Equality of tropical rank and dimension for semimodules of tropical rational functions, and computational aspects

The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has remained elusive due to the challenges of computing it in practice. We establish that the tropical rank is, in fact, precisely equal to the topological dimension of the semimodule, one more than the dimension of the associated linear system of divisors. This implies that the equality of divisorial and tropical ranks in the definition of tropical linear series is equivalent to the pure dimensionality of the corresponding linear system. We then address the question of computing the tropical rank. In particular, we show that checking whether a given family of tropical rational functions is tropically independent is equivalent to solving a turn-based stochastic mean-payoff game, whereas calculating the tropical rank of a finitely generated semimodule of tropical rational functions is NP-hard. We conclude with several complementary results and questions regarding combinatorial and topological properties of the tropical rank.

💡 Research Summary

The paper investigates the relationship between two fundamental invariants associated with a semimodule M of tropical rational functions on a metric graph Γ: the tropical rank r_trop(M) and the topological dimension dim(M) of the linear system generated by M. The tropical rank is defined as the maximal size of a tropically independent subset of M, where tropical independence means that no non‑trivial tropical linear combination can achieve the minimum at every point of Γ only once. The topological dimension is defined as the dimension of the associated linear system |(D,M)| (a polyhedral subset of the symmetric product Γ(d) where d = deg D) plus one.

The authors first establish that for any divisor D on Γ, the space R(D) of rational functions with divisor ≥ −D is a finitely generated tropical semimodule. They then show that for any sub‑semimodule M⊆R(D), the linear system |(D,M)| inherits a polyhedral structure via piecewise‑linear maps induced by tropical matrix multiplication. This geometric description allows them to compare the combinatorial notion of tropical independence with the polyhedral dimension.

Theorem 1.1 proves the central equality r_trop(M) = dim(M) for all sub‑semimodules M⊆R(D). The proof proceeds by first handling the finitely generated case, where the maximal number of independent generators coincides with the maximal dimension of a face of the polyhedral complex |(D,M)|. For arbitrary M, the tropical rank is shown to be the supremum of the ranks of its finitely generated sub‑semimodules, and the dimension is defined as the supremum of the dimensions of the corresponding polyhedral complexes; the two suprema agree, yielding the equality. Consequently, the tropical rank of the complete linear system |D| equals the dimension of the classical linear system |D|.

The paper then turns to algorithmic aspects. Using a non‑linear operator T defined on ℝⁿ by
 T_i(c) = sup_{x∈X} min_{j≠i} (f_j(x) – f_i(x) + c_j),
the authors show (Theorem 2.5) that tropical independence of functions {f₁,…,fₙ} is equivalent to the existence of a vector c and a positive scalar ρ such that T(c) = c + ρ·e (where e is the all‑ones vector). This condition can be interpreted as a fixed‑point problem for a monotone, additively homogeneous map. By standard reductions, this fixed‑point problem is equivalent to solving a turn‑based stochastic mean‑payoff game.

Theorem 1.3 establishes a polynomial‑time Turing reduction between (i) testing tropical independence of a given family of tropical rational functions and (ii) solving a turn‑based stochastic mean‑payoff game. Since such games are known to lie in NP ∩ coNP (Condon 1992) but are not known to be in P, the same complexity status transfers to the tropical independence problem. The authors also note that the independence problem is unlikely to be NP‑hard, and they provide a certificate based on the eigenvector c from the fixed‑point formulation.

In contrast, computing the tropical rank of a finitely generated semimodule is shown to be NP‑hard (Theorem 5.17). The reduction is from classic NP‑hard problems such as 3‑SAT or maximum clique: given an instance, one constructs a set of tropical rational functions whose maximal independent subset size encodes the solution to the original problem. Hence, while testing independence of a fixed family is “easy” (NP ∩ coNP), determining the maximal size of an independent family is computationally intractable in general.

The paper further connects its results with prior work. It generalizes earlier observations by Devlin, Santos, and Sturmfels on tropical rank of matrices over the tropical semiring, and by Butković on tropical linear algebra. It also clarifies the relationship between the divisorial rank r(D,M) (the classical Baker–Norine rank) and the tropical rank, showing that equality r(D,M) = r_trop| (D,M) | holds precisely when the linear system |(D,M)| is pure of dimension r(D,M) (Theorem 1.2). This links a purely combinatorial condition (pure dimensionality) with the algebraic notion of rank equality.

The final section presents complementary observations and open questions concerning the combinatorial structure of tropical semimodules, stability under tropical linear combinations, and the geometry of the associated polyhedral complexes. The authors suggest further study of special graph families where the rank‑dimension equality may admit more refined algorithmic treatment, and of potential approximation algorithms for the tropical rank problem.

Overall, the work provides a comprehensive bridge between tropical linear algebra on metric graphs, polyhedral geometry, and algorithmic game theory. It resolves a fundamental conjecture—tropical rank equals topological dimension—while delineating the precise computational boundaries: independence testing lies in NP ∩ coNP, whereas computing the maximal rank is NP‑hard. These insights open new avenues for both theoretical exploration and practical computation in tropical geometry, combinatorial optimization, and related fields.