On the Approximability of NP-complete Optimization Problems

24 Chapter 3. Reductions
The two last inequalities imply that
opt F (x) − mF (x, t2(t1(x),y)) ≤ c3 (opt G(t1(x)) − mG(t1(x),y))
and thus the second requirement of the L-reduction is satisfied with β = c3.
✷
3.6 Parameter dependent reductions
The relative error preserving reductions, like the L-reduction and the P-reduction,
work well when reducing to problems with bounded approximation, but
there are several problems which cannot be approximated within a constant,
unless P = NP. When analyzing approximation algorithms for such problems
one usually specifies the approximability using a one variable function where the
parameter concerns the size of the input instance. For example the maximum
independent set problem can be approximated within O n/(log n) 2 where the
parameter n is the number of nodes in the input graph. Which quantity of
the input instance to choose as the parameter depends on the problem and the
algorithm. In the example above the most relevant quantity turned out to be
the number of nodes.
When reducing between two such problems, say from F to G, therelative
error preserving reductions are not perfect. The trouble is that these reductions
may transform an input instance of F to a much larger input instance of G.
One purpose of a reduction is to be able to use an approximation algorithm for
G to construct an equally good (within a constant) approximation algorithm
for F . Because of the size amplification the constructed algorithm will not be
as good as the original algorithm.
As an example (from Section 6.2.2) we take Max Ind Set as G, someother
problem on graphs as F and transform the input graph to a graph where each
node corresponds to a pair of nodes in the input graph. Thus, if the input
graph of F contains n nodes, the input graph of G will contain O(n2 )nodes,
so the above mentioned approximation algorithm of Max Ind Set will only
give us an algorithm approximating F within
2 n
O
(log n2 ) 2
2 n
= O
(log n) 2
.
In order to tell how the approximability, when given as a function, will be
changed by a reduction, we have to specify how the size of the input instance
will be amplified. The situation is complicated by not knowing in what parameter
the input instances will be measured. For graphs both the number of
nodes and the number of edges may be possible choices.
For every reduction mentioned in earlier sections in this chapter we may
add a statement with size amplification f(n) in order to specify this. If the
size amplification is O(n), i.e. if the size of the constructed structure is a
constant times the size of the original structure, we say that the reduction is