Let $N$ be a finite set. Let $\prec$ be a strict partial order over $N$. I am interested in designing a function $f : N \to \mathbb{R}$ such that:
- $\sum_{n_i \in N} f(n_i) = 1$
- $\forall i,j : n_i \prec n_j \implies f(n_i) < f(n_j)$
...and minimizing the maximum of pairwise differences $| f(n_i) - f(n_j) |$.
I am also interested in the case with an additional constraint:
- $\forall i,j : | f(n_i) - f(n_j) | > k$
In this second case, an important question is, given $(N, \prec)$, for which values of $k$ does the problem have a solution.
Consider the set $\{0,1\}$ with order $0\prec 1$. Then we can construct an $f$ with an arbitrarily small pairwise difference $2\epsilon$ by letting $$f(x) = \left\{ \begin{array}{lr} .5-\epsilon & x=0 \\ .5+\epsilon & x=1 \end{array} \right. $$
So there is no unique optimum.
For the second part, suppose we have a total order. Then we imagine placing $x_1,x_2,\dots$ at $0,k,2k,\dots$. This is clearly the solution which has the smallest total size.
The total size is $\sum_{i=0}^{|N|}ik$, so a solution will exist for $k$ where this sum is less than 1.