On the Approximability of NP-complete Optimization Problems

18 Chapter 3. Reductions
Proposition 3.5 A reduction which is a P-reduction is at the same time an
A-reduction (with the same functions t1 and t2 and another function c). If the
c-function in an A- or P-reduction is of the form c(ε) =k ·ε,for some constant
k,the reduction is both an A-reduction and a P-reduction. If the c-function in
an A- or P-reduction is c(ε) =ε,the reduction is a strict reduction with respect
to the relative error E r .
The proof is trivial.
3.4.3 R-reduction
The R-reduction is a seldom used randomized variant of the P-reduction. In
this thesis it will only be used in Section 4.11 to show that the maximum
independent set is hard to approximate. See Section 4.5 for a definition of
Rptas.
Definition 3.5 [12] Given two NPO problems F and G, anR-reduction (an
Rptas preserving reduction) from F to G is a tuple f =(t1,t2,c,p) such that
i) t1, t2 are random polynomial time computable functions, p is a real constant
between zero and one, and c : QF → QG is a computable function.
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)), E r G (t1(x),y) ≤ c(ε) ⇒ E r F (x, t2(x, y)) ≤ ε
with probability at least p.
If F R-reduces to G we write F ≤ p
G ≤ p
R
F we write F ≡p
R G.
R
G and if at the same time F ≤p
R G and
Thus every P-reduction is also an R-reduction with the probability constant
p =1.
Proposition 3.6 Given two NPO problems F and G,if F ≤ p
R G and G has
an Rptas,then so does F .
The proof is similar to that of Proposition 3.4.
3.4.4 L-reduction
The L-reduction was introduced in 1988 by Papadimitriou and Yannakakis
in order to prove completeness in the classes Max NP and Max SNP, see
Section 4.10.
Definition 3.6 [88] Given two NPO problems F and G, anL-reduction (linear
reduction) from F to G is a tuple f =(t1,t2,α,β) such that
i) t1, t2 are polynomial time computable functions and α and β are positive
constants.
ii) t1 : IF →IG and ∀x ∈IF and ∀y ∈ SG(t1(x)), t2(x, y) ∈ SF (x).