A Characterization of Approximability for Biased CSPs
Abstract
A $\mu$biased MaxCSP 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 biasapproximation 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 NPhard to approximate to a factor of $\Omega(r^3\mu^{r1}\log(1/\mu))$ for every $r \geq 2$ and $\mu < 2^{r}$. We also give a $O(\mu^{r1}\log(1/\mu))$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.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.04617
 Bibcode:
 2022arXiv220104617G
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 68 Pages