On the Approximability of NP-complete Optimization Problems

42 Chapter 4. Approximability classes
4.10.3 Closures of Max NP and Max SNP
Definition 4.17 [88] Max NP is the closure under P-reductions of Syntactic
Max NP, thatis
{F ∈ NPO : ∃G ∈ Syntactic Max NP ∃ P-reduction f : F ≤ p
P G}
Max NP was originally defined by Papadimitriou and Yannakakis [88], but
their definition has been interpreted both as Definition 4.17 (which was their
intention [86]) and as Definition 4.15 [84].
Definition 4.18 [88] Max SNP is the closure under P-reductions of Syntactic
Max SNP, thatis
{F ∈ NPO : ∃G ∈ Syntactic Max SNP ∃ P-reduction f : F ≤ p
P G}
Not only maximization problems, but also some minimization problems can
be placed in Max NP and Max SNP with these definitions.
Example 4.5 Show that Min Node Cover −B (minimum node cover in a
graph with bounded degree) is in Max SNP.
Let us (as in Section 2.6) use that given a graph without free nodes (nodes
which do not have any neighbours), a set of nodes is a node cover if and only if
its complement is an independent set. We observe that free nodes are trivially
included in any maximal independent set. We would like to find a P-reduction
from Min Node Cover −B to Max Ind Set −B, which we in Example 4.4
showed is included in Syntactic Max SNP.
Often it is more convenient to show that a reduction is an L-reduction.
Recall from Proposition 3.8 that an L-reduction satisfies the conditions of a
P-reduction. Let F be Min Node Cover −B, G be Max Ind Set −B
and f : IF →IG be the identity function which takes a graph and returns it
immediately. This function is certainly polynomial time computable.
1. Show that there is a constant α such that for every instance x ∈IF
opt G(f(x)) ≤ α · opt F (x),
i.e. that the size of the maximum independent set is bounded above by
a constant times the size of the minimum node cover of a graph.
Since every node in a maximum independent set (apart from the free
nodes) must be connected to at least one node in the node cover and
since the degree of the graph is bounded by B, the size of the maximum
independent set cannot be larger that B times the size of the minimum
node cover. Thus α = B.
2. Show that there is a constant β such that for every instance x ∈IF
and for every solution of f(x) withmeasurec2 wecaninpolynomial
time find a solution of x with measure c1 such that |opt F (x) − c1| ≤
β |opt G(f(x)) − c2|.