We illustrate the limitations of the hyperplane separation bound, a non-combinatorial lower bound on the extension complexity of a polytope. Most notably, this bounding technique is used by Rothvo{\ss} (J ACM 64.6:41, 2017) to establish an exponential lower bound for the perfect matching polytope. We point out that the technique is sensitive to the particular choice of slack matrix. For the canonical slack matrices of the spanning tree polytope and the completion time polytope, we show that the lower bounds produced by the hyperplane separation method are trivial. These bounds may, however, be strengthened by normalizing rows and columns of the slack matrices.
翻译:我们举例说明了超高平板分离约束的局限性,这是对聚域的扩展复杂性的非combinator式下限。最明显的是,Rothvols}(J ACM 64.6:41, 2017)使用这一约束技术来为完美匹配的聚域建立指数性下限。我们指出,该技术对松软矩阵的特定选择十分敏感。对于横跨树形多ope和完成时间聚域的金字裤短网,我们表明超高平板分离方法产生的下限是微不足道的。然而,这些界限可以通过松懈矩阵的行和列的正常化而得到加强。