Probabilistically Checkable Proofs (PCPs) encapsulate the striking idea that verification of proofs becomes nearly trivial if one is willing to use randomness. The PCP theorem, proven in the early 90's, is a cornerstone of modern computational complexity theory. It completely revises our notion of a proof, leading to an amazingly robust behavior: A PCP proof is guaranteed to have an abundance of errors if attempting to prove a falsity. This stands in sharp contrast to our classical notion of a proof whose correctness can collapse due to one wrong step. An important drive in the development of PCP theory is the revolutionary effect it had on the field of approximation. Feige et. al. [JACM, 1996] discovered that the PCP theorem is *equivalent* to the inapproximability of several classical optimization problems. Thus, PCP theory has resulted in a leap in our understanding of approximability and opened the gate to a flood of results. To date, virtually all inapproximability results are based on the PCP theorem, and while there is an impressive body of work on hardness-of-approximation, much work still lies ahead. The central goal of this proposal is to obtain stronger PCPs than currently known, leading towards optimal inapproximability results and novel notions of robustness in computation and in proofs. This study will build upon (i) new directions opened up by my novel proof of the PCP theorem [JACM, 2007]; and on (ii) state-of-the-art PCP machinery involving techniques from algebra, functional and harmonic analysis, probability, combinatorics, and coding theory. The broader impact of this study spans a better understanding of limits for approximation algorithms saving time and resources for algorithm designers; and new understanding of robustness in a variety of mathematical contexts, arising from the many connections between PCPs and stability questions in combinatorics, functional analysis, metric embeddings, probability, and more.
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