On the Approximability of NP-complete Optimization Problems

16 Chapter 3. Reductions
Since a clique of size k in a graph corresponds to an independent set of size
k in the complement graph and vice versa, f =(t1,t2) defined as follows is a
reencoding of F to G.
Let t1 transform an input graph to its complement and let t2 be the identity
function. These functions are obviously invertible and both the functions and
their inverses are polynomial time computable.
3.3 Strict reduction
The following reduction, introduced by Orponen and Mannila, is applicable to
every measure of approximation. It completely preserves the approximability
with respect to the measure of approximation.
Definition 3.2 [83] Given two NPO problems F and G, astrict reduction
from F to G with respect to a measure E of approximation quality is a pair
f =(t1,t2) such that
i) t1, t2 are polynomial time computable functions.
ii) t1 : IF →IG and ∀x ∈IF and ∀y ∈ SG(t1(x)), t2(x, y) ∈ SF (x).
iii) ∀x ∈IF and ∀y ∈ SG(t1(x)), EF (x, t2(x, y)) ≤EG(t1(x),y).
If there is a strict reduction from F to G we write F ≤p s G and if at the same
time F ≤p s G and G ≤p s F we write F ≡p s G.
Proposition 3.2 (Orponen and Mannila [83])
Given three NPO problems F , G and H. IfF ≤ p s G and G ≤p s H then F ≤p s H.
3.4 Relative error preservingreductions
Reductions which preserve the relative error in some way are the most commonly
used. We shall here present five types of reductions which preserve
the relative error to different extent. The definitions of the A-reduction, Preduction,
R-reduction and F-reduction are very similar. The reductions are
approximation preserving, Ptas-preserving, Rptas-preserving and Fptas-preserving
respectively, see Chapter 4. The fifth reduction, the L-reduction, is
often most practical to use to show that a problem is as hard to approximate
as another. If there is an L-reduction between two problems there is also a Preduction,
and if there is a P-reduction between two problems there is also an
A-reduction. Unfortunately there is no such relation between the F-reduction
and for example the P-reduction as we will see.
3.4.1 A-reduction
Definition 3.3 Given two NPO problems F and G, anA-reduction (approximation
preserving reduction) from F to G is a triple f =(t1,t2,c) such that