半定规划的Lasserre下界与可定义性
Lasserre Lower Bounds and Definability of Semidefinite Programming
摘要：
对于一个大类的优化问题，即那些可以表示为有限值约束满足问题（VCSPs），我们建立了一个二分法的Lasserre层次的水平的数量的半定规划（SDPs），需要解决的问题。特别是，我们表明，如果一个有限值的约束问题是不完全解决其基本的线性规划松弛，它也不完全解决任何次线性数量的Lasserre层次。下界是通过逻辑上的不可定义性结果建立的。我们表明，线性规划放松的问题，以及SDP对应于任何固定级别的Lasserre层次结构是可解释的VCSP实例通过公式的不动点逻辑计数。我们还表明，解决方案的SDP可以表示在这个逻辑。总之，这些结果给出了一种方法，翻译下限的计数逻辑中所需的变量的数量表示VCSP到下限的Lasserre层次结构中所需的水平，以消除完整性差距。作为一种特殊情况，我们得到了相同的二分法类MAXCSP问题，推广了一些早期的Lasserre下界Schoenebeck的结果。
英文摘要：
For a large class of optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs), we establish a dichotomy on the number of levels of the Lasserre hierarchy of semi-definite programs (SDPs) that are required to solve the problem exactly. In particular, we show that if a finite-valued constraint problem is not solved exactly by its basic linear programming relaxation, it is also not solved exactly by any sub-linear number of levels of the Lasserre hierarchy. The lower bounds are established through logical undefinability results. We show that the linear programming relaxation of the problem, as well as the SDP corresponding to any fixed level of the Lasserre hierarchy is interpretable in a VCSP instance by means of formulas of fixed-point logic with counting. We also show that the solution of an SDP can be expressed in this logic. Together, these results give a way of translating lower bounds on the number of variables required in counting logic to express a VCSP into lower bounds on the number of levels required in the Lasserre hierarchy to eliminate the integrality gap. As a special case, we obtain the same dichotomy for the class of MAXCSP problems, generalizing some earlier Lasserre lower bound results of Schoenebeck.
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