多项式理想的θ体
Theta Bodies for Polynomial Ideals
摘要：
受Lovász问题的启发，我们引入了一个层次的嵌套半定松弛的实解的凸包的任意多项式理想，称为theta机构的理想。对于图中的独立集问题，这个层次中的第一个theta体就是图的Lovász theta体。我们证明，θ机构，关闭，拉塞尔的松弛的一个版本的真正的解决方案，理想，他们可以明确地计算使用组合矩矩阵。θ体为最大割问题提供了一个新的半定松弛的规范集。对于有限点集的消失理想，给出了第一θ体等于点的凸包的几个等价刻画。我们还确定了所有理想的第一个θ体的结构。
英文摘要：
Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.
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