On the Approximability of NP-complete Optimization Problems

26 Chapter 3. Reductions
Definition 3.13 [90] Given two NPO problems F and G (either two maximization
problems or two minimization problems), a non-constructive measure
preserving reduction from F to G is a polynomial time computable function
g : IF →IG such that ∀x ∈IF , opt F (x) =opt G(g(x))
We can see that the non-constructive measure preserving reduction is a
non-constructive ratio preserving reduction with c1 = c2 =1.
Suppose that A ′ is a non-constructive approximation algorithm for G, that
is A ′ is a polynomial time computable function A ′ : IG → N such that ∀x ′ ∈
IG,A ′ (x ′ ) ≤ opt G (x ′ )ifopt G = max and A ′ (x ′ ) ≥ opt G (x ′ )ifopt G =min.
If g is a non-constructive ratio preserving reduction from F to G, then a nonconstructive
approximation algorithm A for F is an algorithm which for an
input x ∈IF computes A ′ (g(x))
c1
Proposition 3.19 (Paz and Moran [90])
Given a non-constructive ratio preserving reduction g from F to G and a nonconstructive
approximation algorithm A ′ for G. If A ′ approximates G within
a constant,then A (defined as above) approximates F within a constant. If
c1 = c2 =1(i.e. g is a non-constructive measure preserving reduction) and
A ′ approximates G within any constant,then A approximates F within any
constant.
Thus the non-constructive ratio preserving reduction is a non-constructive
variant of the A-reduction, and the non-constructive measure preserving reduction
is a non-constructive variant of the P-reduction.
3.8 Structure preservingreduction
The last reduction in this chapter, the structure preserving reduction, differs
from all the other reductions defined here because it preserves much more
information about the instances but it is not immediately applicable to approximation.
The reduction was invented by Ausiello, D’Atri and Protasi in
1977.
Definition 3.14 [4] Given an NPO problem F and an input instance x ∈IF ,
the structure of x is the list structF (x) =〈a0,...,ac〉 where
c = |opt F (x) − worstF (x)|, ai = |{y ∈ SF (x) :i = |mF (x, y) − worstF (x)|}|.
Example 3.2 [4] Let F be the minimum set cover problem (given a number
of sets, find the minimum number of sets covering all elements) and x1 =
{S1,S2,S3,S4,S5} where S1 = {1, 2, 3, 4}, S2 = {1, 2, 5}, S3 = {3, 4, 6}, S4 =
{5, 6, 7}, S5 = {1, 2, 3, 4, 5, 6, 7}. Now we have worstF (x1) =5, opt F (x1) =
1, structF (x1) =〈1, 5, 9, 5, 1〉.
Example 3.3 [4] Let F be the minimum set cover problem and define x2 as
the family of sets {S1,S2,S3,S4} where the sets are defined as in the previous
example. In this case we have worstF (x2) =4, opt F (x2) =2, structF (x2) =
〈1, 3, 1〉.
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