On the Approximability of NP-complete Optimization Problems

34 Chapter 4. Approximability classes
Apparently Ptas ⊆ Rptas. This class is used very little in the literature.
The most interesting result is by Berman and Schnitger [12] and says that if any
Max SNP-complete problem (see Section 4.10) is not in Rptas, then there is
a constant c>0 such that the maximum independent set problem cannot be
approximated within n c ,wheren is the number of nodes in the input graph,
see Section 4.11.
4.6 Apx
The next class is very important: the class of problems which have bounded
approximation, that is which can be approximated within a constant. Here are
some famous examples of such problems.
1. Maximum satisfiability (Max Sat): given a set of disjunctive clauses,
find the truth assignment that satisfies the largest number of clauses.
Max Sat can be approximated within the constant 4/3 [110].
2. Maximum three-set packing (Max 3SP): given a collection of sets of size
three, find the largest number of pairwise disjunctive sets. Max 3SP can
be approximated within the constant 3 [52].
3. Euclidean travelling salesperson problem (Min ETsp): given a set of
points in the plane, find the shortest tour which visits all the points. Min
ETsp can be approximated within 3/2[21].
Definition 4.13
Apx = {F ∈ NPO : ∃c ≥ 1 such that F can be approximated within c}
Equivalently we can define Apx as the union of the set of minimization problems
which can be approximated within a constant relative error and the set of
maximization problems which can be approximated within a constant relative
error less than 1.
Obviously Ptas ⊆ Apx.
Proposition 4.7 (Crescenzi and Panconesi [24])
The problem Max Weighted Sat with Bound is Apx-complete under Preductions.
Max Weighted Sat with Bound isthesameproblemasMax Weighted
Sat with Small Bound but with the more generous condition on B that
w(u) ≤ 2 B.
u∈U
A startling result by Crescenzi and Panconesi is that every problem in Apx
can be F-reduced to a problem which is Ptas-complete under F-reductions
[24], in other terms: Apx ⊆ Ptas under F-reductions. The reason for this is
that the F-reduction is not a P-reduction with some additional constraint, see
Proposition 3.10.