The tropical double description method

TL;DR

This paper introduces a tropical analogue of the double description method for polyhedra, enabling efficient computation of vertices from inequalities, with applications in static analysis and significant performance improvements.

Contribution

The paper presents a novel tropical double description algorithm that efficiently identifies extreme points using hypergraph connectivity, outperforming previous methods.

Findings

Method outperforms previous algorithms by orders of magnitude

Reduces extremality checking to hypergraph connectivity problem

Provides improved worst-case bounds for polyhedral computations

Abstract

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.