[Paper Review] Computing the Extreme Points of Tropical Polyhedra

Xavier Allamigeon, Stéphane Gaubert+1 more|arXiv (Cornell University)|Apr 22, 2009

Polynomial and algebraic computation34 references15 citations

TL;DR

This paper presents an efficient algorithm to compute all extreme points of tropical polyhedra defined by max-plus inequalities. By reducing extremality checking to computing the least model of a compact Horn formula, the method enables a novel double description approach that eliminates redundant generators a priori, significantly outperforming prior algorithms in both theory and practice.

ABSTRACT

We present an efficient algorithm to compute all the extreme elements of a max-plus or tropical polyhedron. This algorithm relies on a combinatorial characterization of these extreme elements, when the polyhedron is defined by inequalities. We show that checking the extremality of an element of such a polyhedron reduces to computing the least model of a compact Horn formula, the latter being a factorized representation of a Horn formula. This allows us to develop an analogue of Motzkin’s double description method in which redundant generators are eliminated a priori. We give theoretical bounds and experimental results showing that the algorithm outperforms the previous ones.

Motivation & Objective

• To develop an efficient algorithm for computing all extreme points of max-plus or tropical polyhedra.
• To provide a combinatorial characterization of extreme elements in tropical polyhedra defined by inequalities.
• To reduce the problem of checking extremality to computing the least model of a compact Horn formula.
• To enable a variant of Motzkin’s double description method that eliminates redundant generators in advance.
• To achieve theoretical and empirical improvements over existing algorithms for computing tropical polyhedra extreme points.

Proposed method

• The algorithm uses a combinatorial characterization of extreme elements in tropical polyhedra defined by max-plus inequalities.
• It reduces the extremality check of a candidate point to solving the least model problem for a compact Horn formula representation.
• The compact Horn formula is a factorized form of a Horn formula derived from the polyhedral inequalities.
• The method applies a modified double description method where redundant generators are filtered out before generation.
• The approach leverages the duality between tropical polyhedra and Horn logic to ensure correctness and efficiency.
• Theoretical bounds and experimental evaluation validate the performance gains over previous methods.

Experimental results

Research questions

• RQ1 How can extreme points of tropical polyhedra be characterized combinatorially when defined by max-plus inequalities?
• RQ2 Can the extremality of a candidate point in a tropical polyhedron be reduced to a known computational logic problem?
• RQ3 To what extent can redundant generators be eliminated a priori in tropical polyhedron computation?
• RQ4 How does the proposed method compare in performance to existing algorithms for computing tropical polyhedra extreme points?
• RQ5 What are the theoretical and practical bounds of the new algorithm in terms of scalability and correctness?

Key findings

• The algorithm achieves significant performance improvements over previous methods through a priori elimination of redundant generators.
• Checking extremality reduces to computing the least model of a compact Horn formula, enabling efficient computation.
• The method provides a novel variant of Motzkin’s double description method that avoids generating redundant elements.
• Theoretical bounds show improved complexity compared to earlier approaches for tropical polyhedra computation.
• Experimental results confirm the algorithm outperforms existing techniques in both runtime and scalability.
• The approach establishes a strong link between tropical polyhedra and Horn logic, enabling new algorithmic insights.