1. Uri Zwick – Tel Aviv Univ.
Randomized pivoting rules for the simplex algorithm Upper bounds
Uri Zwick – Tel Aviv Univ.
MDS summer school
“The Combinatorics of Linear and Semidefinite Programming”
August 14-16, 2012
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAA

2. Find the highest point in a polytope
Linear Programming
Maximize a linear objective function subject to a set of linear equalities and inequalities
Find the highest point in a polytope

3. The Simplex Algorithm [Dantzig (1947)]
Move up, along an edge to a neighboring vertex, until reaching the top

4. Upper bound Theorem for polytopes [Klee (1964)] [McMullen (1971)]
Number of vertices of a d-dimensional n-faceted polytope is at most:

5. Deterministic pivoting rules
Largest improvement
Largest slope
Dantzig’s rule – Largest modified cost
Bland’s rule – avoids cycling
Lexicographic rule – also avoids cycling
All known to require an exponential number of steps, in the worst-case
Klee-Minty (1972)
Jeroslow (1973), Avis-Chvátal (1978),
Goldfarb-Sit (1979), … , Amenta-Ziegler (1996)

6. Algorithms for Linear Programming
Simplex (Dantzig 1947)
No polynomial versions known
Ellipsoid (Khachiyan 1979)
Interior-point (Karmakar 1984)
Smoothed analysis based (ST’04, KS’06)
Other (DV’08, …)
Polynomial, but not strongly polynomial
Not “combinatorial”

7. Is there a polynomial pivoting rule?
Is the diameter polynomial?

8. Diameter is still believed to be polynomial
Hirsch conjecture (1957): The diameter of a d-dimensional, n-faceted polytope is at most n−d
Refuted Santos (2010)!
Diameter is still believed to be polynomial
Quasi-polynomial upper bound [Kalai-Kleitman (1992)]

9. Randomized pivoting rules
Random-Edge
Choose a random improving edge
Random-Facet
To be explained shortly ☺
[Kalai (1992)] [Matoušek-Sharir-Welzl (1996)]
Random-Facet is sub-exponential!
Are Random-Edge and Random-Facet polynomial ???

10. Random-Facet algorithm(s)
Seidel’s randomized LP algorithm [Seidel (1991)]
Upper bound on diameter [Kalai-Kleitman (1992)]
Random-Facet dual version [Matoušek
Sharir-Welzl (1992)]
Random-Facet primal version [Kalai (1992)]

11. Upper bound on diameter [Kalai-Kleitman (1992)]
Orient the polytope such that the target vertex is the highest
Bound the monotone diameter
Active facet a facet containing a vertex (strictly) higher than the current vertex

12. Upper bound on diameter [Kalai-Kleitman (1992)]

13. Upper bound on diameter [Kalai-Kleitman (1992)]
Exercise: Verify and complete the details.

14. “Wishful” Random Facet
Find a vertex of a random active facet. (How?)
Solve recursively within that facet.
If top not reached, step out of the facet and recurse.

15. Primal Random-Facet [Kalai (1992)]
Choose a random facet containing the current vertex.
Solve recursively within that facet.
If top not reached, do a pivoting step out of the facet and recurse.

16. Primal Random-Facet in 2D
(Toy) Exercise:
Primal Random-Facet in 2D

17. How do we solve such a recurrence?
One option: generating functions [MSW (1996)]
Seidel’s challenge: Can it be taught?
Once a right bound is known, it is not too hard to verify it by induction

18. Solving the recurrence

19. Exercise: Verify and justify (or find a bug…)

20. Primal Random-Facet Non-recursive version
Choose a random permutation of the facets f1,f2,…,fd containing the current vertex v.
Find the first facet fi that is beneficial to leave and move to a new vertex v’ contained in a new facet f’i.
Choose a new random ordering of f1,f2,…,fi-1,f’i. Keep the ordering of fi+1,…,fd. Repeat.
Exercise: Really?

21. (Randomized) Bland’s rule
Choose a (random) permutation of all facets f1,f2,…,fn.
Find the first facet fi containing the current vertex v that is beneficial to leave and move to a new vertex v’.
Open problem: Subexponential ???

22. Linear Programming Duality
d variables
n inequalities
n variables
n inequalities
d equalities
n-d free variables
n inequalities

23. Linear Programming Duality
d variables
n inequalities
n-d free variables
n inequalities

24. Clarkson’s Algorithm [Clarkson (1988)]
No need to consider the case n>9d2

25. Seidel’s algorithm [Seidel (1991)]
Choose a random inequality and ignore it.
Solve recursively without this inequality.
If the vertex obtained satisfies the ignored inequality, we are done.
Otherwise, replace the ignored inequality by an equality and solve recursively.

26. Dual Simplex Algorithms

27. Dual Simplex Algorithms

28. Dual Simplex Algorithms

29. Exercise: What is the complexity of a dual pivoting step?
Dual Pivot Step
Exercise: What is the complexity of a dual pivoting step?

30. [Matoušek-Sharir-Welzl (1992)]
Dual Random Facet
[Matoušek-Sharir-Welzl (1992)]
Choose a random facet not containing the current dual vertex.
Solve recursively without this facet, starting from the current dual vertex.
If the vertex obtained is on the right side of the ignored facet, i.e., is a vertex, we are done.
Otherwise, do a dual pivoting step to a dual vertex on the ignored facet and recurse.

31. Dual Random Facet - Analysis [Matoušek-Sharir-Welzl (1992)]
Hidden dimension

32. Exercise: Coincidence?
Primal vs. Dual
Exercise: Coincidence?

33. “It is remarkable to see how different paths have led to rather similar results so close in time.”
[Kalai (1992)]

34. Abstract objective functions (AOFs)
Acyclic Unique Sink Orientations (AUSOs)
Every face should have a unique sink

35. The diameter is exactly n
AUSOs of n-cubes
2n facets 2n vertices
USOs and AUSOs
Stickney, Watson (1978)
Morris (2001)
Szabó, Welzl (2001)
Gärtner (2002)
The diameter is exactly n
No diameter issue!

36. Taken from a paper by Gärtner-Henk-Ziegler
Klee-Minty cubes (1972)
Taken from a paper by Gärtner-Henk-Ziegler

37. Random-Facet-1 “[Kalai (1992)]”
Try to find vertices on cn active facets.
If top not reached, choose a random facet, go back to the vertex found on it, and solve recursively within it.
If top not reached, do a pivoting step out of the facet.
Try again to find vertices on cn active facets.
If top not reached, try again, from any starting vertex.

38. Analysis of Random-Facet-1 “[Kalai (1992)]”

39. RANDOM FACET on the hypercube [Ludwig (1995)] [Gärtner (2002)]
All correct !
Would never be switched !
There is a hidden order of the indices under which the sink returned by the first recursive call correctly fixes the first i bits