Transcript 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,

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
Linear Programming
Maximize a linear objective function subject
to a set of linear equalities and inequalities
Find the highest point in a polytope
The Simplex Algorithm
[Dantzig (1947)]
Move up, along an edge to a neighboring
vertex, until reaching the top
Upper bound Theorem for polytopes
[Klee (1964)] [McMullen (1971)]
Number of vertices of a d-dimensional
n-faceted polytope is at most:
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)
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”
Is there a
polynomial pivoting rule?
Is the diameter 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)]
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 ???
Random-Facet algorithm(s)
Upper bound
on diameter
[Kalai-Kleitman
(1992)]
Random-Facet
primal version
[Kalai (1992)]
Seidel’s
randomized
LP algorithm
[Seidel (1991)]
Random-Facet
dual version
[Matoušek
Sharir-Welzl
(1992)]
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
Upper bound on diameter
[Kalai-Kleitman (1992)]
Upper bound on diameter
[Kalai-Kleitman (1992)]
Exercise: Verify and complete the details.
“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.
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.
(Toy) Exercise:
Primal Random-Facet in 2D
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
Solving the recurrence
Exercise: Verify and justify (or find a bug…)
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?
(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 ???
Linear Programming Duality
d variables
n inequalities
n variables
n inequalities
d equalities
n-d free variables
n inequalities
Linear Programming Duality
d variables
n inequalities
n-d free variables
n inequalities
Clarkson’s Algorithm [Clarkson (1988)]
No need to consider the case n>9d2
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.
Dual Simplex Algorithms
Dual Simplex Algorithms
Dual Simplex Algorithms
Dual Pivot Step
Exercise: What is the complexity
of a dual pivoting step?
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.
Dual Random Facet - Analysis
[Matoušek-Sharir-Welzl (1992)]
Hidden
dimension
Primal vs. Dual
Exercise: Coincidence?
“It is remarkable to see how
different paths have led to
rather similar results so
close in time.”
[Kalai (1992)]
Abstract objective functions (AOFs)
Acyclic Unique Sink Orientations (AUSOs)
Every face should
have a unique sink
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!
Klee-Minty cubes (1972)
Taken from a paper by Gärtner-Henk-Ziegler
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.
Analysis of Random-Facet-1
“[Kalai (1992)]”
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