On the volumes of the elliptope, metric polytope, and cut polytope
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. After an affine change of coordinates for the elliptope, 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 for large n 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, we prove that the converse is true. Volume estimates place the crossover near n = 13. We also give exact formulae for the volumes of several families of sparse cut polytopes. In particular, we give an exact formula for the volume of the elliptope of every cycle and show that its volume ratio with the corresponding cut polytope converges rapidly to one.