On the Approximability of NP-complete Optimization Problems

22 Chapter 3. Reductions
Proof The functions t1 and t2 satisfy the requirements i) and ii).
It only remains to show that given x ∈IF ,y ∈ SG(t1(x)) and ε>0thereisa
function c(ε) such that E r G (t1(x),y) ≤ ε ⇒ E r F (x, t2(x, y)) ≤ c(ε).
We distinguish between the cases of minimization and maximization, and
start with supposing that F and G are minimization problems. We know that
the performance ratio expansion of f is bounded by a constant e = c2c3 since
f is a ratio preserving reduction.
RF (x, t2(x, y)) ≤ e · RG(t1(x),y)
E r F (x, t2(x, y)) + 1 ≤ e · (E r G (t1(x),y)+1)
E r F (x, t2(x, y)) ≤ e ·E r G(t1(x),y)+e − 1
E r G (t1(x),y) ≤ ε ⇒E r F (x, t2(x, y)) ≤ eε + e − 1.
Thus c(ε) =eε + e − 1.
Now suppose that F and G are maximization problems.
RF (x, t2(x, y)) ≤ e · RG(t1(x),y)
1
1 −E r F (x, t2(x, y))
≤ e ·
1
1 −E r G (t1(x),y)
1 −E r G (t1(x),y) ≤ e · (1 −E r F (x, t2(x, y)))
E r F (x, t2(x, y)) ≤ 1 − 1 1
+
e e Er G(t1(x),y)
E r G (t1(x),y) ≤ ε ⇒E r F (x, t2(x, y)) ≤ 1+(ε − 1)/e.
Thus c(ε) =1+(ε − 1)/e < 1whenε0 some reduction in the
sequence has an expansion less than ε.
Proposition 3.16 If there is a ratio preserving reduction scheme from F to
G and G has a Ptas,then F has a Ptas.
The proof is trivial.
Proposition 3.17 Given a ratio preserving reduction f = (t1,t2,t3,c2,c3)
from F to G with c2c3 = 1. Then t1 is an L-reduction with α = c2 and
β = c3 if opt F = opt G =maxand α = c3 and β = c2 if opt F = opt G =min.
Thus we in both cases have a strict reduction with respect to the relative error.