On the Approximability of NP-complete Optimization Problems

10 Chapter 2. Definitions
For most of the NP-complete optimization problems there is no hope of
finding a solution with constant absolute error. Therefore this measure is seldom
used. Still there are a few NPO problems which have approximation
algorithms with constant absolute error guarantee, see Section 8.3.
Orponen and Mannila have generalized these measures and defined a general
measure of approximation quality.
Definition 2.6 [83] A general measure of approximation quality of a feasible
solution of an NPO problem F is a function EF which, for each x ∈ IF ,
satisfies EF (x, y) ≥ 0 if and only if y ∈ SF (x) andEF (x, y) = 0 if and only if
mF (x, y) =opt F (x). We say that the measure is cost-respecting if
mF (x, y1) ≤ mF (x, y2) ⇒EF (x, y1) ≥EF (x, y2) if opt F =max,
mF (x, y1) ≤ mF (x, y2) ⇒EF (x, y1) ≤EF (x, y2) if opt F =min.
All the measures defined above are cost-respecting.
When analyzing an approximation algorithm the most common way to state
the approximability is to give the performance ratio.
Definition 2.7 The performance ratio of a feasible solution with respect to
the optimum of an NPO problem F is defined as
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).
The performance ratio and the relative error are obviously related in the
following way.
⎧
⎪⎨ 1
RF (x, y) = 1 −E
⎪⎩
r ∞
= (E
F (x, y)
i=0
r F (x, y)) i
if optF =max,
1+Er F (x, y) if optF =min.
Definition 2.8 We say that an optimization problem F can be approximated
within p for a constant p if there exists a polynomial time algorithm A such
that for all instances x ∈IF , A(x) ∈ SF (x) andRF (x, A(x)) ≤ p.
For a few problems it is more interesting to consider the performance ratio
just for input instances with large optimal value. This asymptotical behaviour
is covered by the following definition.
Definition 2.9 We say that an optimization problem F can be approximated
asymptotically within p for a constant p if there exists a polynomial time algorithm
A and a positive constant N such that for all instances x ∈IF with
opt F (x) ≥ N, A(x) ∈ SF (x) andRF (x, A(x)) ≤ p.
Definition 2.10 The best performance ratio R[F ] for an optimization problem
F is defined as
R[F ]=inf{p : F can be approximated within p}.