SIAM Journal on Optimization, Volume 30, Issue 1, Page 798-822, January 2020.
Chvátal--Gomory cuts (CG-cuts) and the Bienstock--Zuckerberg hierarchy capture useful linear programs that the standard bounded degree sum-of-squares (SoS) hierarchy fails to capture. In this paper we present a novel polynomial time SoS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low degree polynomials). We show that the new SoS hierarchy recovers the Bienstock--Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock--Zuckerberg hierarchy as a polynomial-sized, efficiently constructible extended formulation that satisfies all constant pitch inequalities. The construction is also very simple, and it is fully defined by giving the supporting polynomials. Moreover, for a class of polytopes (e.g., set cover and packing problems), the resulting SoS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up to an arbitrarily small error in the solution value. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.

中文翻译：

---

平方证明，Bienstock-Zuckerberg层次结构和Chvátal-Gomory割的高和

SIAM优化杂志，第30卷，第1期，第798-822页，2020年1月。
Chvátal-Gomory切割（CG-cuts）和Bienstock-Zuckerberg层次结构捕获了有用的线性程序，而标准有界平方和（SoS）层次结构无法捕获。在本文中，我们针对高阶多项式的自定义子空间（不是低阶多项式的标准子空间）提出了一种针对0/1问题的新型多项式时间SoS层次结构。我们表明，新的SoS层次结构恢复了Bienstock-Zuckerberg层次结构。我们的结果暗示了一个线性程序，该程序将Bienstock-Zuckerberg层次结构重现为满足所有恒定音高不等式的多项式大小，可有效构造的扩展公式。构造也非常简单，并且通过给出支持多项式来完全定义。此外，对于一类多面体（例如，设置覆盖物和包装问题），最终的SoS层次结构会优化由CG剪切的任何恒定轮次产生的多面体的多项式时间，直到求解值的误差很小。可以说，这是第一个示例，其中在不对称情况下不同的基函数可用于获得松弛层次。