The tropical double description method

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.

💡 Research Summary

The paper introduces a tropical analogue of the classical double description method, enabling the conversion of a polyhedron given by tropical inequalities (external representation) into a set of its extreme points (internal representation). The authors observe that the main difficulty in the tropical setting is the rapid growth of redundant generators and the lack of an efficient extremality test. To address this, they formulate a novel characterization of extreme points: a point x is extreme for a tropical polyhedron P if and only if the hypergraph constructed from the constraints that are tight at x possesses exactly one minimal strongly connected component (SCC). This hypergraph captures the dependency relations among the defining inequalities, and the extremality condition reduces to a purely combinatorial property that can be checked in almost linear time using standard SCC algorithms (Tarjan or Kosaraju).

Building on this insight, the authors embed the SCC test into each iteration of the double description algorithm. When a new candidate generator is produced by intersecting two existing generators, the algorithm immediately builds the associated hypergraph and discards the candidate if the hypergraph contains more than one minimal SCC. Consequently, redundant generators are eliminated on the fly, avoiding the costly post‑processing steps required by earlier tropical algorithms.

The theoretical analysis shows that, whereas previous tropical double description procedures exhibited worst‑case cubic time complexity and quadratic space usage, the new method achieves a worst‑case bound of O(n² log n) time and O(n²) space, where n is the number of input inequalities. The dominant cost per iteration is the linear‑time SCC detection, making the overall procedure almost linear in the size of the hypergraph.

Extensive experiments validate the practical impact of the approach. Benchmarks derived from static analysis of real programs (thousands of tropical constraints) and synthetic high‑dimensional tropical polyhedra (dimensions up to 200) demonstrate speed‑ups ranging from one order of magnitude to two orders of magnitude compared with the best existing tropical double description implementations. Memory consumption is also reduced substantially, and the number of final generators is markedly lower because many duplicates are eliminated early.

Beyond performance, the paper contributes a conceptual bridge between tropical geometry and hypergraph theory, suggesting that many tropical extremality questions can be recast as graph‑theoretic problems. The authors discuss potential extensions, including parallelization of the SCC checks, integration with tropical linear programming solvers, and application to dynamic systems where constraints evolve over time.

In summary, the work delivers a robust, theoretically grounded, and experimentally verified algorithm for computing the internal representation of tropical polyhedra. By reducing extremality testing to a minimal‑SCC condition in a hypergraph, it achieves near‑linear runtime for each generator test, dramatically outperforms prior methods, and opens new avenues for efficient tropical computations in static analysis, optimization, and beyond.