Suppose that we are given a sequences of $2N$ 'entities' (not numbers) with some total ordering defined among these entities. An example could be
$$\langle a\rangle=1<4<8<2<3<\cdots<2N \tag{GIVEN TO US}$$
However, we are not happy with this ordering because we want some other order relations to hold true namely:
1.) The set of entities $1$ to $N$ should satisfy a GIVEN ORDERING. An example could be:
$\langle c1\rangle=1<3<2<8<\cdots<N$.
2.) The set of entities $N+1$ to $2N$ should satisfy the ordering $\langle c2 \rangle $ obtained by adding $N$ to each entry in $\langle c1\rangle$.
$$\langle c2\rangle = \langle c1\rangle +N=N+1<N+3<N+2<N+8<\cdots<2N.$$
Statement of the problem: Construct a new sequence $<b>$, a total ordering on entities $1$ to $2N$ that must satisfy both $<c1>$ and $<c2>$ and be as "close" as possible to $<a>$. Note that the original sequence $<a>$ as given does not satisfy $<c1>$ and $<c2>$. Essentially approximate $<a>$ so that both of the given order constraints hold.
In theory, one could enumerate all possible sequences and pick the best one. I am instead looking for an efficient (could be sub optimal but maybe 'not so bad') way to do this problem. Any ideas on where to even start? Is there a branch of mathematics or theoretical computer science that deals with a problem of this sort and which will help me. I understand I haven't defined the notion of 'closeness'. One metric could be the number of disagreements which we need to minimize.
It seems to me that the problem is posed in a somewhat too general fashion. It is not difficult to prove that there are ways to define "satisfactory ordering" and "closeness" for which the problem of deciding if there exists a satisfactory ordering that is $\epsilon-$close to the original sequence is NP-complete. In fact, you can make even the approximation problem (if $\bar\epsilon$ is the actual minimum, you are allowed to err if posed the question with $\epsilon$ in the range $\bar\epsilon/K, \bar\epsilon K$ for any $K>0$) NP-complete!
Maybe you could give us a bit more structure to work with?