#### Theory Seminar

# A Characterization of Approximability for Biased CSPs

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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.