On the Approximability of NP-complete Optimization Problems

8 Chapter 2. Definitions
Definition 2.1 An NPO problem (over an alphabet Σ) is a four-tuple
F =(IF ,SF ,mF , opt F )where
•IF ⊆ Σ ∗ is the space of input instances. It is recognizable in polynomial
time.
• SF (x) ⊆ Σ ∗ is the space of feasible solutions on input x ∈IF .Theonly
requirement on SF is that there exist a polynomial q and a polynomial
time computable predicate π such that for all x in IF , SF can be expressed
as SF (x) ={y : |y| ≤q(|x|) ∧ π(x, y)} where q and π only depend on F .
• mF : IF × Σ ∗ → N, theobjective function, is a polynomial time computable
function. mF (x, y) is defined only when y ∈ SF (x).
• opt F ∈{max, min} tells if F is a maximization or a minimization problem.
Solving an optimization problem F given the input x ∈IF means finding a
y ∈ SF (x) such that mF (x, y) is optimum, that is as big as possible if opt F =
max and as small as possible if opt F =min. Letopt F (x) denote this optimal
value of mF .
A maximum solution of a maximization problem is an optimal solution.
A maximal solution is a locally optimal solution, thus an approximation to
the optimal solution. Often it is easy to find a maximal solution but hard to
find a maximum solution. Analogously, a minimum solution of a minimization
problem is an optimal solution and a minimal solution is a locally optimal
solution.
Example 2.1 Maximum three dimensional matching (Max 3DM) isanexample
of a problem in NPO. It is a tuple (I,S,m,opt) where the space of input
instances I contains all sets of triples T ⊆ X × Y × Z from pairwise disjunctive
sets X, Y and Z. The set of feasible solutions S(T ) is the set of matchings of
T ,whereamatchingM ∈ S(T ) is a subset of T where no two triples agree
in any coordinate. The objective function m(T,M) is the cardinality |M| of
the solution M and opt =max. 3DM is a well-known NP-complete problem.
Thus Max 3DM can probably not be solved exactly in polynomial time.
Definition 2.2 PO is the class of NPO problems which can be solved in
polynomial time.
Maximum matching in a bipartite graph (Max 2DM) isanexampleofa
problem in PO.
2.3 How approximation is measured
Approximating an optimization problem F given the input x ∈IF means finding
any y ′ ∈ SF (x). How good the approximation is depends on the relation
between mF (x, y ′ )andopt F (x). There are a number of ways to measure the approximability.
Different problems have different approximabilities which must