# APX

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In complexity theory the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer.

Computational complexity theory focuses on classifying computational problems according to their inherent difficulty, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

In computational complexity theory, NP is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time.

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete variables is known as a discrete optimization. In a discrete optimization problem, we are looking for an object such as an integer, permutation or graph from a countable set. Problems with continuous variables include constrained problems and multimodal problems.

## Contents

An approximation algorithm is called an ${\displaystyle f(n)}$-approximation algorithm for input size ${\displaystyle n}$ if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of ${\displaystyle f(n)}$ times worse than the optimal solution. Here, ${\displaystyle f(n)}$ is called the approximation ratio. Problems in APX are those with algorithms for which the approximation ratio ${\displaystyle f(n)}$ is a constant ${\displaystyle c}$. The approximation ratio is conventionally stated greater than 1. In the case of minimization problems, ${\displaystyle f(n)}$ is the found solution's score divided by the optimum solution's score, while for maximization problems the reverse is the case. For maximization problems, where an inferior solution has a smaller score, ${\displaystyle f(n)}$ is sometimes stated as less than 1; in such cases, the reciprocal of ${\displaystyle f(n)}$ is the ratio of the score of the found solution to the score of the optimum solution.

A problem is said to have a polynomial-time approximation scheme (PTAS) if for every multiplicative factor of the optimum worse than 1 there is a polynomial-time algorithm to solve the problem to within that factor. Unless P = NP there exist problems that are in APX but without a PTAS, so the class of problems with a PTAS is strictly contained in APX. One such problem is the bin packing problem.

In computer science, a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems.

In the bin packing problem, items of different volumes must be packed into a finite number of bins or containers each of volume V in a way that minimizes the number of bins used. In computational complexity theory, it is a combinatorial NP-hard problem. The decision problem is NP-complete.

## APX-hardness and APX-completeness

A problem is said to be APX-hard if there is a PTAS reduction from every problem in APX to that problem, and to be APX-complete if the problem is APX-hard and also in APX. As a consequence of P ≠ NP ⇒ PTAS ≠ APX, if P ≠ NP is assumed, no APX-hard problem has a PTAS. In practice, reducing one problem to another to demonstrate APX-completeness is often done using other reduction schemes, such as L-reductions, which imply PTAS reductions.

In computational complexity theory, a PTAS reduction is an approximation-preserving reduction that is often used to perform reductions between solutions to optimization problems. It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization problems such as APX. Notationally, if there is a PTAS reduction from a problem A to a problem B, we write .

In computer science, particularly the study of approximation algorithms, an L-reduction is a transformation of optimization problems which linearly preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximability of optimization problems play a similar role to that of polynomial reductions in the studies of computational complexity of decision problems.

### Examples

One of the simplest APX-complete problems is MAX-3SAT-3, a variation of the boolean satisfiability problem. In this problem, we have a boolean formula in conjunctive normal form where each variable appears at most 3 times, and we wish to know the maximum number of clauses that can be simultaneously satisfied by a single assignment of true/false values to the variables.

MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as:

In computer science, the Boolean satisfiability problem is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make = TRUE. In contrast, "a AND NOT a" is unsatisfiable.

In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. As a canonical normal form, it is useful in automated theorem proving and circuit theory.

Other APX-complete problems include:

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

In computational complexity theory and combinatorics, the token reconfiguration problem is an optimization problem on a graph with both an initial and desired state for tokens.

### PTAS

PTAS (polynomial time approximation scheme) consists of problems that can be approximated to within any constant factor besides 1 in time that is polynomial to the input size, but the polynomial depends on such factor. This class is a subset of APX.

### APX-intermediate

Unless P = NP, there exist problems in APX that are neither in PTAS nor APX-complete. Such problems can be thought of as having a hardness between PTAS problems and APX-complete problems, and may be called APX-intermediate. The bin packing problem is thought to be APX-intermediate. Despite not having a known PTAS, the bin packing problem has several "asymptotic PTAS" algorithms, which behave like a PTAS when the optimum solution is large, so intuitively it may be easier than problems that are APX-hard.

One other example of a potentially APX-intermediate problem is min edge coloring.

### f(n)-APX

One can also define a family of complexity classes ${\displaystyle f(n)}$-APX, where ${\displaystyle f(n)}$-APX contains problems with a polynomial time approximation algorithm with a ${\displaystyle O(f(n))}$ approximation ratio. One can analogously define ${\displaystyle f(n)}$-APX-complete classes; some such classes contain well-known optimization problems. Log-APX-completeness and poly-APX-completeness are defined in terms of AP-reductions rather than PTAS-reductions; this is because PTAS-reductions are not strong enough to preserve membership in Log-APX and Poly-APX, even though they suffice for APX.

Log-APX-complete, consisting of the hardest problems that can be approximated efficiently to within a factor logarithmic in the input size, includes min dominating set when degree is unbounded.

Poly-APX-complete, consisting of the hardest problems that can be approximated efficiently to within a factor polynomial in the input size, includes max independent set in the general case.

There also exist problems that are exp-APX-complete, where the approximation ratio is exponential in the input size. This may occur when the approximation is dependent on the value of numbers within the problem instance; these numbers may be expressed in space logarithmic in their value, hence the exponential factor.

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