A greedoid and a matroid inspired by Bhargava's $p$-orderings

TL;DR

This paper introduces a new combinatorial structure inspired by Bhargava's p-orderings, showing that maximum-perimeter subsets under certain conditions form a strong greedoid and a matroid, with applications to diversity and factorials.

Contribution

It generalizes Bhargava's p-orderings to a greedy algorithm framework, establishing new greedoid and matroid structures based on ultrametric distances and weights.

Findings

Maximum-perimeter subsets can be found via a greedy algorithm.

These subsets form a strong greedoid and a matroid.

The construction generalizes p-orderings and relates to diversity in phylogenetics.

Abstract

Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w \left(e\right) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \in E$ have a "distance" $d \left(e, f\right) = d \left(f, e\right) \in \mathbb{R}$ assigned to them, such that the distances satisfy the ultrametric triangle inequality $d(a,b)\leqslant \max \left\{d(a,c),d(b,c)\right\}$. We look for a subset of $E$ of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances). We show that any such subset can be found by a greedy algorithm (which starts with the empty set, and then adds new elements one by one, maximizing the perimeter at each step). We use this to define numerical invariants, and also to show that the maximum-perimeter subsets of all sizes form a strong greedoid, and the maximum-perimeter subsets of any given size are the bases of a matroid. This essentially generalizes the "$P$-orderings" constructed by Bhargava in order to define his generalized factorials, and is also similar to the strong greedoid of maximum diversity subsets in phylogenetic trees studied by Moulton, Semple and Steel. We further discuss some numerical invariants of $E, w, d$ stemming from this construction, as well as an analogue where maximum-perimeter subsets are replaced by maximum-perimeter tuples (i.e., elements can appear multiple times).