Meta-Algorithms versus Circuit Lower Bounds

计算机戏剧性地改变了我们的生活。我们现在可以有效地解决许多计算问题，这些问题如果没有计算机是不可能解决的。然而，许多重要的问题似乎仍然超出了我们最强大的超级计算机的能力范围。这些问题表面上的困难是真的吗(问题本身)，或者这些问题确实有我们还没有发现的有效算法解决方案？*计算复杂性领域研究的正是这个问题：什么问题需要过多的计算资源(计算时间、内存等)？除了澄清可以有效解决的问题的边界外，识别计算上的困难问题还具有重要的实践意义。例如，今天使用的几乎所有密码系统(包括电子银行)的安全性都依赖于未经证实的假设，即某些计算问题很难解决。因此，证明这些问题实际上很难，就可以证明这些密码协议是真正安全的。*现代复杂性理论的重要发现之一是证明计算问题的难度与为相关计算问题设计高效算法之间的深层联系。这两项任务就像计算机科学的英和杨一样：一个任务的进步不可能有另一个任务的进步。*拟议研究的主要目标是更好地了解这种联系，并利用它在计算难度和高效算法设计方面取得进一步进展。*

Computers have dramatically changed our lives. We can now efficiently solve many computational problems that would be impossible to solve without computers. Yet, many important problems still seem beyond the reach of even our most powerful super-computers. Is the apparent difficulty of these problems real (intrinsic to the problem), or these problems do have efficient algorithmic solutions that we haven't been able to discover yet?****The field of computational complexity studies precisely this question: what problems require excessive computational resources (computation time, memory, etc.)? In addition to clarifying the boundary of what can be solved efficiently, identifying computationally hard problems also has important practical consequences. For instance, the security of virtually all cryptographic systems in use today (including electronic banking) relies on the unproven assumptions that certain computational problems are very hard to solve. Thus, proving that such problems are actually hard would prove that these cryptographic protocols are truly secure. ****One of important discoveries of modern complexity theory is the deep connection**between proving the hardness of computational problems and designing efficient algorithms for related computational problems. The two tasks are like Ying and Yang of Computer Science: progress in one is impossible without progress in the other. ****The main goal of the proposed research is to gain better insight into this connection, and exploit it to make further progress in both computational hardness and efficient algorithm design.****