Linear Programming, the Simplex Algorithm and Simple Polytopes
Linear Programming the Simplex Algorithm and Simple Polytop es
Gil Kalai
Institute of Mathematics Hebrew University of Jerusalem Jerusalem Israel
e mail kalai math huji ac il
May
Abstract
In the rst part of the pap er we survey some far reaching applications of the basic facts
of linear programming to the combinatorial theory of simple p olytop es In the second part we
discuss some recent developments concerning the simplex algorithm We describ e sub exp onential
randomized pivot rules and upp er b ounds on the diameter of graphs of p olytop es
Intro duction
d
A convex p olyhedron is the intersection P of a nite numb er of closed halfspaces in R P is a
d
d dimensional p olyhedron brie y a d p olyhedron if the p oints in P a nely span R A convex
d dimensional p olytop e brie y a d p olytop e is a bounded convex d p olyhedron Alternatively a
d
convex d p olytop e is the convex hull of a nite set of p oints which a nely span R
A nontrivial face F of a d p olyhedron P is the intersection of P with a supp orting hyp erplane
F itself is a p olyhedron of some lower dimension If the dimension of F is k we call F a k face of
P The empty set and P itself are regarded as trivial faces faces of P are called vertices faces
are called edges and d faces are called facets For material on convex p olytop es and for many
references see Ziegler s recent b o ok
The set of vertices and b ounded edges of P can b e regarded as an abstract graph called the
graph of P and denoted by G P
We will denote by f P the numb er of k faces of P The vector f P f P f P is
k d
called the f vector of P Euler s famous formula V E F gives a connection b etween the
numb ers V E F of vertices edges and faces of every p olytop e
A convex d p olytop e or p olyhedron is called simple if every vertex of P b elongs to precisely
d edges Simple p olyhedra corresp ond to non degenerate linear programming problems When you
cut a simple p olytop e P near a vertex v with a hyp erplane H which intersect the interior of P the
intersection P H is a d dimensional simplex S The vertices of S are the intersections of
edges of P which contain v with H and the k dimensional faces of S are the intersection of
k faces of P with H The following basic prop erty of simple p olytop es follows
Let P b e a simple d p olytop e and let v b e a vertex of P Every set of k edges adjacent to v
determines a k dimensional face of P which contains the vertex v
d
d
In particular there are precisely k faces in P containing v and altogether faces of all
k
dimensions which contain v
Linear programming and the simplex algorithm
Linear programming is the problem of maximizing a linear ob jective function sub ject to a nite
set of linear inequalities The relevance of convex p olyhedra to linear programming is clear The
set P of feasible solutions for a linear programming problem is a p olyhedron
There are two fundamental facts concerning linear programming the reader should keep in mind
If is b ounded from ab ove on P then the maximum of on P is attained at a face of P in
particular there is a vertex v for which the maximum is attained If is not b ounded from
ab ove on P then there is an edge of P on which is not b ounded from ab ove
A su cient condition for v to b e a vertex of P on which is maximal is that v is a local
maximum namely v is bigger or equal than w for every vertex w which is a neighb or
of v
The simplex algorithm is a metho d to solve a linear programming problem by rep eatedly moving
from one vertex v to an adjacent vertex w of the feasible p olyhedron so that in each step the value
of the ob jective function is increased The sp eci c way to cho ose w given v is called the pivot rule
The d dimensional simplex and the d dimensional cub e
d
The d dimensional simplex S is the convex hull of d a nely indep endent p oints in R The
d
faces of S are themselves simplices In fact the convex hull of every subset of vertices of a simplex
d
d
is a face and therefore f S The graph of S is the complete graph on d vertices
k d d
k
d
The d dimensional cub e C is the set of all p oints x x x in R such that for every i
d d
x The vertices of C are all the vectors of length d and two vertices are adjacent
i
d
d
d k
in the graph of C if they agree in all but one co ordinates f C
