Challenging Computational Infeasibility: PCP and Boolean functions

ended

Computer Science, in particular, Analysis of Algorithms and Computational-Complexity theory, classify algorithmic-problems into feasible ones and those that cannot be efficiently-solved. Many fundamental problems were shown NP-hard, therefore, unless P=NP, they are infeasible. Consequently, research efforts shifted towards approximation algorithms, which find close-to-optimal solutions for NP-hard optimization problems. The PCP Theorem and its application to infeasibility of approximation establish that, unless P=NP, there are no efficient approximation algorithms for numerous classical problems; research that won the authors --the PI included-- the 2001 Godel prize. To show infeasibility of approximation of some fundamental problems, however, a stronger PCP was postulated in 2002, namely, Khot's Unique-Games Conjecture. It has transformed our understanding of optimization problems, provoked new tools in order to refute it and motivating new sophisticated techniques aimed at proving it. Recently Khot, Minzer (a student of the PI) and the PI proved a related conjecture: the 2-to-2-Games conjecture (our paper just won Best Paper award at FOCS'18). In light of that progress, recognized by the community as half the distance towards the Unique-Games conjecture, resolving the Unique-Games conjecture seems much more likely. A field that plays a crucial role in this progress is Analysis of Boolean-functions. For the recent breakthrough we had to dive deep into expansion properties of the Grassmann-graph. The insight was subsequently applied to achieve much awaited progress on fundamental properties of the Johnson-graph. With the emergence of cloud-computing, cryptocurrency, public-ledger and Blockchain technologies, the PCP methodology has found new and exciting applications. This framework governs SNARKs, which is a new, emerging technology, and the ZCASH technology on top of Blockchain. This is a thriving research area, but also an extremely vibrant High-Tech sector.