Como dijiste, no hay que tomar una decisión, por lo que se necesitan nuevas clases de complejidad y nuevos tipos de reducciones para llegar a una definición adecuada de dureza NP para problemas de optimización .
Una forma de hacerlo es tener dos nuevas clases NPO y PO que contengan problemas de optimización y, por supuesto, imitan las clases NP y P para problemas de decisión. También se necesitan nuevas reducciones. Entonces podemos recrear una versión de dureza NP para problemas de optimización en la línea que fue exitosa para problemas de decisión. Pero primero tenemos que acordar qué es un problema de optimización .
Definición: Sea O = ( X , L , f , o p t )O=(X,L,f,opt) un problema de optimización . XX es el conjunto de entradas o instancias adecuadas codificadas como cadenas. LL es una función que asigna cada instancia x ∈ Xx∈X a un conjunto de cadenas, las soluciones factibles de la instancia xx . Es un conjunto porque hay muchas soluciones para un problema de optimización. Por lo tanto, tenemos una función objetivo ff que nos dice para cada par (x , y ) (x,y) y ∈ L ( x )y∈L(x) de instancia y solución sucostoovalor. o p topt nos dice si estamos maximizando o minimizando.
Esto nos permite definir qué es una solución óptima : Sea y o p t ∈ L ( x )yopt∈L(x) la solución óptima de una instancia x ∈ Xx∈X de un problema de optimización O = ( X , L , f , o p t )O=(X,L,f,opt) con f ( x , y o p t ) = o p t { f ( x , y′)∣y′∈L(x)}.
f(x,yopt)=opt{f(x,y′)∣y′∈L(x)}.
The optimal solution is often denoted by
y∗y∗.
Now we can define the class NPO: Let NPONPO be the set of all optimization-problems O=(X,L,f,opt)O=(X,L,f,opt) with:
- X∈PX∈P
- There is a polynomial pp with |y|≤p(|x|)|y|≤p(|x|) for all instances x∈Xx∈X and all feasible solutions y∈L(x)y∈L(x). Furthermore there is an deterministic algorithm that decides in polynomial time whether y∈L(x)y∈L(x).
- ff can be evaluated in polynomial time.
The intuition behind it is:
- We can verify efficiently if xx is actually a valid instance of our optimization problem.
- The size of the feasible solutions is bounded polynomially in the size of the inputs, And we can verify efficiently if y∈L(x)y∈L(x) is a fesible solution of the instance xx.
- The value of a solution y∈L(x)y∈L(x) can be determined efficiently.
This mirrors how NPNP is defined, now for PO: Let POPO be the set of all problems from NPONPO that can be solved by an deterministic algorithm in polynomial time.
Now we are able to define what we want to call an approximation-algorithm: An approximation-algorithm of an optimization-problem O=(X,L,f,opt)O=(X,L,f,opt) is an algorithm that computes a feasible solution y∈L(x)y∈L(x) for an instance x∈Xx∈X.
Note: That we don’t ask for an optimal solution we only what to have a feasible one.
Now we have two types of errors: The absolute error of a feasible solution y∈L(x)y∈L(x) of an instance x∈Xx∈X of the optimization-problem O=(X,L,f,opt)O=(X,L,f,opt) is |f(x,y)−f(x,y∗)||f(x,y)−f(x,y∗)|.
We call the absolute error of an approximation-algorithm AA for the optimization-problem OO bounded by kk if the algorithm AA computes for every instance x∈Xx∈X a feasible solution with an absolute error bounded by kk.
Example: According to the Theorem of Vizing the chromatic index of a graph (the number of colours in the edge coloring with the fewest number of colors used) is either ΔΔ or Δ+1Δ+1, where ΔΔ is the maximal node degree. From the proof of the theorem an approximation-algorithm can be devised that computes an edge coloring with Δ+1Δ+1 colours. Accordingly we have an approximation-algorithm for the Minimum−EdgeColoringMinimum−EdgeColoring-Problem where the absolute error is bounded by 11.
This example is an exception, small absolute errors are rare, thus we define the relative error ϵA(x)ϵA(x) of the approximation-algorithm AA on instance xx of the optimization-problem O=(X,L,f,opt)O=(X,L,f,opt) with f(x,y)>0f(x,y)>0 for all x∈Xx∈X and y∈L(x)y∈L(x) to be
ϵA(x):={0f(x,A(x))=f(x,y∗)|f(x,A(x))−f(x,y∗)|max{f(x,A(x)),f(x,y∗)}f(x,A(x))≠f(x,y∗)
ϵA(x):={0|f(x,A(x))−f(x,y∗)|max{f(x,A(x)),f(x,y∗)}f(x,A(x))=f(x,y∗)f(x,A(x))≠f(x,y∗)
where A(x)=y∈L(x)A(x)=y∈L(x) is the feasible solution computed by the approximation-algorithm AA.
