantee of ρ is equivalent to a performance guarantee of r=ρ1. In the literature, both definitions are common but it is clear which definition is used since, for maximization problems, as ρ ≤ 1 while r ≥ 1.

The absolute performance guarantee PA of some approximation algorithm A, where x refers to an instance of a problem, and where RA(x) is the performance guarantee of A on x (i.e. ρ for problem instance x) is:

PA=inf{r1RA(x)r,x}.

That is to say that PA is the largest bound on the approximation ratio, r, that one sees over all possible instances of the problem. Likewise, the asymptotic performance ratio RA is:

RA=inf{r1n+,RA(x)r,x,|x|n}.

That is to say that it is the same as the absolute performance ratio, with a lower bound n on the size of problem instances. These two types of ratios are used because there exist algorithms where the difference between these two is significant.

Performance guarantees
r-approx[1][2] ρ-approx rel. error[2] rel. error[1] norm. rel. error[1][2] abs. error[1][2]
max: f(x) r1OPT ρOPT (1c)OPT (1c)OPT (1c)OPT+cWORST OPTc
min: f(x) rOPT ρOPT (1+c)OPT (1c)1OPT (1c)1OPT+cWORST OPT+c

Algorithm design techniques

By now there are several standard techniques that one tries to design an approximation algorithm. These include the following ones.

  1. Greedy algorithm
  2. Local search
  3. Enumeration and dynamic programming
  4. Solving a convex programming relaxation to get a fractional solution. Then converting this fractional solution into a feasible solution by some appropriate rounding. The popular relaxations include the following.
    1. Linear programming relaxation
    2. Semidefinite programming relaxation
  5. Embedding the problem in some simple metric and then solving the problem on the metric. This is also known as metric embedding.

Epsilon terms

In the literature, an approximation ratio for a maximization (minimization) problem of c - ϵ (min: c + ϵ) means that the algorithm has an approximation ratio of c ∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable MAX-3SAT instances due to Johan Håstad.[3] As mentioned previously, when c = 1, the problem is said to have a polynomial-time approximation scheme.

An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size n goes to infinity as n does. In this case, the approximation ratio is ck / OPT = c ∓ o(1) for some constants c and k. Given arbitrary ϵ > 0, one can choose a large enough N such that the term k / OPT < ϵ for every n ≥ N. For every fixed ϵ, instances of size n < N can be solved by brute force , thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of c ∓ ϵ for every ϵ > 0.

See also

Citations

Template:More footnotes 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 35: Approximation Algorithms, pp. 1022–1056.
  • Dorit H. Hochbaum, ed. Approximation Algorithms for NP-Hard problems, PWS Publishing Company, 1997. ISBN 0-534-94968-1. Chapter 9: Various Notions of Approximations: Good, Better, Best, and More
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External links

Template:Optimization algorithms

  1. 1.0 1.1 1.2 1.3 1.4 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.

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  2. 2.0 2.1 2.2 2.3 2.4 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.

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  3. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang