Analysis of Complex Networks

12.6 A Basis System of Problems

First let us introduce a few necessary notations.

An inequality of the form x′ ≤ x″ between two 4-dimensional vectors x′, x″ ∈ X will mean the validity of four inequalities x′_i ≤ x″_i, i =1,...,4. We say that vectors x, x ∈X are incomparable if neither x′ ≤ x″ nor x′ ≤ x″ holds.

We denote D⁻(x′) {x ∈ X| x ≥ X}, D⁺(x′) {x ∈ X| x ≥ x′}, D⁻ (X′) ∪_x∈X, D⁻(x), D⁺(X′) ∪_x∈X′ D⁺(x); X(I) (T, Δ_V, Δ_V,C, Δ_C,V).

Letter I will denote the set of all inputs of the CLC problem.

In the previous section eight CLC(x) problems for eight different values of the constraining vector x were analyzed. Yet we recall that in Section 12.4 an infinite family CLC(X) of CLC(x) problems over all possible vectors x ∈ X was defined. Does this mean that, once we have committed ourselves to obtaining the whole picture of complexity over all problems in CLC(X), we have to perform a similar complexity analysis for every CLC(x) problem from CLC(X)?

If this were so, it would be very unfortunate, ...