1. 1 A polynomial relaxation-type algorithm for linear programming Sergei Chubanov University of Siegen, Germany sergei.chubanov@uni-siegen.de

2. 2 Relaxation method Project the current point onto the half-space generated by a constraint which is not satisfied: Agmon, and Motzkin and Schoenberg (1954)

3. 3 Relaxation method is exponential

4. 4 Relaxation method

5. 5 Outline A strongly polynomial algorithm which either finds a solution or proves that there are no 0,1-solutions A polynomial algorithm for linear programming

6. 6 Linear system is induced by the system if and only if

7. 7 an induced inequality (ii) (i) Given, construct one of the two objects: Task such that

8. 8 Elementary case  is a row of of max. length

9. 9 Elementary case

10. 10 Divide-and-conquer algorithm 1. If, then the elementary case. 2. D&C returns or 3. Calculate D&C returns or 4. Calculate with

11. 11 Recursion

12. 12 Recursion Recursive call for the same center and a smaller radius

13. 13 Recursion Recursive call either produces an approximate solution or a valid inequality

14. 14 Recursion Recursive call either produces an approximate solution or a valid inequality

15. 15 Recursion Recursive call for the same radius and another center which is the projection of the current center onto the half-space.

16. 16 Recursion The second recursive call either produces an approximate solution or an induced inequality

17. 17 Recursion

18. 18 Recursion The algorithm may fail to construct an induced inequality

19. 19 … … … Depth of recursion

20. 20 At most recursive calls Running time

21. 21 nonzero components of equations variables Running time

22. 22 D&C algorithm Not faster than the relaxation method Can solve the task, but not always

23. 23 Parameterized system

24. 24 Strengthened parameterized system D&C is applied to

25. 25 (I) is induced by the strengthened parameterized system (II) Task Given

26. 26 If D&C finds an approximate solution to the strengthened parameterized system is an exact solution to the parameterized system  If D&C finds a solution  is a solution to the system in question

27. 27 The two recursive calls at the iteration where it fails produce the inequalities where are linear combinations of the rows of and If D&C fails = 0

28. 28  contradiction  infeasible or   If D&C fails

29. 29 is a linear combination of the rows of is induced by the original parameterized system If D&C returns an inequality

30. 30 The smaller ball does not contain any solution of the original parameterized system If D&C returns an inequality

31. 31 If D&C returns an inequality

32. 32  If D&C returns an inequality

33. 33 If D&C returns an inequality

34. 34 Case 1. If D&C returns an inequality

35. 35 Case 2. If D&C returns an inequality

36. 36 or    If D&C returns an inequality

37. 37   If D&C returns an inequality

38. 38 Algorithm If D&C fails, then either no solutions or If D&C generates an induced inequality, either no solutions or  Repeated application of the following argument:

39. 39 Algorithm The algorithm either finds a solution or decides that there are no 0,1- solutions in strongly polynomial time

40. 40 Algorithm If the system is feasible and the bounds are tight, a solution can be found in strongly polynomial time

41. 41 (1) (2) (1) is feasible if and only if (2) has an integer solution General case

42. 42 (2) (3) (2) has an integer solution if and only if (3) has an integer solution By solving (3) we also solve (1)  a polynomial algorithm for linear programming Polynomial algorithm