AI 中文总结
研究0/1打包问题，构造多面体且多项式的扩展公式近似Gomory闭包\(P'\)，与Mastrolilli方法类似且适用更高迭代，还通过通信复杂性获得拟多项式多面体扩展公式，在某些参数范围更优，描述扩展协议判断集合是否相交。
AI 中文摘要
我们考虑0/1打包问题\(\max\{c^T x \colon Ax \leq 1, \, x \in \{0,1\}^n\}\)，其中\(A \in \mathbb{R}_{\geq 0}^{m \times n}\)。解决此类问题的一种方法是通过Gomory割平面来收紧线性规划松弛\(P\)。\(P\)的Gomory闭包\(P'\)是\(P\)与其所有割平面的交集。在\(P'\)上的优化问题是NP难的。Mastrolilli（2020）表明，对于固定的\(\epsilon>0\)，Lasserre层次结构产生一个多项式大小的凸但非多面体的扩展公式，它以\(1+\epsilon\)的因子近似\(P'\)。我们的主要结果是构造一个多面体且多项式的扩展公式，以相同的近似保证来近似\(P'\)。我们的构造基于第一原理。与Mastrolilli的方法一样，我们的方法也适用于固定\(t\)和\(\epsilon>0\)的更高迭代\(P^{(t)}\)。与显式构造不同，通信复杂性提供了一种描述扩展公式的替代方法。使用这种方法，我们为上述问题获得了一个在某些参数范围内更优的拟多项式多面体扩展公式。为了实现这一点，我们描述了一个扩展Yannakakis协议的通信协议，以决定Alice的团和Bob的稳定集是否相交。
英文摘要
We consider $0/1$ packing problems $\max\{c^T x \colon Ax \leq 1, \, x \in \{0,1\}^n\}$, with $A \in \mathbb{R}_{\geq 0}^{m \times n}$. A way to solve such problems is via tightening the linear programming relaxation $P$ with Gomory \emph{cutting-planes}. The Gomory-closure $P'$ of $P$ is the intersection of $P$ with all its cutting planes. The optimization problem over $P'$ is NP-hard. Mastrolilli (2020) has shown that for fixed $ε>0$, the Lasserre hierarchy yields a polynomial-size convex but non-polyhedral extended formulation that approximates $P'$ up to a factor of $1+ε$. Our main result is the construction of a polyhedral and polynomial extended formulation that approximates $P'$ with the same approximation guarantee. Our construction is based on first principles. Like Mastrolilli's approach, ours also applies to higher iterates $P^{(t)}$ for fixed $t$ and $ε>0$. In contrast to an explicit construction, communication complexity provides an alternative way to describe extended formulations. Using this approach we obtain a quasi-polynomial polyhedral extended formulation for the above problem that is superior in some parameter regimes. To achieve this, we describe a communication protocol extending Yannakakis' protocol to decide whether the clique of Alice and the stable set of Bob intersect.
