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Many rays of the submodular cone

Georg Loho, Arnau Padrol, Germain Poullot

TL;DR

The paper advances the understanding of the submodular cone by introducing a constructive inductive framework that builds new rays from existing ones through generalized polymatroid sums and admissibility cones. By establishing that the deformation cone of a GPsum is isomorphic to an admissibility cone and leveraging independence polytopes, the authors derive a doubling-exponential lower bound $t_n \ge 2^{2^{n-2}}$ for the number of rays and an upper bound $t_n \le n^{2^n}$, improving prior estimates. They develop seeds and fertility to systematically generate many rays, providing explicit fertile families (11 polytopes for $n=4$ and 656 for $n=5$) and showing how to lift rays to higher dimensions. The work connects submodular ray enumeration to generalized polymatroids, deformed permutahedra, and toric geometry, and offers a versatile inductive approach that could sharpen upper/lower bounds and illuminate higher-dimensional faces of submodular cones with practical combinatorial constructions.

Abstract

The study of the cone of submodular functions goes back to Jack Edmonds' seminal 1970 paper, which already highlighted the difficulty of characterizing its extreme rays. Since then, researchers from diverse fields have sought to characterize, enumerate, and bound the number of such rays. In this paper, we introduce an inductive construction that generates new rays of the submodular cone. This allows us to establish that the $n$-th submodular cone has at least $2^{2^{n-2}}$ rays, which improves upon the lower bound obtained from Hien Q. Nguyen's 1986 characterization of indecomposable matroid polytopes by a factor of order $\sqrt{n^3}$ in the exponent.

Many rays of the submodular cone

TL;DR

The paper advances the understanding of the submodular cone by introducing a constructive inductive framework that builds new rays from existing ones through generalized polymatroid sums and admissibility cones. By establishing that the deformation cone of a GPsum is isomorphic to an admissibility cone and leveraging independence polytopes, the authors derive a doubling-exponential lower bound for the number of rays and an upper bound , improving prior estimates. They develop seeds and fertility to systematically generate many rays, providing explicit fertile families (11 polytopes for and 656 for ) and showing how to lift rays to higher dimensions. The work connects submodular ray enumeration to generalized polymatroids, deformed permutahedra, and toric geometry, and offers a versatile inductive approach that could sharpen upper/lower bounds and illuminate higher-dimensional faces of submodular cones with practical combinatorial constructions.

Abstract

The study of the cone of submodular functions goes back to Jack Edmonds' seminal 1970 paper, which already highlighted the difficulty of characterizing its extreme rays. Since then, researchers from diverse fields have sought to characterize, enumerate, and bound the number of such rays. In this paper, we introduce an inductive construction that generates new rays of the submodular cone. This allows us to establish that the -th submodular cone has at least rays, which improves upon the lower bound obtained from Hien Q. Nguyen's 1986 characterization of indecomposable matroid polytopes by a factor of order in the exponent.

Paper Structure

This paper contains 12 sections, 28 theorems, 22 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Generalized-polymatroid sum and admissibility cone
  5. Polytopes, fans, and deformations
  6. Deformed permutahedra and generalized polymatroids
  7. Quotients and GP-sums
  8. Admissibility cones
  9. Many rays of the submodular cone
  10. Independence polytopes
  11. Inductive construction of rays
  12. Upper and lower bounds on the number of rays of the submodular cone

Key Result

Theorem A

Given a suitably chosen subset of the rays of the submodular cone $\mathbb{SC}_{n}$, each pair gives rise to (at least one) ray of the submodular cone $\mathbb{SC}_{n+1}$. Applying the same construction to the resulting subset of rays of $\mathbb{SC}_{n+1}$ yields rays of $\mathbb{SC}_{n+2}$, and so

Figures (8)

  • Figure 1: The construction of $\GPsum$: for $\mathsf{P}$ and ${\mathsf{Q}}$ two segments, the corresponding $\textcolor{blue}{\mathsf{P}+\mathbb{R}_-^n}$ and $\textcolor{red}{{\mathsf{Q}}+\mathbb{R}_+^n}$ are unbounded polyhedra. Computing their intersection yields the GP-sum $\textcolor{violet}{\GPsum}$, whose top and bottom faces are $\mathsf{P}$ and ${\mathsf{Q}}$ respectively. Note that $\GPsum$ is a polytope, and that the combinatorics of $\GPsum$ heavily depend on the exact coordinates of $\mathsf{P}$ and ${\mathsf{Q}}$, not on their sole combinatorics.
  • Figure 2: (Top left) The braid fan of dimension 3 intersected with the plane spanned by $({\boldsymbol{e}}_1, {\boldsymbol{e}}_2)$: each maximal chamber ${\mathsf{C}}_\sigma$ is decorated with the corresponding permutation $\sigma\in {\mathcal{S}}_3$. (Top right & Bottom) Various $\GPsum[{\mathsf{Q}}][\mathsf{P}_i]$ for normally equivalent $\mathsf{P}_1,\dots, \mathsf{P}_4$. The normal fan of each $\GPsum[{\mathsf{Q}}][\mathsf{P}_i]$ is different.
  • Figure 3: The submodular cone $\mathbb{SC}_{3}$ has dimension $7$. As its lineality space has dimension $3$, it is a cone over a polytope of dimension $3\, (=7-3-1)$. We depict this polytope (a bi-pyramid over a vertical triangle, shown at the center of the figure). Each point of this polytope corresponds to the GP-sum of two deformed $1$-permutahedra (which are segments or points). Each face corresponds to a class of normally equivalent GP-sums. We illustrate some of these GP-sums.
  • Figure 4: The triangle ${\mathsf{T}}_1$ in blue, and the segment ${\mathsf{S}}_1$ in red. Both lie in $\mathbb{R}^3$, but as they belong to the same plane, we picture it in 2-dimensions. In blue and red are the coordinates of the vertices. In black is depicted the braid fan: the dashed lines are the hyperplanes $\{{\boldsymbol{x}}\in \mathbb{R}^3 ~;~ x_i = x_j\}$ for $i\ne j$ (intersected with our plane of embedding), and the each maximal region is a cone ${\mathsf{C}}_\sigma$ labeled by its corresponding permutation $\sigma\in {\mathcal{S}}_3$.
  • Figure 5: Fertile pairs among indecomposable deformed $3$-permutahedra
  • ...and 3 more figures

Theorems & Definitions (79)

  • Theorem A: \ref{['cor:CollectionOfRays']}
  • Theorem B: \ref{['thm:UpperLowerBoundsTnSn']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 69 more