  A $\mu$-biased Max-CSP instance with predicate $\psi:\{0,1\}^r \to \{0,1\}$
is an instance of Constraint Satisfaction Problem (CSP) where the objective is
to find a labeling of relative weight at most $\mu$ which satisfies the maximum
fraction of constraints. Biased CSPs are versatile and express several well
studied problems such as Densest-$k$-Sub(Hyper)graph and SmallSetExpansion.
  In this work, we explore the role played by the bias parameter $\mu$ on the
approximability of biased CSPs. We show that the approximability of such CSPs
can be characterized (up to loss of factors of arity $r$) using the
bias-approximation curve of Densest-$k$-SubHypergraph (DkSH). In particular,
this gives a tight characterization of predicates which admit approximation
guarantees that are independent of the bias parameter $\mu$.
  Motivated by the above, we give new approximation and hardness results for
DkSH. In particular, assuming the Small Set Expansion Hypothesis (SSEH), we
show that DkSH with arity $r$ and $k = \mu n$ is NP-hard to approximate to a
factor of $\Omega(r^3\mu^{r-1}\log(1/\mu))$ for every $r \geq 2$ and $\mu <
2^{-r}$. We also give a $O(\mu^{r-1}\log(1/\mu))$-approximation algorithm for
the same setting. Our upper and lower bounds are tight up to constant factors,
when the arity $r$ is a constant, and in particular, imply the first tight
approximation bounds for the Densest-$k$-Subgraph problem in the linear bias
regime. Furthermore, using the above characterization, our results also imply
matching algorithms and hardness for every biased CSP of constant arity.