d k d
k
Applications of the fundamental prop erties of linear program
ming to the combinatorial theory of simple p olytop es
Let P b e a simple d p olytop e and let b e linear ob jective function which attains di erent values
on di erent vertices of P Call such a linear ob jective function generic Actually it will b e enough
to assume only that is not constant on any edge of P
The fundamental fact concerning linear programming is that the maximum of on P is attained
at a vertex v and that a su cient condition for v to b e the vertex of P on which is maximal is
that v is a local maximum namely v is strictly bigger than w for every vertex w which is a
neighb or of v
Every face F of P is itself a p olytop e and attains di erent values on distinct vertices of F
Among the vertices of F there is a vertex on which is maximal and again this vertex is the only
vertex in F which is a lo cal maximum of in the face F
These considerations have far reaching applications on the understanding of the combinatorial
structure of simple p olytop es We refer the reader to Ziegler s b o ok for historical notes and for
references to the original pap ers Our presentation is also quite close to that in We hop e that
the theory of h numb ers describ ed b elow will re ect back on linear programming but this is left to
b e seen
Degrees and h numb ers
Let P b e a simple d p olytop e and let b e a generic linear ob jective function For a vertex v of P
de ne the degree of v denoted by deg v to b e the numb er of its neighb oring vertices with smaller
value of the ob jective function Clearly deg v d
De ne now h P to b e the numb er of vertices of P of degree k This numb er as we de ned it
k
dep ends on the ob jective function but we will so on see that it is actually indep endent from
We can see one sign for this already no matter what is there will always b e precisely one vertex
of degree d on which attains the maximum and one vertex of degree on which attains the
minimum This follows at once from the fact that lo cal maximum global maximum
To continue we will count pairs of the form F v where F is a k face of P and v is a vertex of
F which is a lo cal maximum hence a global maximum of in F
On the one hand the numb er of such pairs is precisely f P the numb er of k faces of P This
k
is b ecause every k face has a unique lo cal maximum
On the other hand let us compute how many pairs contain a given vertex v of P This dep ends
only on the degree of v Assume that deg v r and consider the set of edges of P
T f v w v w g
Thus jT j r As we mentioned ab ove every set S of k edges containing v determines a k face F S
containing v In this face the set of edges containing v is precisely S In order for v to b e a lo cal
maximum in this face it is necessary and su cient that for every edge v w in S v w
This o ccurs if and only if S T Therefore the numb er of k faces containing v for which v is a
r
lo cal maximum is precisely the numb er of subsets of T of size k namely
k
Summing over all vertices v of P and recalling that h P denotes the numb er of vertices of
k
degree k we obtain
d
X
r
h P f P k d
r
k
k
r
Note that this formula describ es the f vector of P f P f P f P as an upp er
d
triangular matrix with ones on the diagonal times the h vector of P h P h P h P
d
Therefore the h numb ers are in fact linear combinations of the face numb ers and in particular
they do not dep end on the linear ob jective function
Put
d
X
k
F x f P x
P
k
k
and
d
X
k
H x h P x
P
k
k
Relation gives
d
X
r
H x h P x
P r
r
d d d
X X X
r
k k
h P x f P x F x
r P
k
k
r
k k
Therefore H x F x and
P P
d
X
r
r k
h P f P
r
k
k
r
In particular
d
h P f P f P f P f P
d
d
h P f P f P f P df P
d
d
d
h P f P f P f P f P
d
h P f P
d d
h P f P d
d d
d
h P f P d f P
d d d
For the simplex S h for every k The graph of S is the complete graph on d vertices
d k d
and for every generic ob jective function there will by precisely one vertex of degree k for k d
d
For the cub e C h To see this consider the ob jective function which is the sum of the
d k
k
co ordinates This is not a generic ob jective function but it is not constant on edges of the p olytop e
and this is su cient for our purp oses The vertices of degree k are precisely those having v k
d
and there are such vertices
k
Euler Formula and the Dehn Sommerville Relations
