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Understanding Sparse Cutting-Planes: Approximating Integer Hulls with Efficient Methods

This study explores the theory and practical applications of sparse cutting-planes in the context of approximating integer hulls. It highlights the limitations of current approaches, particularly the reliance on sparse inequalities in commercial solvers, and discusses the importance of selecting cutting-planes strategically. The research investigates geometric abstractions and density considerations, providing upper and lower bounds for sparse cuts and analyzing their effectiveness through various cases. The goal is to refine methods for generating and utilizing sparse cuts while examining when denser cuts might be warranted.

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Understanding Sparse Cutting-Planes: Approximating Integer Hulls with Efficient Methods

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  1. Sparse Cutting-Planes Marco Molinaro SantanuDey, Andres Iroume, Qianyi WangGeorgia Tech

  2. CuttinG-planes Better approximation of the integer hull IN THEORY • Can use any cutting-plane • Putting all gives exactly the integer hull • Many families of cuts, large literature, since 60’s

  3. Cutting-Planes IN PRACTICE • Only want to use sparseinequalities • Solvers use sparsity to filter out cuts (to solve LPs fast) • Very limited theoretical investigation [Andersen-Weismantel10] • Do not give integer hull 1-sparse at most non-zero entries

  4. Cutting-Planes IN PRACTICE • Only want to use sparse cutting planes • Most commercial solvers use sparsity to filter out cuts • Very limited theoretical investigation [Andersen-Weismantel10] • Do not give integer hull • GOAL: Understand sparse cutting-planes • Always good, if we select cuts smartly? • Really bad, even if uses ? • How to generate sparse cuts? • … at most non-zero entries

  5. Cutting-Planes • GOAL: Understand sparse cutting-planes • Always good, if we select cuts smartly? • Really bad, even if uses ? • How to generate sparse cuts? • …

  6. Geometric abstraction polytope in (e.g. integer hull) intersection of all -sparse inequalities Well defined for every polytope

  7. Geometric abstraction polytope in (e.g. integer hull) intersection of all -sparse inequalities at most GOAL: How does behave?

  8. good Ex 1: = k-subset of Ex 2: = Ex 3: – convex hull of random 0/1 points (computational) bad (density)

  9. Results • General upper bound • Matching lower bounds • Extended formulations • Extensions: allowing “few” dense cuts First three results appear in How good are sparse cutting-planes?Dey, M., Wang, IPCO 14

  10. 1- General upper bound Thm: For all polytopes in , is at most: (density)

  11. 1- General upper bound Thm: For all polytopes in , is at most: many vertices (density) Sparse cuts are good if number of vertices is “small”

  12. 1- General upper bound Thm: For all polytopes in , is at most… Idea: randomly sparsifyinequalities (dense) inequality randomly sparsify existence concentration + union bound with strictly positiveprob. is sparse, valid and has similar effect as so there exists such ineq.

  13. 2-Matching Lower Bounds Thm1: Conv random 0/1 points matches upper bound with prob ¼ depends on how many points (density)

  14. 2-Matching Lower Bounds Thm1: Conv random 0/1 points matches upper bound with prob ¼ Main element:anticoncentration far from expectation Lemma: For independent uniform 0/1 RVs for up to

  15. 2-Matching Lower Bounds Thm1: Conv random 0/1 points matches upper bound with prob ¼ Main element:anticoncentration far from expectation Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ (density)

  16. 2-Matching Lower Bounds Thm1: Conv random 0/1 points matches upper bound with prob ¼ Main element:anticoncentration far from expectation Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ 0/1 with prob 1/2 Used often in computational experiments, hard Frevilleand Plateau 96, Chu and Beasly 98, Kaparis and Letchford 08 and 10, …

  17. 2-Matching Lower Bounds Thm1: Conv random 0/1 points matches upper bound with prob ¼ Main element:anticoncentration far from expectation Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ New element: order statistics of uniform distribution

  18. 3- Extended formulations coordinate projection

  19. 3- Extended formulations Thm1: (Extended formulations help sparsity) If is ext formulation of , then • Thm2: (Extended formulations can helpsparsitya lot) • A bad polytope where • for all • Has extended formulation with

  20. 4-extensions What if we also allow “few” dense cuts? Thm: There is a polytope such that adding all50-sparse + dense cuts still leaves distance Idea: bad polytope for sparse cuts in each orthant In worst case, really need to use a lot of dense cuts

  21. What’s next Push for understanding of sparse cutting-planes QUESTIONS • When should we use denser cuts? • If starts with sparse LP formulation? Almost block structure? • Sparsifycutting planes? • Reformulationsthat allow good sparse cuts • Sparse + few dense cuts for packing problems • Better model?

  22. Thank you!

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