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Sparse Cutting-Planes

Sparse Cutting-Planes. Marco Molinaro Santanu Dey , Andres Iroume , Qianyi Wang Georgia Tech. CuttinG -planes. Better approximation of the integer hull. IN THEORY Can use any cutting-plane Putting all gives exactly the integer hull

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Sparse Cutting-Planes

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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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