We can now define approximation-algorithm AA for the optimization-problem O=(X,L,f,opt)O=(X,L,f,opt) to be a δδ-approximation-algorithm for OO if the relative error ϵA(x)ϵA(x) is bounded by δ≥0δ≥0 for every instance x∈Xx∈X, thus
ϵA(x)≤δ∀x∈X.
ϵA(x)≤δ∀x∈X.
The choice of max{f(x,A(x)),f(x,y∗)}max{f(x,A(x)),f(x,y∗)} in the denominator of the definition of the relative error was selected to make the definition symmetric for maximizing and minimizing. The value of the relative error ϵA(x)∈[0,1]ϵA(x)∈[0,1]. In case of a maximizing problem the value of the solution is never less than (1−ϵA(x))⋅f(x,y∗)(1−ϵA(x))⋅f(x,y∗) and never larger than 1/(1−ϵA(x))⋅f(x,y∗)1/(1−ϵA(x))⋅f(x,y∗) for a minimizing problem.
Now we can call an optimization-problem δδ-approximable if there is a δδ-approximation-algorithm AA for OO that runs in polynomial time.
We do not want to look at the error for every instance xx, we look only at the worst-case. Thus we define ϵA(n)ϵA(n), the maximal relativ error of the approximation-algorithm AA for the optimization-problem OO to be
ϵA(n)=sup{ϵA(x)∣|x|≤n}.
ϵA(n)=sup{ϵA(x)∣|x|≤n}.
Where |x||x| should be the size of the instance.
Example: A maximal matching in a graph can be transformed in to a minimal node cover CC by adding all incident nodes from the matching to the vertex cover. Thus 1/2⋅|C|1/2⋅|C| edges are covered. As each vertex cover including the optimal one must have one of the nodes of each covered edge, otherwise it could be improved, we have 1/2⋅|C|⋅f(x,y∗)1/2⋅|C|⋅f(x,y∗). It follows that |C|−f(x,y∗)|C|≤12
|C|−f(x,y∗)|C|≤12
Thus the greedy algorithm for a maximal matching is a
1/21/2-approximatio-algorithm for
Minimal−VertexCoverMinimal−VertexCover. Hence
Minimal−VertexCoverMinimal−VertexCover is
1/21/2-approximable.
Unfortunately the relative error is not always the best notion of quality for an approximation as the following example demonstrates:
Example: A simple greedy-algorithm can approximate Minimum−SetCoverMinimum−SetCover. An analysis shows that |C||C∗|≤Hn≤1+ln(n)
|C||C∗|≤Hn≤1+ln(n)
and thus
Minimum−SetCoverMinimum−SetCover would be
ln(n)1+ln(n)ln(n)1+ln(n)-approximable.
If the relative error is close to 11 the following definition is advantageous.
Let O=(X,L,f,opt)O=(X,L,f,opt) be an optimization-problem with f(x,y)>0f(x,y)>0 for all x∈Xx∈X and y∈L(x)y∈L(x) and AA an approximation-algorithm for OO. The approximation-ratio rA(x)rA(x) of feasible solution A(x)=y∈L(x)A(x)=y∈L(x) of the instance x∈Xx∈X is
rA(x)={1f(x,A(x))=f(x,y∗)max{f(x,A(x))f(x,y∗),f(x,y∗)f(x,A(x))}f(x,A(x))≠f(x,y∗)
rA(x)={1max{f(x,A(x))f(x,y∗),f(x,y∗)f(x,A(x))}f(x,A(x))=f(x,y∗)f(x,A(x))≠f(x,y∗)
As before we call an approximation-algorithm AA an rr-approximation-algorithm for the optimization-problem OO if the approximation-ratio rA(x)rA(x) is bounded by r≥1r≥1 for every input x∈Xx∈X.
rA(x)≤r
rA(x)≤r
And yet again if we have an
rr-approximation-algorithm
AA for the optimization-problem
OO then
OO is called
rr-approximable. Again we only care about to the worst-case and define the
maximal approximation-ratio rA(n)rA(n) to be
rA(n)=sup{rA(x)∣|x|≤n}.rA(n)=sup{rA(x)∣|x|≤n}.