For a generic linear ob jective function there is a unique maximal vertex and a unique minimal
vertex Therefore h P h P and by the formulas ab ove we obtain f P f P f P
d
d
f P which is Euler Formula usually written
d
d d
f P f P f P f P
d
More generally if is a generic linear ob jective function then so is However if v is a vertex
of a simple p olytop e P and v has degree k w r t then v has degree d k w r t
This gives the Dehn Sommerville relations
h P h P
k d k
The Dehn Sommerville relations are the only linear equalities among face numb ers of simple
d p olytop es
The cyclic p olytop es
The cyclic d p olytop e with n vertices C d n is the convex hull of n distinct p oint on the moment
d d
curve x t t t t R This is a remarkable class of p olytop es and the reader should
consult for their prop erties C d n will denote a p olar p olytop e to C d n For the
de nition of p olarity see C d n is a simple d p olytop e with n facets
The upp er b ound theorem
Motzkin conjectured that the maximal numb er of vertices and more generally of k dimensional
faces for d p olytop es with n facets is attained by C d n the p olar to cyclic d p olytop es with n
facets This conjecture was proved by McMullen It is easy to reduce this conjecture to simple
p olytop es and to calculate the h numb ers of C d n see This gives
n d k
h C d n h C d n
k d k
k
for k d
Since the face numb ers are linear combination of h numb ers with nonnegative co e cients the
upp er b ound theorem follows from the following relations and the Dehn Somerville relations
n d k
h P k d
d k
k
Pro of Consider a generic linear ob jective function which gives higher values to vertices in a
facet F than to all other vertices To construct such an ob jective function start with an ob jective
function whose maximum is attained precisely on the facet F and then make a slight p erturbation
to make it generic Every vertex v of degree k in F has precisely one neighb or not in F and
therefore the degree of v in P is k This gives
h F h P
k k
Next
X
h F k h P d k h P
k k k
where the sum is over all facets F of P
To prove consider a vertex v of degree k in P The vertex v is adjacent to d edges and every
subset of d out of them determine a facet The degree of v is k in every facet determined
by d edges adjacent to v where one of the k edges p ointing down w r t is deleted and there
are k such facets The degree of v is k in the remaining d k facets
n d k
and gives the upp er b ound relations h P by induction on k For k
d k
k
we have equality h n d For k we obtain
d
X
d k h P k h P h F n h P
d k d k d k d k
n d k
Therefore k h P n d k h P i e h P h P And
d k d k d k d k
k
assuming the upp er b ound relation for k we obtain for k
n d k n d k n d k
h P
d k
k k k
Abstract ob jective functions and telling the p olytop e from its graph
Consider an ordering of the vertices of a simple d p olytop e P For a nonempty face F we say that
a vertex v of F is a lo cal maximum in F if v is larger w r t the ordering than all its neighb oring
vertices in F An abstract objective function AOF of a simple d p olytop e is an ordering which
satis es the basic prop erty of linear ob jective functions
Every nonempty face F of P has a unique lo cal maximum vertex
If P is a simple d p olytop e and is a linear ordering of the vertices we de ne as b efore the
degree of a vertex v w r t the ordering as the numb er of adjacent vertices to v that are smaller
b e than v w r t Thus the degree of a vertex is a nonnegative numb er b etween and d Let h
k
the numb er of vertices of degree k Finally put F P to b e the total numb er of nonempty faces of
P
Claim
d
X
k
h F P
k
r
and equality hold if and only if the ordering is a AOF
Pro of Count pairs F v where F is a non empty face of P of any dimension and v is a
vertex which is lo cal maximum in F w r t the ordering On the one hand every vertex v of
k
degree k contributes precisely pairs F v corresp onding to all subsets of edges from v leading
P
d
k
h On the other to smaller vertices w r t Therefore the numb er of pairs is precisely
r k
hand the numb er of such pairs is at least F P every face has at least one lo cal maximum and
it is equal to F P i every face has exactly one lo cal maximum i e if the ordering is an AOF
Claim A connected k regular subgraph H of G P is the graph of a k face if and only if
