A Family of Dictatorship Tests with Perfect Completeness for 2-to-2 Label Cover

We give a family of dictatorship tests with perfect completeness and low-soundness for 2–to–2 constraints. The associated 2–to–2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. Our result provides some indication of the expressiveness and non-triviality of 2–to–2 constraints, given the close connections between dictatorship tests and satisfiability and approximability of CSPs.

[1]  Subhash Khot,et al.  Hardness of Finding Independent Sets in 2-Colorable and Almost 2-Colorable Hypergraphs , 2014, SODA.

[2]  A KhotSubhash,et al.  The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ1 , 2015 .

[3]  Johan Håstad.  On the NP-Hardness of Max-Not-2 , 2014, SIAM J. Comput..

[4]  Subhash Khot,et al.  An Improved Dictatorship Test with Perfect Completeness , 2017, FSTTCS.

[5]  Ryan O'Donnell,et al.  Conditional hardness for satisfiable 3-CSPs , 2009, STOC '09.

[6]  Moses Charikar,et al.  Near-optimal algorithms for unique games , 2006, STOC '06.

[7]  Venkatesan Guruswami,et al.  SDP Gaps for 2-to-1 and Other Label-Cover Variants , 2010, ICALP.

[8]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[9]  Elchanan Mossel,et al.  Conditional Hardness for Approximate Coloring , 2009, SIAM J. Comput..

[10]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[11]  Yuichi Yoshida,et al.  A Query Efficient Non-adaptive Long Code Test with Perfect Completeness , 2010, APPROX-RANDOM.

[12]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[13]  Venkatesan Guruswami,et al.  Improved Inapproximability Results for Maximum k-Colorable Subgraph , 2009, APPROX-RANDOM.

[14]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[15]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[16]  Ryan O'Donnell,et al.  3-bit Dictator Testing: 1 vs. 5/8 , 2009, SODA.

[17]  Prasad Raghavendra,et al.  Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[18]  Ryan O'Donnell,et al.  Analysis of Boolean Functions , 2014, ArXiv.

[19]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[20]  Venkatesan Guruswami,et al.  The Quest for Strong Inapproximability Results with Perfect Completeness , 2021, Electron. Colloquium Comput. Complex..

[21]  Elchanan Mossel,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, IEEE Annual Symposium on Foundations of Computer Science.

[22]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[23]  Hubie Chen.  A rendezvous of logic, complexity, and algebra , 2006, SIGA.

[24]  Guy Kindler,et al.  Towards a proof of the 2-to-1 games conjecture? , 2018, Electron. Colloquium Comput. Complex..

[25]  J. Littlewood,et al.  On the Number of Real Roots of a Random Algebraic Equation , 1938 .

[26]  Subhash Khot,et al.  On independent sets, 2-to-2 games, and Grassmann graphs , 2017, Electron. Colloquium Comput. Complex..