On the Approximability of NP-complete Optimization Problems

20 Chapter 3. Reductions
iii) ∀x ∈IF and ∀y ∈ SG(t1(x)),
E r G (t1(x),y) ≤ c(ε, x) ⇒ E r F (x, t2(x, y)) ≤ ε.
iv) The time complexity of c is p1(1/ε, |x|) wherep1 is a polynomial.
v) ∀x ∈IF , c(ε, x) ≤ 1/p2(1/ε, |x|) wherep2 is a polynomial.
If F F-reduces to G we write F ≤ p
G ≤ p
F
F we write F ≡p
F G.
F
G and if at the same time F ≤p
F
G and
Proposition 3.9 (Crescenzi and Panconesi [24])
Given two NPO problems F and G,if F ≤ p
F G and G has a Fptas,then so
does F .
Proposition 3.10 (Crescenzi and Panconesi [24])
An F-reduction is not definable as a P-reduction with some additional constraint.
The reason for this is that the function c in the definition of the P-reduction
must be independent of |x| but in the definition of the F-reduction c can be
polynomially dependent on |x|. A surprising fact shown by Crescenzi and Panconesi
is that every problem which can be approximated within some constant
can be F-reduced to a problem which has a Ptas [24]. Thus F ≤ p
F G and G
has a Ptas ⇒ F has a Ptas. See Section 4.6.
3.4.6 Transitivity
All relative error preserving reductions defined in this section are transitive.
Proposition 3.11 ([12, 24, 83, 88])
Given three NPO problems F , G and H.
If F ≤ p
A G and G ≤p
A H then F ≤p
If F ≤ p
P G and G ≤p
P H then F ≤p
If F ≤ p
R G and G ≤p
R H then F ≤p
If F ≤ p
G and G ≤p H then F ≤p
L
If F ≤ p
F
L
G and G ≤p
F
A H.
P H.
R H.
L H.
H then F ≤p
F H.
3.5 Ratio preservingreduction
In this chapter we have seen a lot of definitions of reduction which preserve
the relative error. Now we will switch to another type of reduction, introduced
by Simon in 1990, which preserves the performance ratio. Recall from Definition
2.7 that the performance ratio of a solution y to an instance x of a problem
F is
optF (x)/mF (x, y) if optF =max,
RF (x, y) =
mF (x, y)/optF (x) if optF =min.
where x ∈IF and y ∈ SF (x).