On the volume of the elliptope and related metric polytopes

TL;DR

This paper studies the relative volumes of convex bodies related to graph optimization relaxations, revealing how their sizes compare as the number of vertices grows, with implications for the tightness of these relaxations.

Contribution

It provides volume ratio bounds for the cut, metric, rooted metric, and elliptope polytopes, including exact formulas for some sparse graphs, enhancing understanding of relaxation tightness.

Findings

Rooted metric polytope volume exceeds elliptope volume for complete graphs.

For small n, metric polytope volume is less than elliptope; for large n, the opposite is suggested.

Exact volume formulas are derived for certain sparse graphs.

Abstract

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.