半定规划的乘积定理
Product theorems via semidefinite programming
摘要：
半定程序在乘积下完美合成的倾向在复杂性理论中被利用了很多次：例如，洛瓦兹确定了五边形的香农容量;证明了非确定性通信复杂性的直和定理和差异的直积定理;在交互式证明系统中证明了限制类博弈的并行重复定理。尽管所有这些例子的产品定理-一些可以追溯到近30年-它只是最近，米塔尔和Szegedy开始制定一个一般的理论来解释何时以及为什么半定节目表现完美的产品。这一理论在文献中抓住了许多例子，但也有一些值得注意的例外，它不能解释-即早期的平行重复结果Feige和Lovasz，和直接产品定理的差异方法的通信复杂性的李，Shraibman和Spalek。我们扩展了米塔尔和Szegedy的理论来解释这些情况。事实上，据我们所知，我们的理论抓住了文献中半定乘积定理的所有例子。
英文摘要：
The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovasz to determine the Shannon capacity of the pentagon; to show a direct sum theorem for non-deterministic communication complexity and direct product theorems for discrepancy; and in interactive proof systems to show parallel repetition theorems for restricted classes of games. Despite all these examples of product theorems--some going back nearly thirty years--it was only recently that Mittal and Szegedy began to develop a general theory to explain when and why semidefinite programs behave perfectly under product. This theory captured many examples in the literature, but there were also some notable exceptions which it could not explain--namely, an early parallel repetition result of Feige and Lovasz, and a direct product theorem for the discrepancy method of communication complexity by Lee, Shraibman, and Spalek. We extend the theory of Mittal and Szegedy to explain these cases as well. Indeed, to the best of our knowledge, our theory captures all examples of semidefinite product theorems in the literature.
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