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# Challenging Computational Infeasibility: PCP and Boolean functions

## Periodic Reporting for period 2 - PCPABF (Challenging Computational Infeasibility: PCP and Boolean functions)

Our research into the mathematics of computation and, in particular, computational complexity theory, revolves around classifying computational problems as feasible or alternatively as infeasible, typically in the worst-case regime. A paradigm that transformed research in this direction is the PCP characterization of NP, which allowed proofs that approximation problems are NP-hard (there is practically no other strategy for proving such statements). PCP has become a vast and deep research area, closely interacting with other areas of mathematics such as analysis of Boolean functions and computational problems over Lattices.

Unique Games Conjecture
An important step was made by Subhash Khot in 2002, who proposed the Unique Games Conjecture (UGC). Unique-games (UG) is a particular approximation problem whose complexity is not yet settled, despite major efforts. Its complexity has been shown to be strongly related to a whole class of approximation problems. The research efforts on this conjecture have been in both directions: trying to show iUG is hard, as well as, trying to come up with even a truly subexponential algorithm for UG. In recent years, the UGC and its variants received significant attention and are recognized as the most important problem in the area.
One of our main goals in this project is to solve the UGC. As part of this effort we have proposed an attack which could lead to showing UG NP-hard, thus proving the Unique Games Conjecture. This approach has led to an infeasibility result for a related problem, namely, the 2-to-1-games (albeit, a variant of it with imperfect completeness). This result is recognized as going half the way toward UGC (UG can be rephrased as 1-to-1-games) .
One of the major mathematical directions that came about in that proof relates to expansion properties of the Grassmann graph, in particular, structure theorems about sets of its vertices. The Grassmann graph turned out to be crucial in other related settings (it is the graph of the future), such as testing a function to be a low-degree polynomial. This is one of the directions our project pursues.

Lattices
The project has been taken on finding connections between classical constraint satisfaction problems and computational problems over Lattices—the main goal is to resolve the complexity of computational lattice problems.
Infeasibility results for computational problems over lattices have natural application for designing cryptographic primitives whose security proof relies on the hardness of any of the above mentioned problems. In cryptography one needs assumptions stating that some computational problems are infeasible, however, in the average-case regime. Computational problems over Lattices turn out to be almost the only source for such assumptions as they allow worst-case to average-case reductions.
Alternatively, such connections may allow a proof that approximation problems of the typical PCP form—for example the 2-to-1-games with perfect completeness—are in coNP. Such a statement gives a very strong indication that the problem is not NP-hard, as if it were NP-hard it would imply NP=coNP, which would change our entire perspective on computation.
The approach could lead to better cryptography, and, optionally, one may be able to resolve these lattice problems—for weak enough approximation ratios—using a quantum computer; this might fundamentally change our perspective on computation and could have far reaching practical applications.

Analysis of Boolean functions
Analysis of Boolean function has become central for PCP and infeasibility proofs. Still, the insights gained have far reaching consequences to other areas. For example, with regards to phenomena that have a sharp-threshold or phase-transition (that is, they change extremely fast from one phase to another depending on some parameter). One such prime type of theorem is in regards to graph properties—some properties change very quickly as the graph becomes denser, near the critical-probability, from being false to being true. A prime example for such a property is the existence of a giant component, namely, a set of vertices that is connected and comprise a large portion of the vertices. These type of theorems require a refinement of the Analysis of Boolean functions technology, and involve more general spectral Methods as they apply to Branching processes for Random Graphs, for example, the Galton-Watson process.

We have been working on these problems and quite fortunately have found application of them to the spread of pandemics, which became very relevant in real-life due to the COVID pandemic.
One of our main results is that we have come very close to settling 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 Boolean functions.
The project has been laying the foundation for resolving the basic questions regarding the complexity of approximation problems whose complexity is not yet determined.
Classifying approximation problems into infeasible requires synergy of three areas and technologies:

- PCP, in particular, composition recursion and parallel repetition, as applied
- Boolean functions, in particular, structure theorems and expansion
- Complexity of computational Lattice problems—for problems whose complexity falls between P and NP-hard

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 regarding the complexity of computational problems over lattices. This will hopefully lead to major breakthroughs and cryptographic applications.