A characterization of approximability for biased CSPs

@article{Ghoshal2022ACO,
  title={A characterization of approximability for biased CSPs},
  author={Suprovat Ghoshal and Euiwoong Lee},
  journal={Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing},
  year={2022}
}
  • Suprovat GhoshalEuiwoong Lee
  • Published 12 January 2022
  • Computer Science
  • Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
A µ-biased Max-CSP instance with predicate ψ:{0,1}r → {0,1} is an instance of Constraint Satisfaction Problem (CSP) where the objective is to find a labeling of relative weight at most µ 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 µ on the approximability of biased CSPs. We show that the… 
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References

SHOWING 1-10 OF 43 REFERENCES

Approximation of non-boolean 2CSP

We develop a polynomial time Ω (1/R log R) approximate algorithm for Max 2CSP-R, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges

Approximating CSPs with global cardinality constraints using SDP hierarchies

This work forms a general approach towards approximating CSPs with global cardinality constraints using SDP hierarchies and presents a generic conversion from integrality gap instances for the Lasserre hierarchy to a dictatorship test whose soundness is at most integralities gap.

Optimal algorithms and inapproximability results for every CSP?

A generic conversion from SDP integrality gaps to UGC hardness results for every CSP is shown, which achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph

The results indicate that approximating Densest k-subgraph within a polynomial factor might be a harder problem than Unique Games or Small Set Expansion, since these problems were recently shown to be solvable using neΩ(1) rounds of the Lasserre hierarchy, where e is the completeness parameter in Unique Games and Small Set expansion.

Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection

This work conjecture that Max Bisection is approximable within αGW -- e, i.e., the bisection constraint (essentially) does not make Max Cut harder and obtains an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat.

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

The hardness for Max-2-Sat applies to monotone Max- 2-Sat instances, meaning that it is proved that tight inapproximability for the Max-k-Vertex-Cover problem is obtained.

Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique

  • Subhash Khot
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
It is shown that graph min-bisection, densest subgraph and bipartite clique have no PTAS, and a way of certifying that a given polynomial belongs to a given subspace of polynomials is given.

Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph

It is shown, assuming the exponential time hypothesis (ETH), that there is no polynomial-time algorithm that approximates Densest k-Subgraph to within n1/(loglogn)c factor of the optimum, where c > 0 is a universal constant independent of n.

Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

Though it is unable to prove the majority is stablest conjecture, some partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture.

Better Balance by Being Biased

This work conjecture that Max Bisection is approximable within αGW − ε, that is, that the bisection constraint (essentially) does not make Max Cut harder and obtains an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat.