By Rolf Wanka

Viele kombinatorische Optimierungsprobleme haben sich als schwierig exakt lösbar herausgestellt, weshalb guy sich mit Näherungslösungen zufrieden geben muss. In diesem Buch werden Approximationsalgorithmen vorgestellt, die für eine Reihe populärer Optimierungsprobleme beweisbar gute Lösungen in vertretbarer Zeit berechnen. Im ersten Teil werden die grundlegenden Begriffe vorgestellt, mit Beispielalgorithmen ausgeführt und jeweils die Grenzen aufgezeigt. Im zweiten Teil werden allgemeine Techniken eingeführt und anhand instruktiver Beispiele mit Leben erfüllt.

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**Example text**

By the way it is well-known (cf. [2]) that the following converse of the previous fact is true: If ϕ(Z) is antimonotone in Z in all finite and infinite structures, then ϕ(Z) is logically equivalent to a formula ψ(Z) negative in Z. Nevertheless, we do not know whether, in Theorem 2, we can replace “ϕ(Z) a Πt -formula negative in Z” by “ϕ(Z) a Πt -formula antimonotone in Z;” in fact, for t ≥ 3, it is not known if every Πt -formula antimonotone in Z is equivalent to 34 Y. Chen and J. Flum a formula ψ(Z) in Πt negative in Z.

Therefore we simply need to compare c with γx + k to check whether x, k ∈ LQ . The total time complexity of the above algorithm is O(kα · (|f (x)| + h(kα)) + p1 (|x|) + p2 (|f (x)|)), where p1 (·) is the time taken by algorithm f to transform an instance of Q to an instance of M AX 3-S AT, and p2 (·) is the time taken by g to output its answer. Thus the algorithm that we outlined is indeed an FPT algorithm for LQ . Note that the proof of Proposition 1 also shows that every minimization problem in MAX SNP has a M AX 3-S AT-lower bound.

Definition 3 (Dense Set). Let Q = {I , S, V, opt} be an NPO problem. A set of instances I ⊆ I is said to be dense with respect to a set of conditions C if there exists a constant c ∈ N such that for all closed intervals [a, b] ⊆ R+ of length |b − a| ≥ c, there exists an instance I ∈ I with |I| ∈ [a, b] such that I satisfies all the conditions in C. Further, if such an I can be found in polynomial time (polynomial in b), then I is said to be dense poly-time uniform with respect to C. For example, for the M AXIMUM ACYCLIC S UBGRAPH problem, the set of all oriented digraphs is dense (poly-time uniform) with respect to the condition: opt(G) = |E(G)|.