1. Secular session of 2nd FILOFOCS April 10, 2013
Improved upper bounds for Random-Edge and Random-Jump on abstract cubes
Thomas Dueholm Hansen – Aarhus Univ. Mike Paterson – Univ. of Warwick Uri Zwick – Tel Aviv Univ.
Secular session of 2nd FILOFOCS April 10, 2013
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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. Figure taken from a paper by Gärtner-Henk-Ziegler
Klee-Minty cubes (1972)
Figure taken from a paper by Gärtner-Henk-Ziegler

5. Randomized pivoting rules
Random-Edge
Choose a random improving edge
Random-Facet
Choose a random facet containing the current vertex. Recursively find the optimum within that facet. If the vertex found is not the optimum, choose an edge that leaves the chosen facet.
[Kalai (1992)] [Matoušek-Sharir-Welzl (1996)]
Sub-exponential !!!

6. Abstract objective functions (AOFs)
Acyclic Unique Sink Orientations (AUSOs)
Many pivoting rules only look at orientation of edges
Every face should have a unique sink

7. AUSOs of n-cubes 2n facets 2n vertices USOs and AUSOs
Adler, Saigal (1976)
Stickney, Watson (1978)
Williamson Hoke (1988)
The directed diameter is exactly n
Vertex queries / Oracle access

8. Markov Decision Processes Turn-based Stochastic Games
AUSO LP
Cube- AUSO
Cube- LP SG
MDP
Markov Decision Processes
Turn-based Stochastic Games

9. [Friedmann-Hansen-Z ’11]
Random-Facet
[Kalai (1992)] [MSW (1996)]
on AUSOs
[Matoušek (1994)]
on Cube-AUSOs
[Friedmann-Hansen-Z ’11]
on LPs

10. [Friedmann-Hansen-Z ’11]
Random-Edge
[Gärtner-Kaibel (2007)]
on cube-AUSOs
[Matoušek-Szabó (2006)]
on Cube-AUSOs
[Friedmann-Hansen-Z ’11]
on Cube-LPs

11. Random-Edge on cube-AUSOs [Hansen-Paterson-Z (2013)] on Cube-AUSOs
[Matoušek-Szabó (2006)]
on Cube-AUSOs
[Friedmann-Hansen-Z ’11]
on Cube-LPs

12. Jump
Jump – Jump to the antipodal vertex in the sub-cube spanned by the outgoing edges

13. Random-Jump
Random-Jump – Jump to a random vertex in the sub-cube spanned by the outgoing edges

14. [Friedmann-Hansen-Z ’11]
Random-Jump
[Mansour-Singh (1999)]
on cube-AUSOs
[Friedmann-Hansen-Z ’11]
on Cube-LPs

15. Random-Jump on cube-AUSOs [Hansen-Paterson-Z (2013)] on Cube-LPs
[Friedmann-Hansen-Z ’11]
on Cube-LPs

16. Cube-(A)USO properties
Out-map of vertex – Indices of outgoing edges
Out-maps of all 2n vertices are distinct
There are exactly vertices of out-degree k
Most out-degrees are (n)

17. On average a jump bypasses at least 2k1 vertices
Jumping on a DAG
Out-degree k  2k
On average a jump bypasses at least 2k1 vertices
[Mansour-Singh (1999)]
Can we do better?

18. Jumping on a DAG

19. Jumping on a DAG Riser: Faller:
Lemma: The expected length of a run of risers is at most 2n.
Lemma: If u is a faller, then , w.p
Theorem: The expected number of jumps is

20. Jumping on a DAG Theorem: The expected number of jumps is .
Challenge: Use more AUSO properties

21. Random-Edge Consider t consecutive steps of Random-Edge
If the degrees of intermediate vertices are high, a “jump” can reach many different vertices
But,
Some intermediate degrees may be low
The distribution over vertices reached is not uniform

22. [Hansen-Paterson-Z (2013)]
Random-Edge
A vertex u is (t,k)-good if any vertex that can be reached from it in t steps has degree  k
[Gärtner-Kaibel (2007)]
A vertex u is (t,k)-clean if with high enough probability the vertices on a random path of length t have increasing Hamming distance from u and are all of degree  k
[Hansen-Paterson-Z (2013)]

23. Open problems
Are Random-Edge and Random-Jump exponential or sub-exponential?
Is there an algorithm that is faster than Random-Facet?
Is there a polynomial algorithm for finding the sink of an AUSO?
Is there a polynomial time pivoting rule for the simplex algorithm?

24. THE END