Aarhus Univ Uri Zwick Tel Aviv Univ RandomEdge is slower than RandomFacet on abstract cubes Hebrew University April 11 2016 Maximize a ID: 806532
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Slide1
Thomas Dueholm Hansen – Aarhus Univ.Uri Zwick – Tel Aviv Univ.
Random-Edge is slower than Random-Facet on abstract cubes
Hebrew University
April 11, 2016
Slide2Maximize a linear objective function subject to a set of linear equalities and inequalities
Linear ProgrammingFind the highest point in a polytope/polyhedron
Slide3Move
up, from vertex to vertex, along edges, until reaching the top.The Simplex Algorithm[Dantzig (1947)]
Slide4Largest improvementLargest slopeDantzig’s rule – Largest modified costBland’s rule – avoids cycling
Lexicographic rule – also avoids cyclingZadeh’s rule – “least recently entered”Deterministic pivoting rulesAll known to require an exponential number of steps, in the worst-case
[Klee-Minty (1972)] ,
… ,
[
Amenta
-Ziegler
(1996
)]
[
Friedmann
(2011)]
Slide5Random-EdgeChoose a random improving edgeRandomized pivoting rules
Random-Facet is sub-exponential! Random-Facet[Kalai (1992)] [Matoušek-Sharir-Welzl (1992)]To be described shortlyIs
Random-Edge
sub-exponential
???
Is
Random-Edge
faster than
Random-Facet
?
Slide6Random-FacetAssume
[
Kalai
(1992)]
[
MSW (1992)]
= number of inequalities
= number of variables = dimension
[Hansen-Z (2015)]
(following
[
Kalai
(1992)]
)
Fastest known
(randomized)
pivoting rule
Slide7Random-FacetAssume
[
Kalai
(1992)]
[
MSW (1992)]
= number of inequalities
= number of variables = dimension
[Hansen-Z (2015)]
(following
[
Kalai
(1992)]
)
Fastest known
randomized
pivoting rule
Slide8The diameter of a d-dimensional, n-faceted polytope is at most
n−dRefuted by [Santos (2010)].Diameter is still believed to be polynomial.
Quasi-polynomial
upper bound
[
Kalai-Kleitman
(1992)] ([Todd (2014)])
Hirsch
Conjecture
(1957)
Slide9Abstract objective functions (AOFs)
Every
face
has a unique sink
Acyclic Unique Sink Orientations (AUSOs)
Slide10Klee-Minty cubes (1972)Figure taken from a paper by Gärtner-Henk-Ziegler
Slide11AUSOs of n-cubesThe diameter
is exactly n
Williamon
-Hoke
(1988)
Kalai
(1998)
Szabó
,
Welzl
(2001)
Gärtner
(2002)
AUSOs
No diameter issue!
2
n
facets
2
n
vertices
Every
subcube
has a unique
sink
Slide12Klee-Minty
cubes (1972)
000
100
001
010
110
…
has a
Hamiltonian path
(Gray code)
.
Random-Edge
on
takes
,
w.h.p
.
[
Balogh-Pemantle
(2007)]
011
111
101
Slide13Turn-based
2-Player
0-Sum
Stochastic Games
[Shapley ’53] [Gillette ’57] … [Condon ’92]
Both players have optimal
positional
strategies.
Can optimal strategies be found in
polynomial
time?
Limiting average
cost
Discounted
cost
Total cost
Game (two actions per state)
AUSO (of a cube)
Random-Facet
is the fastest known algorithm.
Slide14Parity Games (PGs) A simple example214
132
EVEN
wins if largest priority
seen
infinitely often
is
even
Priorities
Slide15Random-Facet on the
-cube[Ludwig (1995)] [Gärtner (2002)] Start at some vertex of the -cube
.
Split the
-cube into
two
-cubes
along a
random
coordinate
.
Recursively find the sink
of the
-cube
containing
.
If
is not the global sink,
move to
.
Recursively find the sink
of the
-cube
containing
,
starting from
.
…
-
th
coordinate
Exponential?
Sub-exponential!
The starting point
of the
second recursive call contains
valuable information
.
Slide16All 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.
(After reordering the coordinates according
to the
hidden
order.)
Random-Facet
on the
-cube
[Ludwig (1995)] [
Gärtner
(2002)]
The algorithm does not know the
hidden
order.
But, choosing a coordinate that was
fixed
has no effect.
Thus, the second recursive call is effectively on an
-cube.
Slide17All correct !
Would never be switched !
Random-Facet
on the
-cube
[Ludwig (1995)] [
Gärtner
(2002)]
(After reordering the coordinates according
to the
hidden
order.)
Slide18Primal Random-Facet Non-recursive versionChoose a random permutation of the facets
containing the current vertex v.Find the first facet that is beneficial to leave and move to a new vertex contained in a new facet .
Choose a new random ordering of
. Keep the ordering
of
. Repeat.
Slide19Upper boundLower boundAlgorithmRANDOM EDGERANDOM FACET
Randomized Pivoting Rules
[
Kalai
’92]
[
Matousek-Sharir-Welzl
’92]
[
Friedmann
-Hansen-Z ’11]
Slide20Lower bounds for Random-Edge
for AUSOs [Matoušek-Szabó (2006)]
for
LP
s
[
Friedmann
-Hansen-Z (2011)]
for
AUSOs
[here]
The new lower bound is a
simplification
of the lower
bound of
Matoušek
and
Szabó
obtained by replacing the
Klee-Minty
cube, used as a building block, by a
path AUSO
.
Two main building blocks:
Product
of AUSOs
,
Hypersink
replacement
Main part of the technical analysis:
Random Walk with reshuffles
(on a path AUSO)
Slide21A
product
(
blowup
) construction
Slide taken from a presentation by
Tibor
Szabó
.
