CAREER: New Directions in Inapproximability and Probabilistically Checkable Proofs

理论计算机科学研究的中心问题是：如何有效（快速）地解决计算问题？虽然许多问题确实有有效的算法，但有一大类重要问题（称为 NP 完全问题）不太可能有有效的算法。但实际上，大概解决这些问题可能就足够了。本研究探讨是否可以有效地找到 NP 完全问题的近似解，以及近似的质量如何。这项研究的主要贡献是负面结果，即证明对于某些 NP 完全问题，有效找到近似解的可能性很小。为了说明证明这种负面结果的重要性，研究者证明了它不太可能闯入某个密码系统，从而保证其安全性免受恶意攻击。这项研究的另一个相关方面是概率可检查证明，这是一种指定数学陈述的证明格式的方法，这样可以通过仅查看证明中的几个地方而不是阅读整个证明来非常有效地检查证明的有效性。该研究在科学研讨会、几位国际研究人员之间的合作、研究生课程的开发、促进本科生研究和为博士生提供建议方面有可能产生更广泛的影响。学生。理论和实践中出现的许多计算问题都是NP完全的。处理 NP 完整性的一种广泛研究的方法是设计计算近似最优解的多项式时间算法。然而，事实证明，对于许多问题，计算近似解本身就是一个 NP 完全问题，这是 1992 年发现的著名结果，称为 PCP 定理。尽管随着这一发现进行了大量的研究，但对于许多 NP 完全问题来说，最著名的近似结果和最知名的不可逼近性结果之间存在差距。本研究的重点是通过证明严格的不可逼近性结果来填补这一空白。 PCP 定理也可以被视为证明检查的结果（这就是它的发现方式）。它提供了一种指定 NP 语句证明的方法，以便可以非常有效地检查证明的有效性。该研究调查了更有效的 PCP 的构造，并进一步应用于不可近似的结果。开发的技术可能会在度量嵌入和学习理论等领域有新的应用。该研究在科学研讨会、国际研究人员之间的合作、研究生课程的开发、促进本科生研究和为博士生提供建议方面有可能产生更广泛的影响。学生。

The central question studied in theoretical computer science is: how efficiently (fast) can computational problems be solved?While many problems do have efficient algorithms, there is a wide class of important problems (called NP-complete problems) which are very unlikely to have efficient algorithms. In practice however, it may suffice to solve these problems approximately. This research investigates whether approximate solutions to NP-complete problems can be found efficiently, and how good is the quality of approximation. The main contribution of this research is negative results, i.e. proving that for certain NP-complete problems, efficiently finding even approximate solutions is very unlikely. To illustrate the significance of proving such negative results, the investigator proves that it is unlikely to break into a certain cryptosystem, giving a guarantee of its security against malicious attacks. Another related aspect of this research is Proababilistically Checkable Proofs, a method to specify proof formats for mathematical statements, such that the validity of the proof can be checked very efficiently, by looking at only a few places in the proof instead of reading the entire proof. The research has a potential for broader impact in terms of scientific workshops, collaboration between several international researchers, developement of graduate courses, promoting undergraduate research, and advising Ph.D. students. Many computational problems arising in theory and practice are NP-complete. An extensively studied approach to cope with NP-completeness is designing polynomial time algorithms that compute approximately optimal solutions. However, it turns out that for many problems, computing approximate solutions itself is an NP-complete problem, a famous result known as the PCP Theorem, discovered in 1992. In spite of the tremendous body of research that followed this discovery, for many NP-complete problems, there is a gap between the best known approximation result, and the best known inapproximability result. This research focusses on filling this gap by proving tight inapproximability results. The PCP Theorem can also be viewed as a result about proof checking (and that is how it was discovered). It gives a way of specifying proofs for NP-statements such that the validity of the proof can be checked very efficiently. The research investigates constructions of more efficient PCPs, with further applications to inapproximability results. The techniques developed are likely to have new applications to areas like metric embeddings and learning theory. The research has a potential for broader impact in terms of scientific workshops, collaboration between international researchers, developement of graduate courses, promoting undergraduate research and advising Ph.D. students.