Decision problems vs “real” problems that aren’t yes-or-no

I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. Also for each desired approximation factor, there are many answers that are correct and many that are wrong, and this changes with the desired approximation factor!

So how can one say that this problem is NP-hard?

(inspired by the second bullet in How hard is counting the number of simple paths between two nodes in a directed graph?)

Answer

As you said, there is no decision to make, so new complexity classes and new types of reductions are needed to arrive at a suitable definition of NP-hardness for optimization-problems.

One way of doing this is to have two new classes NPO and PO that contain optimizations problems and they mimic of course the classes NP and P for decision problems. New reductions are needed as well. Then we can recreate a version of NP-hardness for optimization problems along the lines that was successful for decision problems. But first we have to agree what an optimization-problem is.

Definition: Let O=(X,L,f,opt) be an optimization-problem. X is the set of inputs or instances suitable encoded as strings. L is a function that maps each instance xX onto a set of strings, the feasible solutions of instance x. It is a set because there are many solutions to an optimization-problem. Thus we haven an objective function f that tells us for every pair (x,y) yL(x) of instance and solution its cost or value. opt tells us whether we are maximizing or minimizing.

This allows us to define what an optimal solution is: Let yoptL(x) be the optimal solution of an instance xX of an optimization-problem O=(X,L,f,opt) with f(x,yopt)=opt{f(x,y)yL(x)}. The optimal solution is often denoted by y.

Now we can define the class NPO: Let NPO be the set of all optimization-problems O=(X,L,f,opt) with:

  1. XP
  2. There is a polynomial p with |y|p(|x|) for all instances xX and all feasible solutions yL(x). Furthermore there is an deterministic algorithm that decides in polynomial time whether yL(x).
  3. f can be evaluated in polynomial time.

The intuition behind it is:

  1. We can verify efficiently if x is actually a valid instance of our optimization problem.
  2. The size of the feasible solutions is bounded polynomially in the size of the inputs, And we can verify efficiently if yL(x) is a fesible solution of the instance x.
  3. The value of a solution yL(x) can be determined efficiently.

This mirrors how NP is defined, now for PO: Let PO be the set of all problems from NPO 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) is an algorithm that computes a feasible solution yL(x) for an instance xX.

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 yL(x) of an instance xX of the optimization-problem O=(X,L,f,opt) is |f(x,y)f(x,y)|.

We call the absolute error of an approximation-algorithm A for the optimization-problem O bounded by k if the algorithm A computes for every instance xX a feasible solution with an absolute error bounded by k.

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 Δ or Δ+1, where Δ is the maximal node degree. From the proof of the theorem an approximation-algorithm can be devised that computes an edge coloring with Δ+1 colours. Accordingly we have an approximation-algorithm for the MinimumEdgeColoring-Problem where the absolute error is bounded by 1.

This example is an exception, small absolute errors are rare, thus we define the relative error ϵA(x) of the approximation-algorithm A on instance x of the optimization-problem O=(X,L,f,opt) with f(x,y)>0 for all xX and yL(x) to be

ϵA(x):={0f(x,A(x))=f(x,y)|f(x,A(x))f(x,y)|max

where A(x)=y\in L(x) is the feasible solution computed by the approximation-algorithm A.

We can now define approximation-algorithm A for the optimization-problem O=(X,L,f,opt) to be a \delta-approximation-algorithm for O if the relative error \epsilon_A(x) is bounded by \delta\ge 0 for every instance x\in X, thus
\epsilon_A(x)\le \delta\qquad \forall x\in X.

The choice of \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 \epsilon_A(x)\in[0,1]. In case of a maximizing problem the value of the solution is never less than (1-\epsilon_A(x))\cdot f(x,y^*) and never larger than 1/(1-\epsilon_A(x))\cdot f(x,y^*) for a minimizing problem.

Now we can call an optimization-problem \delta-approximable if there is a \delta-approximation-algorithm A for O that runs in polynomial time.

We do not want to look at the error for every instance x, we look only at the worst-case. Thus we define \epsilon_A(n), the maximal relativ error of the approximation-algorithm A for the optimization-problem O to be
\epsilon_A(n)=\sup\{\epsilon_A(x)\mid |x|\le n\}.

Where |x| should be the size of the instance.

Example: A maximal matching in a graph can be transformed in to a minimal node cover C by adding all incident nodes from the matching to the vertex cover. Thus 1/2\cdot |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\cdot |C|\cdot f(x,y^*). It follows that \frac{|C|-f(x,y^*)}{|C|}\le\frac{1}{2}
Thus the greedy algorithm for a maximal matching is a 1/2-approximatio-algorithm for \mathsf{Minimal-VertexCover}. Hence \mathsf{Minimal-VertexCover} is 1/2-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 \mathsf{Minimum-SetCover}. An analysis shows that \frac{|C|}{|C^*|}\le H_n\le 1+\ln(n) and thus \mathsf{Minimum-SetCover} would be \frac{\ln(n)}{1+\ln(n)}-approximable.

