In this paper, we investigate the relationships between the volumes of four convex bodies: the cut polytope, metric polytope, rooted metric polytope, and elliptope, defined on graphs with $n$ vertices. The cut polytope is contained in each of the other three, which, for optimization purposes, provide polynomial-time relaxations. It is therefore of interest to see how tight these relaxations are. Worst-case ratio bounds are well known, but these are limited to objective functions with non-negative coefficients. Volume ratios, pioneered by Jon Lee with several co-authors, give global bounds and are the subject of this paper. For the rooted metric polytope over the complete graph, we show that its volume is much greater than that of the elliptope. For the metric polytope, for small values of $n$, we show that its volume is smaller than that of the elliptope; however, for large values, volume estimates suggest the converse is true. We also give exact formulae for the volume of the cut polytope for some families of sparse graphs.

翻译：本文研究了定义在$n$个顶点图上的四个凸体的体积关系：割多面体、度量多面体、有根度量多面体和椭圆体。割多面体包含于其他三个凸体之中，从优化角度而言，这三个凸体提供了多项式时间的松弛方法。因此，考察这些松弛的紧致程度具有重要意义。已知的最坏情况比率边界仅适用于具有非负系数的目标函数。由Jon Lee与多位合作者开创的体积比率方法提供了全局边界，亦是本文的研究主题。对于完全图上的有根度量多面体，我们证明其体积远大于椭圆体。对于度量多面体，在$n$值较小时，其体积小于椭圆体；但当$n$值较大时，体积估计表明结论相反。此外，我们给出了若干稀疏图族的割多面体体积的精确公式。