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）解决计算问题？虽然许多问题确实有有效的算法，但有一类重要的问题（称为NP完全问题）不太可能有有效的算法。然而，在实践中，近似地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近似解NP完全问题，以及近似的质量有多好。这项研究的主要贡献是否定的结果，即证明对于某些NP完全问题，有效地找到近似解是非常不可能的。为了说明证明这种否定结果的重要性，研究人员证明了它不太可能闯入某个密码系统，从而保证了它的安全性免受恶意攻击。这项研究的另一个相关方面是Proababilistically Checkable Proofs，一种为数学陈述指定证明格式的方法，这样证明的有效性可以非常有效地检查，只需查看证明中的几个地方，而不是阅读整个证明。该研究有可能在科学研讨会，几个国际研究人员之间的合作，研究生课程的开发，促进本科生研究和建议博士方面产生更广泛的影响。学生在理论和实践中出现的许多计算问题都是NP完全问题。一个广泛研究的方法来科普NP-完全性是设计多项式时间算法，计算近似最优解。然而，事实证明，对于许多问题，计算近似解本身是一个NP完全问题，这是1992年发现的著名结果，称为PCP定理。尽管在这一发现之后进行了大量的研究，但对于许多NP完全问题，在最著名的近似结果和最著名的不可近似结果之间存在差距。这项研究的重点是填补这一空白，证明紧不可逼近的结果。PCP定理也可以被看作是关于证明检查的结果（这就是它是如何被发现的）。它给出了一种NP-语句证明的方法，使得证明的有效性可以被非常有效地检查。该研究探讨了更有效的PCP的建设，进一步应用于不可逼近的结果。开发的技术可能会有新的应用领域，如度量嵌入和学习理论。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咨询方面具有更广泛的影响力。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.