CAREER: New Directions in Inapproximability and Probabilistically Checkable Proofs

理论计算机科学研究的中心问题是：如何高效（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。然而，在实践中，大致解决这些问题就足够了。本文研究了np完全问题的近似解能否有效地找到，以及近似解的质量有多好。本研究的主要贡献是否定的结果，即证明了对于某些np完全问题，有效地找到甚至近似解是非常不可能的。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，从而保证了其免受恶意攻击的安全性。这项研究的另一个相关方面是概率检验证明（probababilistic Checkable Proofs），这是一种为数学命题指定证明格式的方法，这样就可以非常有效地检查证明的有效性，只需查看证明中的几个地方，而不是阅读整个证明。这项研究在科学研讨会、几位国际研究人员之间的合作、研究生课程的开发、促进本科研究和指导博士生方面具有更广泛的影响。在理论和实践中出现的许多计算问题都是np完全的。一个被广泛研究的处理np完备性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，这是一个著名的结果，被称为PCP定理，于1992年发现。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近似结果之间存在着差距。本研究的重点是通过证明紧密不近似结果来填补这一空白。PCP定理也可以被看作是证明检查的结果（这就是它是如何被发现的）。它给出了一种为np陈述指定证明的方法，使得证明的有效性可以非常有效地检查。该研究探讨了更有效的pcp结构，并进一步应用于非近似结果。所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。这项研究在科学研讨会、国际研究人员之间的合作、研究生课程的开发、促进本科研究和指导博士生方面具有更广泛的影响。

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.