If the relative error is close to 1 the following definition is advantageous.

Let O=(X,L,f,opt) be an optimization-problem with f(x, y)>0 for all x\in X and y\in L(x) and A an approximation-algorithm for O. The approximation-ratio r_A(x) of feasible solution A(x)=y\in L(x) of the instance x\in X is
r_A(x)=\begin{cases}1&f(x,A(x))=f(x,y^*)\\\max\left\{
\frac{f(x,A(x))}{f(x, y^*)},\frac{f(x, y^*)}{f(x, A(x))}\right\}&f(x,A(x))\ne f(x,y^*)\end{cases}

As before we call an approximation-algorithm A an r-approximation-algorithm for the optimization-problem O if the approximation-ratio r_A(x) is bounded by r\ge1 for every input x\in X.
r_A(x)\le r
And yet again if we have an r-approximation-algorithm A for the optimization-problem O then O is called r-approximable. Again we only care about to the worst-case and define the maximal approximation-ratio r_A(n) to be
r_A(n)=\sup\{r_A(x)\mid |x|\le n\}.
Accordingly the approximation-ratio is larger than 1 for suboptimal solutions. Thus better solutions have smaller ratios. For \mathsf{Minimum-SetCover} we can now write that it is (1+\ln(n))-approximable. And in case of \mathsf{Minimum-VertexCover} we know from the previous example that it is 2-approximable. Between relative error and approximation-ratio we have simple relations:
r_A(x)=\frac{1}{1-\epsilon_A(x)}\qquad \epsilon_A(x)=1-\frac{1}{r_A(x)}.

For small deviations from the optimum \epsilon<1/2 and r<2 the relative error is advantageous over the approximation-ratio, that shows its strengths for large deviations \epsilon\ge 1/2 and r\ge 2.

The two versions of \alpha-approximable don’t overlap as one version has always \alpha\le 1 and the other \alpha\ge 1. The case \alpha=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 O from NPO that haven an r-approximation-algorithm with r\ge1 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 O_1 to instances of O_2, but also good solutions from O_2 back to good solutions from O_1.

Hence we define the approximation-preserving-reduction for two optimization-problems O_1=(X_1,L_1,f_1,opt_1) and O_2=(X_2,L_2,f_2,opt_2) from NPO. We call O_1 AP-reducible to O_2, written as O_1\le_{AP} O_2, if there are two functions g and h and a constant c with:

  1. g(x_1, r)\in X_2 for all x_1\in X_1 and rational r>1
  2. L_2(g(x, r_1))\ne\emptyset if L_1(x_1)\ne\emptyset for all x_1\in X_1 and rational r>1
  3. h(x_1, y_2, r)\in L_1(x_1) for all x_1\in X_1 and rational r>1 and for all y_2\in L_2(g(x_1,r))
  4. For fixed r both functions g and h can be computed by two algorithms in polynomial time in the length of their inputs.
  5. We have f_2(g(x_1,r),y_2)\le r \Rightarrow f_1(x_1,h(x_1,y_2,r))\le 1+c\cdot(r-1) for all x_1\in X_1 and rational r>1 and for all y_2\in L_2(g(x_1,r))

In this definition g and h depend on the quality of the solution r. Thus for different qualities the functions can differ. This generality is not always needed and we just work with g(x_1) and h(x_1, y_2).

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 O_2\in APX and we show that O_1\le_{AP} O_2 it follows that O_1\in APX too.

Finally we can define what we mean by \mathcal{C}-hard and \mathcal{C}-complete for optimization-problems:

Let O be an optimization-problem from NPO and \mathcal{C} a class of optimization-problems from NPO then O is called \mathcal{C}-hard with respect to \le_{AP} if for all O'\in\mathcal{C} O'\le_{AP} O holds.

Thus once more we have a notion of a hardest problem in the class. Not surprising a \mathcal{C}-hard problem is called \mathcal{C}-complete with respect to \le_{AP} if it is an element of \mathcal{C}.

Thus we can now talk about NPO-completness and APX-completness etc. And of course we are now asked to exhibit a first NPO-complete problem that takes over the role of \mathsf{SAT}. It comes almost naturally, that \mathsf{Weighted-Satisfiability} can be shown to be NPO-complete. With the help of the PCP-Theorem one can even show that \mathsf{Maximum-3SAT} is APX-complete.

Attribution
Source : Link , Question Author : Ran G. , Answer Author : uli

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