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Challenging Computational Infeasibility: PCP and Boolean functions

Periodic Reporting for period 1 - PCPABF (Challenging Computational Infeasibility: PCP and Boolean functions)

Reporting period: 2019-10-01 to 2021-03-31

One of the main problems addressed here is the classification of algorithmic, computational problems into feasible vs. infeasible.
Within this problematic, an important step was made by Subhash Khot when he introduced the unique games conjecture (UGC). In recent years, the UGC and its variants received significant attention.

One of our main goals in this project is to solve the UGC. Such conjectures have numerous applications to hardness of approximation, and connections to several topics in algorithms, computational complexity, and geometry. However, there is still no consensus regarding the validity of these conjectures. Only recently an approach towards proving the Unique Games Conjecture, or rather a weak form of it, was proposed. We have presented, building on previous work, an approach towards proving the related 2-to-1 Games Conjecture (or rather a variant of it with imperfect completeness).

Specifically, we have proposed a combinatorial conjecture concerning a consistency test on the Grassmann graph and show that it implies the 2-to-1 Games Conjecture with imperfect completeness.

We are now working on the connection between this type of problems and computational problems over lattices with the aim to resolve the complexity of lattice problems. This will provide us with open options for basing cryptographic primitives on the hardness of any of the above mentioned problems.

Focusing on these problems will lead to a better understanding of these as will save human effort on solving infeasible problems. In addition, it will lead to better cryptography, and, optionally, results in quantum computation.
One of our main results is that we have come very close to settle the Influence-entropy conjecture.

This can have far-reaching consequences regarding infeasibility of computational approximation problems as all current PCP effort is based on theorems regarding the analysis of Boolean functions.

In addition, we have utilized our mathematical understanding to figure out ways to overcome the COVID pandemic, in particular random graphs and Fourier in the framework of Boolean analysis.
The Influence-Entropy conjecture is the main conjecture in the field of ABF and we came close to resolving it.
This is a very meaningful progress on which we continue to progress.

We are making strong advances in our work on lattices. This will hopefully lead to major breakthroughs and cryptographic applications.

In our work on Coronavirus, we have suggested that we might reach stability way before most of the population is immune, contrary to the common opinion,
because, as we demonstrate, superspreaders will be getting out of commission sooner than the general population.

In addition, our research has allowed us to make the following recommendation, that is, to apply separation between distinct communities, instead of quarantine.