On the volume of the elliptope and related metric polytopes

2026-04-17

AI Summary

This study investigates the volume relationships among the cut polytope and three polynomial-time solvable relaxations—the metric polytope, the rooted metric polytope, and the elliptope—on graphs with $n$ vertices, using global volume ratios to assess relaxation tightness. Combining tools from convex geometry, combinatorial optimization, and volume estimation, the work establishes—through analytical derivations and numerical experiments—the first rigorous comparison of the volumes of the rooted metric polytope and the elliptope. It reveals a crossover phenomenon wherein the relative volumes of the metric polytope and the elliptope invert as the graph size increases, and provides exact volume formulas for the cut polytope on several families of sparse graphs. Notably, on complete graphs, the rooted metric polytope is significantly larger than the elliptope, while the metric polytope is smaller for small $n$ but eventually surpasses the elliptope as $n$ grows.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• elliptope
• metric polytope
• cut polytope
• volume ratio
• relaxation tightness

Innovation

• elliptope
• metric polytope
• volume ratio
• cut polytope
• convex relaxation