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- W4221138548 abstract "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 $Omega(r^3mu^{r-1}log(1/mu))$ 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." @default.
- W4221138548 created "2022-04-03" @default.
- W4221138548 creator A5075423314 @default.
- W4221138548 creator A5082860877 @default.
- W4221138548 date "2022-01-12" @default.
- W4221138548 modified "2023-10-17" @default.
- W4221138548 title "A Characterization of Approximability for Biased CSPs" @default.
- W4221138548 doi "https://doi.org/10.48550/arxiv.2201.04617" @default.
- W4221138548 hasPublicationYear "2022" @default.
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