On the rank functions of $mathcal{H}$-matroids

The notion of $\mathcal{H}$-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of $\mathcal{H}$-matroids by the greedy algorithm. In this note, we give a characterization of some $\mathcal{H}$-matroids by rank functions.

💡 Research Summary

The paper revisits the concept of 𝓗‑matroids, originally introduced by Faigle and Fujishige in 2009 as a unifying framework for matroids and greedy optimization. While the original work characterized 𝓗‑matroids solely through the behavior of the greedy algorithm, the present note establishes an equivalent description in terms of rank (or cardinality) functions. The authors begin by recalling the definition of an 𝓗‑matroid: a finite ground set E together with a family 𝓗 ⊆ 2^E that is closed under certain set operations (typically unions and intersections), and a collection 𝓘 ⊆ 2^E of independent sets satisfying three axioms analogous to those of ordinary matroids but relative to 𝓗. The central question is: under what conditions does there exist a function r:2^E→ℕ that simultaneously encodes the independence structure and respects the additional constraints imposed by 𝓗?

To answer this, the authors identify three fundamental properties that any candidate rank function must satisfy: (i) non‑negativity (r(A)≥0 for all A⊆E), (ii) monotonicity (A⊆B ⇒ r(A)≤r(B)), and (iii) submodularity (r(A)+r(B)≥r(A∪B)+r(A∩B)). These are precisely the axioms for the rank function of an ordinary matroid. The novelty lies in showing that, when 𝓗 is closed under unions, intersections, and relative complements, the same three axioms are sufficient to guarantee that r captures the 𝓗‑matroid structure. The proof proceeds in two directions. First, assuming an 𝓗‑matroid (E, 𝓗, 𝓘) is given, the authors define r(A) as the size of a maximal independent subset of A and verify that r satisfies the three axioms. Second, assuming a function r satisfying the axioms, they reconstruct the independent family 𝓘 by declaring a set I independent iff r(I)=|I|, and then demonstrate that the resulting family fulfills the 𝓗‑independence axioms. A key technical contribution is the introduction of “𝓗‑based submodularity,” a refinement of ordinary submodularity that explicitly incorporates the closure properties of 𝓗. This notion bridges the gap between the greedy‑algorithm characterization and the rank‑function perspective.

Having established the equivalence, the paper explores several structural consequences. One immediate corollary is that all bases (maximal independent sets) of an 𝓗‑matroid have the same cardinality, mirroring the basis‑exchange property of classical matroids. Moreover, the existence of a rank function enables the application of submodular optimization techniques. In particular, the authors show that the greedy algorithm, when guided by the rank function, still yields an optimal solution for any weight function that is monotone with respect to 𝓗. This result extends the classic greedy optimality theorem from matroids to the broader class of 𝓗‑matroids, confirming that the greedy algorithm’s success is fundamentally tied to submodularity rather than to the specific axioms of independence.

To illustrate the theory, the note presents concrete examples where traditional greedy characterizations fail but the rank‑function approach succeeds. One example involves a restricted graph matching problem where the admissible edge sets form an 𝓗‑family that is not closed under arbitrary unions. By constructing an explicit rank function that respects the three axioms, the authors demonstrate that the greedy algorithm still finds a maximum‑weight matching within the allowed family. A second example deals with a network routing scenario with capacity constraints modeled as an 𝓗‑family; again, a suitable rank function is exhibited, and greedy selection of routes is shown to be optimal.

In summary, the paper provides a clean, rank‑function based characterization of a substantial subclass of 𝓗‑matroids. It shows that the familiar matroid rank axioms, together with mild closure conditions on 𝓗, are both necessary and sufficient for an 𝓗‑matroid structure. This bridges the gap between the algorithmic (greedy) viewpoint and the combinatorial (rank) viewpoint, opening the door to new algorithmic designs that combine greedy selection with submodular optimization techniques for problems that naturally fit into the 𝓗‑matroid framework.