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Related Experiment Videos

Convex dynamics: unavoidable difficulties in bounding some greedy algorithms.

Tomasz Nowicki1, Charles Tresser

  • 1IBM, P.O. Box 218, Yorktown Heights, New York 10598, USA. tnowicki@us.ibm.com

Chaos (Woodbury, N.Y.)
|March 9, 2004
PubMed
Summary
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A greedy algorithm for scheduling and digital printing is bounded for 2D polygons but becomes complex in higher dimensions. Natural generalizations fail above two dimensions, proving complexity is inherent for higher-dimensional polytopes.

Area of Science:

  • Computational Geometry
  • Dynamical Systems
  • Optimization Algorithms

Background:

  • A greedy algorithm for scheduling and digital printing, using polytope vertices as outputs, has proven boundedness in any dimension.
  • This boundedness relies on invariant regions within an equivalent dynamical system.
  • Proofs are simpler in 2D (polygons) but become complex in 3D and higher.

Purpose of the Study:

  • To demonstrate that the complexity of the boundedness proof in higher dimensions is justified.
  • To show that natural generalizations of 2D invariant region constructions fail in dimensions above two.
  • To explore limitations of invariant regions for higher-dimensional polytopes and simplices.

Main Methods:

  • Proving that certain higher-dimensional polytopes lack combinatorially equivalent invariant regions.

Related Experiment Videos

  • Modifying polytope examples to show invariant regions cannot be formed by expanding half-space borders.
  • Investigating mechanisms preventing invariant regions in simplices.

    • Main Results:

    • The natural generalization of 2D invariant region constructions fails in dimensions > 2.
    • Certain polytopes in dimensions > 2 do not admit equivalent invariant regions.
    • Specific simplices also lack admitting such invariant regions, unlike 2D polygons where parallel pushing suffices.

    Conclusions:

    • The complexity of proving boundedness for higher-dimensional polytopes is inherent and not an artifact of the proof method.
    • The geometric constructions for invariant regions that work in 2D do not extend to higher dimensions.
    • Understanding these limitations is crucial for applications in scheduling, digital printing, and chaos theory.