On the randomized simplex algorithm in abstract cubes

Jiři Matoušek Charles University

Prague

Tibor SzabóETH Zürich

Linear Programming ------ the geometric view

• Given a convex polytope P in Rn with m facets and a linear objective function c,

• Find the minimum value of c on P.

• The minimum is taken at a vertex of P.

• A simplex algorithm moves from vertex to vertex along an edge each time decreasing the objective function value.

Pivot Rules

• Which improving edge to choose: the pivot rule • No deterministic pivot rule is known to yield a

polynomial or even subexponential running time. In fact almost all pivot rules are known to have bad instances.

• Randomized pivot rules are a bit more succesful. There is a subexponential randomized pivot rule and there are no known superpolynomial lower bounds for any decent randomized pivot rule.

LP Algorithms

• Simplex method [Dantzig 1947] – very fast in practice– very good “average case” – very bad/unknown “worst-case”

• Ellipsoid method [Khachyian], interior-point methods [Karmakar],…– weakly polynomial but NO (worst-case) bound

in terms of n and m alone

Abstract frameworks

• Abstract objective functions

• Acyclic unique sink orientations• LP-type problems [Sharir, Welzl]

• Abstract optimization problems [Gärtner]

Abstract Objective Functions

• P is a polytope, f : V(P) → R is a function

• f is unimin on P if there is no local minima other than the global minima.

• f is an abstract objective function on P if it is unimin on any face F of P.

Adler and Saigal, 1976.

Williamson Hoke, 1988.

Kalai, 1988.

Unimin functions on the cube

• Any randomized algorithm needs at least queries for some unimin function on the hypercube [Aldous ’84]

• There is a (simple) randomized algorithm which works in steps

• Improvement: [Aaronson, ’04]• Quantum query complexity

)(22non

2122 nn

222 nn nn 42

RandomFacet on AOF

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

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

• Still the best known!

• Matoušek gave AOFs on which Kalai’s analysis is essentially tight.

nOe

RandomEdge• RandomEdge is the simplex algorithm which

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

– on the n-dimensional simplex is Liebling

– on n-dimensional polytopes with n+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

)(logn

)(log2 n

)( 2nO

RandomEdge on AOFs

• RandomEdge is quadratic on Matoušek’s orientations (which kill RandomFacet)

• Williamson Hoke (1988) conjectured that RandomEdge is quadratic on all AOFs. (cf. Tovey, 1997)

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

RandomEdge is slow

Theorem. [Matoušek, Sz., FOCS’04] 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

• Klee-Minty cube• Blowup construction [Schurr-Sz., ‘02]

• Hypersink reorientation [Schurr-Sz., ‘02]

• Randomness

Klee-Minty cube

111

iiixxx

101x

ni 22/10

Blowup Construction

A very special case: the Klee-Minty cube

reversed KMm-1

KMm-1

KMm

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 permute (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 )(2)2( nTnT n

Difficulties…

• 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 .

31cne

m

31nm

ikmn

31Cnk

Ai Rand KMm

Hypersink reorientation to ensure that when the walk enters the sink of any of the small blocks it enters a random copy of Ai on the

first n coordinate

Ai is an (n+ikm)-cube,

m

k

A0 is an arbitrary n-cube

constrcut Ai+1 from Ai recursively

mn

Claim: The first 2i steps visit vertices with outdegree at least k

Ai Rand KMm

When the walk enters the sink of any of the small blocks it enters a random copy of Ai on the first

n coordinate

m

k

1. Phase: first 2i steps (Note: k≥11m)2. Phase: in between (still no KMm is in its sink)

3. Phase: one of the KMm is in its sink

Proof: induction on i

31Cnk 31nm

Conclusion: The first 2t steps of RandomEge

in the 2n-dimensional cube At visit vertices with outdegree at least k

At is a (n+tkm)-cube,

Choose Cnkmnt 31

An upper bound, please!

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

Best known upper bound is ,

where p(n) is an arbitrary polynomial [Gärtner and Kaibel, ’05]

• Find an algorithm which gets to the minima of AOFs on the n-cube faster than exp(n)

)(2 npn

BottomTop

• From v move to the sink in the subcube spanned by the outgoing edges. (Note: BottomTop is NOT an algorithm!) [suggested by Kaibel]

Theorem [Schurr, Sz., IPCO’05]

There is an AUSO of the n-cube on which BottomTop, starting at a random vertex, takes at least c2n/2 steps.

Lower bounds

• Improve on the current modest lower bounds for AUSOs:

Deterministic complexity: Ω(n2/log n)

Randomized complexity: Ω(n)

Realizability

• Can one modify the construction such that the cube is realizable? (Probably not …)

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

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

Unique Sink Orientations of Cubes

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

Find a subexponential algorithm!

THE END