Accordingly the approximation-ratio is larger than
11 for suboptimal solutions. Thus better solutions have smaller ratios. For
Minimum−SetCoverMinimum−SetCover we can now write that it is
(1+ln(n))(1+ln(n))-approximable. And in case of
Minimum−VertexCoverMinimum−VertexCover we know from the previous example that it is
22-approximable. Between relative error and approximation-ratio we have simple relations:
rA(x)=11−ϵA(x)ϵA(x)=1−1rA(x).rA(x)=11−ϵA(x)ϵA(x)=1−1rA(x).
For small deviations from the optimum ϵ<1/2ϵ<1/2 and r<2r<2 the relative error is advantageous over the approximation-ratio, that shows its strengths for large deviations ϵ≥1/2ϵ≥1/2 and r≥2r≥2.
The two versions of αα-approximable don’t overlap as one version has always α≤1α≤1 and the other α≥1α≥1. The case α=1α=1 is not problematic as this is only reached by algorithms that produce an exact solution and consequentially need not be treated as approximation-algorithms.
Another class appears often APX. It is define as the set of all optimization-problems OO from NPONPO that haven an rr-approximation-algorithm with r≥1r≥1 that runs in polynomial time.
We are almost through. We would like to copy the successful ideas of reductions and completness from complexity theory. The observation is that many NP-hard decision variants of optimization-problems are reducible to each other while their optimization variants have different properties regarding their approximability. This is due to the polynomialtime-Karp-reduction used in NP-completness reductions, which does not preserve the objective function. And even if the objective functions is preserved the polynomialtime-Karp-reduction may change the quality of the solution.
What we need is a stronger version of the reduction, which not only maps instances from optimization-problem O1O1 to instances of O2O2, but also good solutions from O2O2 back to good solutions from O1O1.
Hence we define the approximation-preserving-reduction for two optimization-problems O1=(X1,L1,f1,opt1)O1=(X1,L1,f1,opt1) and O2=(X2,L2,f2,opt2)O2=(X2,L2,f2,opt2) from NPONPO. We call O1O1 APAP-reducible to O2O2, written as O1≤APO2O1≤APO2, if there are two functions gg and hh and a constant cc with:
- g(x1,r)∈X2g(x1,r)∈X2 for all x1∈X1x1∈X1 and rational r>1r>1
- L2(g(x,r1))≠∅L2(g(x,r1))≠∅ if L1(x1)≠∅L1(x1)≠∅ for all x1∈X1x1∈X1 and rational r>1r>1
- h(x1,y2,r)∈L1(x1)h(x1,y2,r)∈L1(x1) for all x1∈X1x1∈X1 and rational r>1r>1 and for all y2∈L2(g(x1,r))y2∈L2(g(x1,r))
- For fixed rr both functions gg and hh can be computed by two algorithms in polynomial time in the length of their inputs.
- We have f2(g(x1,r),y2)≤r⇒f1(x1,h(x1,y2,r))≤1+c⋅(r−1)
f2(g(x1,r),y2)≤r⇒f1(x1,h(x1,y2,r))≤1+c⋅(r−1)
for all x1∈X1x1∈X1 and rational r>1r>1 and for all y2∈L2(g(x1,r))y2∈L2(g(x1,r))
In this definition gg and hh depend on the quality of the solution rr. Thus for different qualities the functions can differ. This generality is not always needed and we just work with g(x1)g(x1) and h(x1,y2)h(x1,y2).
Now that we have a notion of a reduction for optimization-problems we finally can transfer many things we know from complexity theory. For example if we know that O2∈APXO2∈APX and we show that O1≤APO2O1≤APO2 it follows that O1∈APXO1∈APX too.
Finally we can define what we mean by CC-hard and CC-complete for optimization-problems:
Let OO be an optimization-problem from NPONPO and CC a class of optimization-problems from NPONPO then OO is called CC-hard with respect to ≤AP≤AP if for all O′∈CO′∈C O′≤APOO′≤APO holds.
Thus once more we have a notion of a hardest problem in the class. Not surprising a CC-hard problem is called CC-complete with respect to ≤AP≤AP if it is an element of CC.
Thus we can now talk about NPONPO-completness and APXAPX-completness etc. And of course we are now asked to exhibit a first NPONPO-complete problem that takes over the role of SATSAT. It comes almost naturally, that Weighted−Satisfiability can be shown to be NPO-complete. With the help of the PCP-Theorem one can even show that Maximum−3SAT is APX-complete.