On the approximation resistance of balanced linear threshold functions

@article{Potechin2019OnTA,
  title={On the approximation resistance of balanced linear threshold functions},
  author={Aaron Potechin},
  journal={Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing},
  year={2019}
}
  • Aaron Potechin
  • Published 12 July 2018
  • Computer Science, Mathematics
  • Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
In this paper, we show that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. We also show that the almost monarchy predicate P(x) = sign((k−4)x1 + ∑i=2kxi) is approximable for sufficiently large k. 
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References

SHOWING 1-10 OF 36 REFERENCES
A characterization of approximation resistance for even k-partite CSPs
TLDR
This work gives a characterization of approximation resistance for k-partite CSPs defined by an even predicate and assumes the Unique Games Conjecture.
Approximation Resistance from Pairwise-Independent Subgroups
We show optimal (up to a constant factor) NP-hardness for a maximum constraint satisfaction problem with k variables per constraint (Max-kCSP) whenever k is larger than the domain size. This follow...
Some optimal inapproximability results
We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations…
Approximation Resistant Predicates from Pairwise Independence
TLDR
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 Characterization of Approximation Resistance
TLDR
A complete characterization of approximation resistant predicates under the Unique Games Conjecture is presented and characterizations in the {\it mixed} linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy are presented.
Proving Weak Approximability Without Algorithms
TLDR
This work uses the recent characterization of strong approximation resistance by Khot et al. to prove that for a given predicate, certain necessary conditions for strong resistance derived from their characterization, are violated, which implies the existence of a good rounding algorithm, proving weak approximability.
A characterization of strong approximation resistance
TLDR
This work presents a characterization of strongly approximation resistant predicates under the Unique Games Conjecture, and presents characterizations in the mixed linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy.
The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory
TLDR
This paper isolates a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit a dichotomy, and explains the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy.
Optimal algorithms and inapproximability results for every CSP?
TLDR
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.
A Proof of CSP Dichotomy Conjecture
  • Dmitriy Zhuk
  • Computer Science, Mathematics
    2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2017
TLDR
An algorithm is presented that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.
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