RandomEdge can be mildly exponential on

abstract cubes

Jiri Matousek Charles University

Prague

Tibor SzabóETH Zürich

Linear Programming

• Given a convex polyhedron P in Rn with at most m facets and a linear objective function c, one would like to determine the minimum value of c on P.

• The minimum is taken at a vertex of P.• The simplex algorithm moves from vertex to

vertex along an edge each time decreasing the objective function value.

• The way to select the next vertex is the pivot rule

RandomEdge• RandomEdge is the simplex algorithm which

selects an improving edge uniformly at random.• Its running time

– on the d-dimensional simplex is Liebling

– on d-dimensional polytopes with d+2 facets is Gärtner et al. (2001)

– on the n-dimensional Klee-Minty cube is Williamson Hoke (1988)

Gärtner, Henk, Ziegler (1995)

Balogh, Pemantle (2004)

)log( 2 nn)( 2n

)(logd

)(log2 d

)( 2nO

Abstract Objective Functions

• P is a polytope

• f : V(P) → R is an abstract objective function if a local minimum of any face F is also the unique global minimum of F. Adler and Saigal, 1976.

Williamson Hoke, 1988.

Kalai, 1988.

RandomFacet on AOF

• Kalai (1992): the simplex algorithm RandomFacet finishes in subexponential time on any AOF.

(also: Matousek, Sharir and Welzl in a dual setting)

• Matousek gave AUSOs on which Kalai’s analysis is essentially tight.

• RandomEdge is quadratic on Matousek’s orientations

• Williamson Hoke (1988) conjectured that RandomEdge is quadratic on all AOFs.

Acyclic Unique Sink Orientations

• Let P be a polytope. An orientation of its graph is called an acyclic unique sink orientation or AUSO if every face has a unique sink (that is a vertex with only incoming edges) and no directed cycle.

• AUSOs and AOFs are the same

Killing RandomEdge

Theorem. There exists an AUSO of the

n-dimensional cube, such that

RandomEdge started at a random vertex,

with probability at least ,

makes at least moves before reaching the sink.

31

1 cne31cne

Ingredients of the good pasta

• The flour:

• The water:

• The eggs:

• The mixing:

Ingredients of a slow cube

Klee-Minty cube

Blowup construction

Hypersink reorientation

Randomness

Klee-Minty cube

reversed KMm-1

KMm-1

KMm

Blowup Construction

Hypersink reorientation

A simpler construction

Let A be an n-dimensional cube, on which RandomEdge is slow.

Let .

• Take the blowup of A with random KMm whose sink is in the same copy of A

• Reorient the hypersink by placing a random copy of A.

nm

A

A

A

A

rand A

A simpler construction

A typical RandomEdge move

• Move in frame:– RandomEdge move in KMm

– Stay put in A

• Move within a hypervertex:– RandomEdge move in A– Move to a random vertex of

KMm on the same level

A

rand A

A

A

v

Random walk with reshuffles on KMm

RandomEdge on A

Walk with reshuffles on KMm

• Start at a random v(0) of KMm

• v(i) is chosen as follows:– With probability pi,step we make a step of RandomEdge from v(i-1).

– With probability pi,resh we reshuffle the coordinates of v(i-1) to obtain v(i) .

– With probability 1- pi,step - pi,resh, v(i) = v(i-1).

Walk with reshuffles on KMm is slow

Proposition. Suppose that

Then with probability at least

The random walk with reshuffles makes

at least steps. (α and β are constants)

stepireshi pp ,, max11min me 1

me

Reaching the hypersink

Either we reach the sink by reaching the sink of a copy of A and then perform RandomEdge on KMm. This takes at least T(n) time.

Or we reach the hypersink without entering the sink of any copy of A. That is the random walk with reshuffles reaches the sink of KMm .

This takes at least time.)(nTe m

The recursion

• RandomEdge arrives to the hypersink at a random vertex. Then it needs T(n) more steps.

So passing from dimension n to n+n the expected running time of RandomEdge doubles.

Iterating n - times gives • In order to guarantee that reshuffles are frequent

enough we need a more complicated construction and that is why we are only able to prove a running time of .

)(2)2( nTnT n

31cne

Open questions

• Obtain any reasonable upper bound on the running time of RandomEdge

• Can one modify the construction such that the cube is realizable? (I don’t think so …)

• Or at least it satisfies the Holt-Klee condition?

• Or at least each three-dimensional subcube satisfies the Holt-Klee condition?

More open questions

• The model of unique sink orientations of cubes (possibly with cycles) include LP on an arbitrary polytope.

Find a subexponential algorithm.

THE END