Slide22A
product
(
blowup
) construction
Adaptation of a slide by
Tibor
Szabó
.
(
y
))
Slide23Hypersink
replacement
Slide
by
Tibor
Szabó
.
Slide24A
A
A
A
rand A
A simpler construction
Slide25R
andomized
product
+
hypersink
replacement
obtained from
by
randomly
permuting
the indices.
Adaptation of a slide by
Tibor
Szabó
.
The copy of
corresponding
to
is a
hypersink
.
Replace it by a
random
translation
of
.
11
Slide26Random-Edge
on a random product[Matoušek-Szabó (2006)]
Each step in
is either
a step in
, i.e.,
, or a step in
, i.e.,
A step
in
, does not change
,
but changes the cube
is in from
to
.
The induced process on
, until
, is exactly
Random-Edge
on
.
T
he induced process on
, is a
Random Walk with Reshuffles
on
.
As
is a random permutation of
,
this corresponds to
randomly
permuting
, a
reshuffle
.
Once
, the orientation on
changes from
to
.
As
is a
random
translation of
,
is a completely
random
in
.
If
reaches
before
reaches
,
we get
two
Random-Edge
walks on
.
Slide27The probability of reshuffle in each step of Random-Reshuffle on depends on
. (To be discussed later.) Lemma: If the probability that Random-Edge on makes less than steps is at most
, and the probability that
Random-Reshuffle
on
makes less than
steps is at most
,
both starting from a random vertex, then the probability that
Random-Edge,
starting at a
random
vertex
,
makes
less than
steps on
is at most
.
Random-Edge
on a
random
product
[
Matoušek-
Szabó
(2006
)]
Slide28Lower bound Construction
[Matoušek-Szabó (2006)]
Let
be an
-AUSO. Compute randomized
powers
of
.
,
, where
, for some
.
If
Random-Reshuffle
on
is
slow enough
, then
Random-Edge
on
, which is an
-AUSO,
makes at least
steps, with high probability.
In the last step, we need
Random-Reshuffle
on
to make at least
steps, with high probability.
Matoušek-Szabó
take
to be the
Klee-Minty
cube.
Random-Reshuffle
on Klee-Minty
cubes is not slow enough…
Further complications. Lower bound becomes
.
Our modification:
Take
to be the
path AUSO
.
Slide29A
path AUSO
A
shortest paths
problem gives rise to an AUSO.
A vertex of the AUSO corresponds to a choice of
an outgoing edge from each vertex of the graph.
In this particular case, the
-
th
bit wants to
stay or become a 1
iff
all preceding bits are 1.
Slide30Random-Reshuffle
on a path AUSO
1111
0
0
1
0
1
in
– number of leading 1s
– number of
non-leading
1s
–
outdegree
of
in
in
Outdegree =
With probability
,
reshuffle
.
Otherwise, with probability
, change the first
0
to 1,
and probability
, change a random non-leading
1
to 0.
The
reshuffle
probability
chosen by an
adversary
.
Slide31Random-Reshuffle
on a path AUSO1111
0
0
1
0
1
– number of leading 1s
– number of
non-leading
1s
in
All states of
type
are obtained with equal probabilities.
type
=
weight
=
A
reshuffle
on a state of
weight
creates a state
of
type
, where
,
with probability
The probability of increasing or decreasing
depends only on
.
We thus obtain induced random walks on
types
, and then on
weights
.
Slide32-cube
Types
0
Constant
drift
to the left
Exponential time to reach
.
Weights
Slide33Induced
random walk on the lineSlight complication:
From
, the probability of
+1
is greater than
½
.
Solution: Consider
two
steps together.
Basic step: Zero or more
reshuffles
, followed by a step in
.
Several
reshuffles
in a row are equivalent to a single
reshuffle.
The worst type of
weight
is
.
Slide34Induced
random walk on the lineProbability of weight increase in one step from
is the reshuffle probability
,
Slide35Induced
random walk on the lineProbability of weight decreasing by 2:
Slide36Concluding remarks and open problemsWe presented an
lower bound for Random-Edge. The bound can be slightly improved to
,
thus
Random-Edge
is
slower
than
Random-Facet
.
This is the best that can be obtained
using
hierarchical
constructions.
Best upper bound known for
Random-Edge
is
.
Is
Random-Edge
sub-exponential?
Is there an algorithm that beats
Random-Facet
?
Is there a
polynomial
time algorithm?
(The algorithm does not necessarily
have to follow a path. It may
jump
around.)
Slide37( Stochastic Shortest Paths (SSPs) )
Minimize the
expected
cost
of getting to the target
Markov Decision Processes (MDPs)
Slide38Random-Facet on the -cube[Kalai (1992)]
[Matoušek-Sharir-Welzl (1992)][Gärtner (2002)] Let
be the current vertex.
Choose a
random
index
.
Recursively
find the sink
of the
subcube
.
If
, then
is the sink of the whole cube.
Otherwise, move from
to
and
run the algorithm recursively from
.
Slide39Analysis of Random-Facet on the -cube
Let be an upper bound on the expected number of steps performed on any -AUSO from any starting point.
Slide40Analysis of Random-Facet on the -cube
Let such that
implies
Let
be the current vertex.
Let
be the optimal vertex
in the
subcube
.
Renumber the indices such that
.
The algorithm chooses a random index
and performs a first recursive call that computes
.
Let
.
Slide41Analysis of Random-Facet on the -cube
Let such that
implies
Let
be the current vertex.
Let
be the optimal vertex
in the
subcube
.
Renumber the indices such that
.
The algorithm chooses a random index
and performs a first recursive call that computes
.
Let
.
If
is not the global sink, the algorithm moves to
.
.