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I'm new to geometric algorithms and computational geometry, so please forgive me if this is an inappropriate question for this forum.

Let $X$ denote the disjoint union of $n$ one-point sets. Let $f:X\rightarrow \mathbb{R}^2$ be a function subject to some set of constraints $C=\{c_1,\cdots,c_m\}$, where $c_i=(x_{i_1},x_{i_2},l_i)$ is a triplet which represents the constraint that $d(f(x_{i_1}),f(x_{i_2}))=l_i$, where $d$ denotes the Euclidean metric on $\mathbb{R}^2$. Suppose $C$ is fixed. I would like to design an algorithm which takes as input an additional constraint, $c'$. The algorithm should efficiently search some representative subset of the space of functions which satisfy the constraints in $C$ to look for one which also satisfies the constraint $c'$. I would be happy with an approximate configuration (i.e. all constraints satisfied within some $\epsilon$).

Has anyone heard of a similar algorithm they can refer me to? I've been browsing the CGAL libraries for something that may be of use to me, but I haven't found anything yet.