Transcript Unique Sinks are Stupid

On the randomized simplex
algorithm in abstract cubes
Jiři Matoušek
Tibor Szabó
Charles University
Prague
ETH 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
n o ( n )
2
• Any randomized algorithm needs at least 2
queries for some unimin function on the
hypercube [Aldous ’84]
• There is a (simple) randomized algorithm which
n 2 12
works in 2 n steps
• Improvement: 2n 2 n 2  [Aaronson, ’04]
n4
• Quantum query complexity  2 n


RandomFacet on AOF
• Kalai (1992): the simplex algorithm
RandomFacet finishes in subexponential time
 
on any AOF. ( e
in cubes.)
O
n
(also: Matoušek, Sharir and Welzl in a dual setting)
• Still the best known!
• Matoušek gave AOFs on which Kalai’s analysis
is essentially tight.
RandomEdge
• RandomEdge is the simplex algorithm which
selects an improving edge uniformly at random.
• Its running time
– on the n-dimensional simplex is
(log n)
Liebling
– on n-dimensional polytopes with n+2 facets is
(log2 n)
Gärtner et al. (2001)
– on the n-dimensional Klee-Minty cube is
O(n 2 )
Williamson Hoke (1988)
(n2 log n) Gärtner, Henk, Ziegler (1995)
(n2 )
Balogh, Pemantle (2004)
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,
 cn1 3
with probability at least 1  e ,
cn1 3
makes at least e
moves before reaching the sink.
Ingredients
•
•
•
•
Klee-Minty cube
Blowup construction
Hypersink reorientation
Randomness
[Schurr-Sz., ‘02]
[Schurr-Sz., ‘02]
Klee-Minty cube
0  x1  1
xi1  xi  1  xi1
2in
0    1/ 2
Blowup Construction
A very special case:
the Klee-Minty cube
reversed
KMm-1
KMm
KMm-1
Hypersink reorientation
A simpler construction
Let A be an n-dimensional cube, on which
RandomEdge is slow.
Let
m  n.
• 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.
A simpler construction
A
A
A
rand
AA
A typical RandomEdge move
v
• Move in frame:
– RandomEdge move in KMm
A
– Stay put in A
• Move within a hypervertex:
– RandomEdge move in A
– Move to a random vertex of
KMm on the same level
A
A
rand A
RandomEdge on A
Random walk with reshuffles on KMm
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
min pi,resh  11 max pi,step
m
Then with probability at least 1  e
the random walk with reshuffles makes
m
at least e steps. (α and β are constants)
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 .
m
This takes at least e  T (n) time.
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 T (2n)  2 n T (n)
Difficulties…
•
In order to guarantee that reshuffles are frequent
enough we need a more complicated
construction and that is why1we
are only able to
3
prove a running time of ecn
.
A0 is an arbitrary n-cube
constrcut Ai+1 from Ai recursively
Ai is an (n+ikm)-cube,
k  Cn
13
Rand KMm
Ai
n  ikm
mn
13
m
m
k
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
Rand KMm
Ai
m
n
m
k
When the walk enters the sink of any of the small blocks
it enters a random copy of Ai on the first n coordinate
Claim: The first 2i steps visit vertices with outdegree at least k
Proof: induction on i
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
At is a (n+tkm)-cube,
k  Cn
13
Choose t  n km  n
13
mn
C
Conclusion:
The first 2t steps of RandomEge in
the 2n-dimensional cube At visit
vertices with outdegree at least k
13
An upper bound, please!
• Obtain any reasonable upper bound on the
running time of RandomEdge
Best known upper bound is 2n p(n) , 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)
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