The tropical double description method

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.

12 pages, prepared for the Proceedings of the Symposium on Theoretical Aspects of Computer Science, 2010, Nancy, France

References in corpus (3)

• Duality and separation theorems in idempotent semimodules
• The Minkowski Theorem for Max-plus Convex Sets
• Minimal half-spaces and external representation of tropical polyhedra

Cited by in corpus (10)

• Computing the vertices of tropical polyhedra using directed hypergraphs
• Minimal half-spaces and external representation of tropical polyhedra
• The number of extreme points of tropical polyhedra
• On the complexity of strongly connected components in directed hypergraphs
• Minimal external representations of tropical polyhedra
• Algebraic solution of tropical polynomial optimization problems
• Basic solutions of systems with two max-linear inequalities
• On $2\times 2$ Tropical Commuting Matrices
• Tropical convex hulls of polyhedral sets
• Tropical pseudolinear and pseudoquadratic optimization as parametric mean-payoff games