Polyhedral extended formulations that approximate the Gomory closure for packing problems

Abstract

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 ${\epsilon}>0$, the Lasserre hierarchy yields a polynomial-size convex but non-polyhedral extended formulation that approximates $P'$ up to a factor of $1+{\epsilon}$. 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 ${\epsilon}>0$. 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.