1. Sparse Cutting-Planes Marco Molinaro Santanu Dey, Andres Iroume, Qianyi Wang Georgia Tech

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

3. IN PRACTICE Only want to use sparse inequalities Solvers use sparsity to filter out cuts (to solve LPs fast) Very limited theoretical investigation [Andersen-Weismantel 10] Do not give integer hull CUTTING-PLANES 1-sparse

4. IN PRACTICE Only want to use sparse cutting planes Most commercial solvers use sparsity to filter out cuts Very limited theoretical investigation [Andersen-Weismantel 10] Do not give integer hull CUTTING-PLANES

6. GEOMETRIC ABSTRACTION Well defined for every polytope

7. GEOMETRIC ABSTRACTION

8. good bad

9. good bad

10. 1.General upper bound 2.Matching lower bounds 3.Extended formulations 4.Extensions: allowing “few” dense cuts RESULTS First three results appear in How good are sparse cutting-planes? Dey, M., Wang, IPCO 14

11. 1- GENERAL UPPER BOUND

12. Sparse cuts are good if number of vertices is “small” many vertices

13. 1- GENERAL UPPER BOUND Idea: randomly sparsify inequalities existence so there exists such ineq. concentration + union bound randomly sparsify (dense) inequality

14. 2-MATCHING LOWER BOUNDS Thm1: Conv random 0/1 points matches upper bound with prob ¼ depends on how many points

15. 2-MATCHING LOWER BOUNDS Main element: anticoncentration far from expectation Thm1: Conv random 0/1 points matches upper bound with prob ¼

16. 2-MATCHING LOWER BOUNDS Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ Main element: anticoncentration far from expectation Thm1: Conv random 0/1 points matches upper bound with prob ¼

17. 2-MATCHING LOWER BOUNDS Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ Main element: anticoncentration far from expectation Thm1: Conv random 0/1 points matches upper bound with prob ¼ 0/1 with prob 1/2 Used often in computational experiments, hard Freville and Plateau 96, Chu and Beasly 98, Kaparis and Letchford 08 and 10, …

18. 2-MATCHING LOWER BOUNDS Thm2: For random packing instances, sparse cuts are as bad as possible with prob ¼ Main element: anticoncentration far from expectation New element: order statistics of uniform distribution Thm1: Conv random 0/1 points matches upper bound with prob ¼

19. 3- EXTENDED FORMULATIONS coordinate projection

20. 3- EXTENDED FORMULATIONS

21. What if we also allow “few” dense cuts? 4-EXTENSIONS Idea: bad polytope for sparse cuts in each orthant In worst case, really need to use a lot of dense cuts

22. Push for understanding of sparse cutting-planes QUESTIONS When should we use denser cuts? If starts with sparse LP formulation? Almost block structure? Sparsify cutting planes? Reformulations that allow good sparse cuts Sparse + few dense cuts for packing problems Better model? WHAT’S NEXT

23. THANK YOU!