This technical report is written as a support material for the reader in [1], detailing the transformations to simplify an initial program to compute a tight bound for a C-optimal assignment into a linear program (LP). In this technical report we show how the initial program to compute a tight bound for a C-optimal assignment, can be transformed into a linear program. Our departure point is the following program. Find R, xC and x∗ that minimize R(x C) R(x∗) subject to xC being a C-optimal for R We start by analyzing what exactly means saying that xC is C-optimal. The condition can be expressed as: for each x inside region C of xC we have that R(x) ≥ R(x). However, instead of considering all the assignments for which x is guaranteed to be optimal, we consider only the subset of assignments such that the set of variables that deviate with respect to x take the same value than in the optimal assignment. If we restrict to this subset of assignments, then each neighborhood covers a 2|C | assignments, one for each subset of variables in the neighborhood. Let 2 α stand for the set of all subsets of the neighborhood C. Then for each A ∈ 2Cα we can define an assignment xk such that for every variable xi in a relation completely covered by A we have that xk i = x ∗ i , and