We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize the weight of satisfied soft constraints while satisfying {\em all} the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework.

Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a {\em robust} satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least $(1 - g(\epsilon))$ fraction of constraints given a $(1-\epsilon)$-satisfiable instance, where $g(\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$, and $g(0) = 0$. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on $(1-\epsilon)$-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm).

We also study hardness results for a certain CSP over larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal --- they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.