# Approximation resistance from pairwise independent subgroups

@article{Chan2013ApproximationRF, title={Approximation resistance from pairwise independent subgroups}, author={Siu On Chan}, journal={Electron. Colloquium Comput. Complex.}, year={2013}, volume={19}, pages={110} }

We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain, whenever k is larger than the domain size. This follows from our main result concerning predicates over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that is balanced pairwise independent. This gives an unconditional analogue of Austrin--Mossel hardness result, bypassing the Unique-Games Conjecture for predicates with an abelian subgroup structure. Our main…

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## 121 Citations

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## References

SHOWING 1-10 OF 112 REFERENCES

Approximation Resistant Predicates from Pairwise Independence

- Mathematics, Computer Science2008 23rd Annual IEEE Conference on Computational Complexity
- 2008

It is shown that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q]k whose support is contained in the set of satisfying assignments to P.

A new point of NP-hardness for unique games

- Computer ScienceSTOC '12
- 2012

For these c, this is the first hardness result showing that it helps to take the alphabet size larger than 2 and the NP-hardness reductions are quasilinear-size and thus show nearly full exponential time is required, assuming the ETH.

Satisfying Degree D Equations over Gf [2]

- Mathematics, Computer Science
- 2011

The hardness results are proved in the form of inapproximability results of Max-CSPs where the predicate in question has the desired form and some immediate results on approximation resistance of some predicates are given.

Satisfying Degree-d Equations over GF[2] n

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2011

The hardness results are proved in the form of inapproximability results of Max-CSPs where the predicate in question has the desired form and some immediate results on approximation resistance of some predicates are given.

Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2008

It is shown that it is NP-hard to approximate the MAX k-CSP problem over non-boolean domains to a ratio greater than q2k/qk, and an approximation algorithm is obtained that achieves a ratio of C(q) ·k/ qk for some constant C( q) depending only on q, via a subroutine for approximating the value of a semidefinite quadratic form.

Derandomized graph products

- Mathematics, Computer Sciencecomputational complexity
- 2005

This paper proves a lower bound for the probability that all steps of a random walk stay within a specified set of vertices of a graph, which extends also to the case where different sets of Vertices are specified for different time steps of the walk.

LS+ Lower Bounds from Pairwise Independence

- Computer Science, Mathematics2013 IEEE Conference on Computational Complexity
- 2013

It is shown that for random instances of such k-CSPs on n variables, even after Ω(n) rounds of the LS+ hierarchy, the integrality gap remains equal to the approximation ratio achieved by a random assignment.

Which problems have strongly exponential complexity?

- Computer Science, MathematicsProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper we address the relative likelihood of sub-exponential algorithms for these…

The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 1997

It is shown that result 2 also holds for the Shortest Lattice Vector Problem in the l norm, and for some of these problems the same result as above is proved, but for a larger factor such as 2 1 & = n or n.

Zero Knowledge and the Chromatic Number

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 1998

This work presents a new technique, inspired by zero-knowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph, and matches (up to low order terms) the known gap for approximation the size of the largest independent set.