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Parallel Redundancy Removal in lrslib with Application to Projections

David Avis, Charles Jordan

TL;DR

The paper tackles efficient redundancy removal and projection for convex polyhedra by parallelizing the classical LP-based reduction and integrating it with Fourier–Motzkin elimination in lrslib. The approach combines parallel LP-based inequality classification, GCD-based deduplication, and handling of hidden linearities, with Clarkson's algorithm as a fast alternative for highly redundant inputs. It demonstrates substantial speedups on multi-core clusters and shows that projecting via a $V$-representation before converting back to an $H$-representation can be advantageous in some cases. The work advances practical polyhedral computation by enabling scalable redundancy removal and offering a hybrid projection strategy that adapts to input redundancy.

Abstract

We describe a parallel implementation in lrslib for removing redundant halfspaces and finding a minimum representation for an H-representation of a convex polyhedron. By a standard transformation, the same code works for V-representations. We use this approach to speed up the redundancy removal step in Fourier-Motzkin elimination. Computational results are given including a comparison with Clarkson's algorithm, which is particularly fast on highly redundant inputs.

Parallel Redundancy Removal in lrslib with Application to Projections

TL;DR

The paper tackles efficient redundancy removal and projection for convex polyhedra by parallelizing the classical LP-based reduction and integrating it with Fourier–Motzkin elimination in lrslib. The approach combines parallel LP-based inequality classification, GCD-based deduplication, and handling of hidden linearities, with Clarkson's algorithm as a fast alternative for highly redundant inputs. It demonstrates substantial speedups on multi-core clusters and shows that projecting via a -representation before converting back to an -representation can be advantageous in some cases. The work advances practical polyhedral computation by enabling scalable redundancy removal and offering a hybrid projection strategy that adapts to input redundancy.

Abstract

We describe a parallel implementation in lrslib for removing redundant halfspaces and finding a minimum representation for an H-representation of a convex polyhedron. By a standard transformation, the same code works for V-representations. We use this approach to speed up the redundancy removal step in Fourier-Motzkin elimination. Computational results are given including a comparison with Clarkson's algorithm, which is particularly fast on highly redundant inputs.
Paper Structure (8 sections, 2 theorems, 8 equations, 1 figure, 3 tables)

This paper contains 8 sections, 2 theorems, 8 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Parallel Redundancy Removal and Finding a Minimum Representation
  4. Projection by the Fourier-Motzkin Method
  5. Computational Results
  6. Redundancy Removal and Minimum Representation
  7. Fourier-Motzkin Elimination
  8. Conclusions and Future Directions

Key Result

Proposition 1

The inequality $b_i+A_ix \ge 0$ is a linearity if $z_{\max}=0$ otherwise it is

Figures (1)

  • Figure 1: Golden square

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof