# If a CSP (over a finite domain) has only linear inequalities as constraints, is it solvable in linear time?

I have an optimization problem in fuzzy logic that I want to model and solve as a CSP. If I could use only linear inequalities in my encoding, is the resulting CSP solvable in linear time?

Problem statement: Given a set of formulas in CNF in Łukasiewicz logic, I want to find an assignment for the variables in some truth set T_k, for some k, such that the assignment maximizes the number of satisfied formulas. I wanted to model this problem into CSP or SAT, hence my question. The SAT problem for CNF Łukasiewicz logic formulas is solvable in linear time. So, maybe we can take advantage of that.