Computing the vertices of tropical polyhedra using directed hypergraphs

TL;DR

This paper introduces a novel method for computing vertices of tropical polyhedra using directed hypergraphs, enabling efficient vertex enumeration and a tropical double description algorithm with improved performance.

Contribution

It presents a new characterization of tropical polyhedron vertices via directed hypergraphs and develops a tropical double description method with theoretical complexity bounds and practical efficiency.

Findings

The vertex characterization can be checked in almost linear time.

The tropical double description method outperforms existing approaches.

Experimental results confirm the efficiency of the proposed algorithm.

Abstract

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.