Uri Zwick – Tel Aviv Univ.

1. Uri Zwick –Tel Aviv Univ. Randomized pivoting rules for the simplex algorithm Upper bounds 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. 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 neighboringvertex, 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(Khachiyan1979) 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 diameterpolynomial?

8. Hirschconjecture(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-Facetis sub-exponential! Are Random-Edge and Random-Facet polynomial ???

10. Random-Facet algorithm(s) Seidel’s randomizedLP algorithm[Seidel (1991)] Upper boundon diameter[Kalai-Kleitman (1992)] Random-Facetdual version[Matoušek Sharir-Welzl (1992)] Random-Facetprimal 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 faceta 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 stepout of the facet and recurse.

16. (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. PrimalRandom-Facet Non-recursive version • Choose a random permutation of the facets f1,f2,…,fdcontaining 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 ficontaining the current vertex vthat 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-dfree variables n inequalities

23. Linear Programming Duality d variables n inequalities n-dfree 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 ignoreit. • 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. Dual Pivot Step Exercise: What is the complexity of a dual pivoting step?

30. 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. 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 shouldhave a unique sink

35. AUSOs of n-cubes 2n facets2n vertices USOs and AUSOs Stickney, Watson (1978) Morris (2001) Szabó, Welzl (2001) Gärtner (2002) The diameter is exactly n No diameter issue!

36. 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