  Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
  To be more precise, we introduce the following notion of a predicate $P$
being \emph{useful} for a (real-valued) objective $Q$: given an almost
satisfiable Max-$P$ instance, there is an algorithm that beats a random
assignment on the corresponding Max-$Q$ instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate $P$ is exactly the same as saying that $P$ is useful for
$P$ itself.
  We say that $P$ is useless if it is not useful for any $Q$. This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max-$P$ it is hard to find an assignment such that the
induced distribution on $k$-bit strings defined by the instance is not
essentially uniform.
  Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
  Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs.
