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

Maximize a linear objective function subject to a set of linear equalities and inequalities

Linear Programming

Find the highest point in a polytope

Move up, along an edge to a neighboringvertex, until reaching the top

The Simplex Algorithm[Dantzig (1947)]

Upper bound Theorem for polytopes[Klee (1964)] [McMullen (1971)]

Number of vertices of a d-dimensional n-faceted polytope is at most:

Largest improvementLargest slope

Dantzig’s rule – Largest modified costBland’s rule – avoids cycling

Lexicographic rule – also avoids cycling

Deterministic pivoting rules

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

Ellipsoid (Khachiyan 1979)Interior-point (Karmakar 1984)

Smoothed analysis based (ST’04, KS’06)Other (DV’08, …)

Simplex (Dantzig 1947)No polynomial versions known

Polynomial, but not strongly polynomialNot “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)]

Random-EdgeChoose a random improving edge

Randomized pivoting rules

Random-Facet is sub-exponential!

Random-FacetTo be explained shortly ☺

[Kalai (1992)] [Matoušek-Sharir-Welzl (1996)]

Are Random-Edge and Random-Facet polynomial ???

Random-Facet algorithm(s)

Upper boundon diameter

[Kalai-Kleitman (1992)]

Random-Facetprimal version[Kalai (1992)]

Seidel’s randomized

LP algorithm[Seidel (1991)]

Random-Facetdual version[Matoušek

Sharir-Welzl (1992)]

Orient the polytope such that the target vertex is the highest

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

Active faceta facet containing a vertex (strictly)

higher than the current vertex

Bound the monotone diameter

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

Exercise: Verify and complete the details.

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

“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)]

Once a right bound is known, it is not too hard to verify it by induction

Seidel’s challenge: Can it be taught?

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 variablesn inequalities

n variablesn inequalitiesd equalities

n-d free variablesn inequalities

Linear Programming Duality

d variablesn inequalities

n-d free variablesn 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.

Hidden dimensio

n

Dual Random Facet - Analysis[Matoušek-Sharir-Welzl (1992)]

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)

Every face shouldhave a unique sink

Acyclic Unique Sink Orientations (AUSOs)

AUSOs of n-cubes

The diameter is exactly n

Stickney, Watson (1978)Morris (2001)

Szabó, Welzl (2001)Gärtner (2002)

USOs and AUSOs

No diameter issue!

2n facets2n vertices

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