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