# 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
• 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.
7 Citations

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## References

SHOWING 1-10 OF 28 REFERENCES
A characterization of approximation resistance for even k-partite CSPs
• Mathematics, Computer Science
ITCS '13
• 2013
This work gives a characterization of approximation resistance for k-partite CSPs defined by an even predicate and assumes the Unique Games Conjecture.
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
• Mathematics, Computer Science
2008 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 Characterization of Approximation Resistance
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2013
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
• Mathematics, Computer Science
APPROX-RANDOM
• 2016
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
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2013
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
• Mathematics
SIAM J. Comput.
• 1998
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?
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
• Mathematics, Computer Science
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
• 2017
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.
A Dichotomy Theorem for Nonuniform CSPs
• A. Bulatov
• Mathematics
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
• 2017
The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP is either solvable in polynomial time or is NP-complete.