Strength of Generalizations of Split Cuts

1. Strength of Generalizations of Split Cuts SanjeebDash OktayGunlukMarco Molinaro

2. Outline • Introduction • Split vs. cross vs. crooked cross closures • Cuts from basic and multi-rowrelaxations • Techniques

3. Introduction

4. IP and Cuts Integer programming Max integer • : linear relaxation • : integer hull • Cut: linear inequality valid for Cutting plane theory: cuts that give good approximation of IP(P) • Crucial for efficiently solving IPs conv

5. Split Cuts Definition • Disjunctive viewpoint [Bal 79] • Every solution has or • Inequality valid for both parts of linear relaxation • Split cut: any cut obtained by split disjunction • Split closure : add all split cuts • even tighter relaxation of • x1 ≥ 1 x1 ≤ 0 Interest • Most important family of cuts for current IP solvers. Hard to beat • Well-developed theory

6. Cross Cuts Definition • Generalization of split cuts • Cross cut: any cut obtained by cross-disjunction • Cross closure : add all cross cuts Interest • The most natural generalization of split cuts • We can almost reach , want more [DG 10, FS 11, Bon] • Revisited by Dash, Gunluk and Vielma with experimental improvement over

7. Crooked Cross Cuts Definition • Also generalization of split cuts • Crooked cross cut: any cut obtained by crooked cross disjunction • Crooked cross closure : add all crooked cross cuts Interest • Stronger than cross cuts [DDG 10] • Solve mixed-integer programs with 2 integer variables [DDG 10]

8. Split vs. Cross vs. crooked cross closures

9. Split vs. Cross vs. Crooked [DDG 10] For all

10. Split vs. Cross Closure • Cross cuts are as strong assplits, • There is such that [CKS 90] • Some important rank-2 split cuts (i.e. ) are cross cuts • 2-step MIR cuts [DG 06] • Used in Dash, Gunluk and Vielma experiments Open question [DGV 11]: Is it always true that ? Result 1: No… We construct such that is strictly contained in Have to show that these infinitely many cuts are not enough!

11. Split vs. Cross vs. Crooked [DDG 10]

12. Cross vs. Crooked Cross Closure • CC cuts are as strong as cross cuts, • There are CC cuts that cannot be obtained by a single cross cut [DDG 10] Open question [DDG 10]: Do we have ? Result 2: No… We construct such that is strictly contained in Have to show that these infinitely many cuts are not enough!

13. Split vs. Cross vs. Crooked [DDG 10]

14. The Complete Picture

15. cuts from basic and multi-rowRelaxations

16. Relaxations • General idea: easier to understand cuts for more specific IPs • original IP • original IP • integral, • integral, • keep linearly indep. rows • aggregate into equations • basic relaxation • -row relaxation

17. Relaxations and Split Cuts Basic relaxation • Theory loves it: nicer structure • Most cuts in practice come from this relaxation: efficiency All split cuts can be obtained from basic relaxations [ACL 05] k-Row relaxation • Things are easier in lower dimensions [GJ 72, DR 08, ALWW 07] All split cuts can be obtained from 1-row relaxations [NW 90]

18. Relaxations and Split Cuts Basic relaxation • Theory loves it: nicer structure • Most cuts in practice come from this relaxation: efficiency All split cuts can be obtained from basic relaxations [ACL 05] k-Row relaxation • Things are easier in lower dimensions [GJ 72, DR 08, ALWW 07] All split cuts can be obtained from 1-row relaxations [NW 90] What about more general cuts?

19. Relaxations and Cross Cuts Basic relaxation Question: Can all cross cuts be obtained from basic relaxations? Result 3: No… • Indicates that does not hold for cuts more complicated than splits • Together with recent works pushes for non-basic cuts [FM 08, CMN] • Shows cross cuts can bypass limitations of basic relaxations

20. Relaxations and Cross Cuts k-Row relaxation • Known that all cross cuts can be obtained from 3-row relaxations [DDG 10] Open question [DDG 10]: Can we obtain all cross cuts from 2-row relaxations? Result 4: No… Construct with 3 equations such that cross cut is stronger than all cuts from all 2-row relaxations together Have to show that these infinitely many relaxations are not enough!

21. Techniques

22. Height Lemma Main tool for handling the effect of intersecting infinitely many cuts/relaxations

23. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base.

24. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base.

25. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base.

26. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights”

27. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base

28. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base Then intersection of all pyramids has point with “height”

29. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base Then intersection of all pyramids has point with “height”

30. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base Then intersection of all pyramids has point with “height”

31. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base Then intersection of all pyramids has point with “height”

32. Height Lemma Lemma:Consider a collection of pyramids in with a common -dimensional base. Sseapexes have: • Uniform lower bound on their “heights” • Uniform upper bound on their distances to base Then intersection of all pyramids has point with “height”

33. Conclusions Results: Better understanding of strength of different generalizations of split cuts • Complete relationship between split, rank-2 split, cross, crooked cross, t-branch split closures • Cross and crooked cross cuts from basic/2-row relaxations are not enough Techniques: Developed general tools for refuting strength of intersection of infinitely many cuts/convex sets.

34. Thank You!