1. Subexponential lower bounds for randomized pivoting rules for the simplex algorithm
Oliver Friedmann – Univ. of Munich Thomas Dueholm Hansen – Aarhus Univ. Uri Zwick – Tel Aviv Univ.
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2. Congratulations to Oliver Friedmann for winning the Tucker prize

3. 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

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

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

7. Randomized pivoting rules
Random-Edge
Choose a random improving edge
Random-Facet
Choose a random facet containing the current vertex and recursively find the optimum within that facet.
If the vertex found is not the optimum, do a pivoting step that leaves the chosen facet.
[Kalai (1992)] [Matoušek-Sharir-Welzl (1996)]
sub-exponential !!!
Are Random-Edge and Random-Facet polynomial ???

8. 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.

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

10. The directed 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 directed diameter is exactly n

11. AUSO results
Random-Facet is sub-exponential [Kalai (1992)] [Matoušek-Sharir-Welzl (1996)]
Sub-exponential lower bound for Random-Facet [Matoušek (1994)]
Sub-exponential lower bound for Random-Edge [Matoušek-Szabó (2006)]
Lower bounds do not correspond to actual linear programs
Can geometry help?

12. Random-Edge , Random-Facet are not polynomial for LPs
Consider LPs that correspond to Markov Decision Processes (MDPs)
Simplex  Policy iteration
Obtain sub-exponential lower bounds for the Random-Edge and Random-Facet variants of the Policy Iteration algorithm for MDPs

13. Randomized Pivoting Rules
Upper bound
Lower bound
Algorithm
RANDOM EDGE
RANDOM FACET
Lower bounds obtained for LPs whose diameter is n
[Kalai ’92] [Matousek-Sharir-Welzl ’92]
[Friedmann-Hansen-Z ’11]

14. 3-bit counter

15. Markov Decision Processes [Shapley ’53] [Bellman ’57] [Howard ’60] …
Total reward version
Discounted version
Limiting average version
Optimal positional policies can be found using LP
Is there a strongly polynomial time algorithm?

16. For the total reward version assume:
Stopping condition
For the total reward version assume:
No matter what the controller does, the terminal is reached with probability 1.

17. Stochastic shortest paths (SSPs)
Minimize the expected cost of getting to the target

18. MDP + policy  Markov Chain
Evaluating a policy
MDP + policy  Markov Chain
Values of a fixed policy can be found by solving a system of linear equations

19. Improving a policy (using a single switch)

20. Basic solution  (positional) Policy Dual LP formulation for MDPs
a is not an improving switch
Basic solution 
(positional) Policy

21. Primal LP formulation for MDPs
Vertex 
Complement of a Policy

22. 3-bit counter
(−N)15

23. 3-bit counter 1

24. 3-bit counter – Improving switches
Random-Edge can choose either one of these improving switches… 1

25. Cycle gadgets Cycles close one edge at a time
Shorter cycles close faster

26. Cycles open “simultaneously”
Cycle gadgets
Cycles open “simultaneously”

27. 3-bit counter 23 1 1

28. From b to b+1 in seven phases
Bk-cycle closes
Ck-cycle closes
U-lane realigns
Ai-cycles and Bi-cycles for i<k open
Ak-cycle closes
W-lane realigns
Ci-cycles of 0-bits open

29. 3-bit counter 34 1 1

30. Size of cycles Various cycles and lanes compete with each other
Some are trying to open while some are trying to close
We need to make sure that our candidates win!
Length of all A-cycles = 8n
Length of all C-cycles = 22n
Length of Bi-cycles = 25i2n
O(n4) vertices for an n-bit counter
Can be improved using a more complicated construction and an improved analysis (work in progress)

31. Concluding remarks and open problems
“Game-theoretic” perspective help understand the behavior of randomized pivoting rules
Polynomial pivoting rule?
Polynomial bound on diameter?
Strongly polynomial algorithms for MDPs?
Polynomial algorithms 2-player games?

32. THE END