there is an AOF in which all vertices in H are smaller than all vertices not in H
Pro of If H is the graph of a k face F of P then consider a linear ob jective function which
attains its minimum precisely at the p oints in F By de nition for every non trivial face such a
linear ob jective function exists Now p erturb a little to get a generic linear ob jective function
in which all vertices of H have smaller values than all other vertices
On the other hand if there is an AOF in which all vertices in H are smaller than all vertices
not in H consider the vertex v of H which is the largest w r t There is a k face F of P
determined by the k edges in H adjacent to v and v is a lo cal maximum in this face Since the
ordering is an AOF v must b e larger than all vertices of F hence the vertices of F are contained
in H and the graph of F is a subgraph of H But the only k regular subgraph of a connected
k regular graph is the graph itself and therefore H is the graph of F
Claims and provide a pro of to a theorem of Blind and Mani
Theorem The combinatorial structure of a simple polytope is determined by its graph
Indeed claim allows us to determine just from the graph all the orderings which are AOF s
Using this claim allows to determine which sets of vertices form the vertices of some k dimensional
face Let us mention that the pro of gives a very p o or algorithm exp onential in the numb er of
vertices and it is an op en problem to nd b etter algorithms
Further facts without such simple geometric pro ofs
One of the most imp ortant development in the theory of convex p olytop es is the complete descrip
tion of h vectors of simple d p olytop es conjectured by McMullen and proved by Stanley and Billera
and Lee See
A crucial part of this characterization is the following For every simple d p olytop e
h P h P h P
d
In words the numb er of vertices of degree k is smaller or equal than the numb er of vertices of
degree k when k d It is a challenging problem to nd a direct geometrical pro of for this
inequality The existing pro ofs have algebraic ingredients and are very di cult
The e ect of a single random pivot step on the degree
One p ossible measure for the progress of a certain pivot rule of the simplex algorithm would b e via
the degree of the vertices Unfortunately it seems di cult to predict how the degrees of vertices
will b ehave in a path of vertices given by some pivot rule
Starting with a random vertex of a simple p olytop e it is p ossible to say what will b e the e ect on
the degree of a single random pivot step By a random pivot step we mean the following Starting
with a vertex v we cho ose at random one of the d neighb oring vertices w If w v we move
to w and otherwise we stay at v
The average degree E P of vertices in a simple d p olytop e which is the exp ected degree of a
random vertex is by the Dehn Sommerville relations d The average degree E P of a vertex
of P obtained by a single random pivot step as describ ed ab ove starting from a random vertex
d
v is f P f P For example for the d cub e E P Similar formulas exist if we
cho ose at random an r face containing v and move from v to its highest vertex
To prove the formula for E P note that the probability that after one random pivot step we
k d Indeed if we start at w this o ccurs with reach a sp eci c vertex w of degree k is
f P
0
probability f P then with probability k d we stay at w If we start with one of the k lower
neighb ors of w altogether this o ccur with probability k f P then we reach w after one step
with probability d It follows that
d
X
E P k d h P
k
f P
k
which equals f P f P by the formulas ab ove Note that E P do es not dep end on the
ob jective function This is no longer true if we are interested in E P the average degree after two
random pivot steps The following problem of indep endent interest naturally arises
Problem Let P b e a simple d p olytop e and b e a generic linear ob jective function Let h
i j
b e the numb ers of pairs of adjacent vertices v w such that v w and deg v i deg w j
What can b e said ab out the collection of numb ers h i j d
i j
This array of numb ers dep ends on the ob jective function and not only on the p olytop e It will
b e interesting to describ e the p ossible h numb ers even for the sp ecial case when the p olytop e is
i j
combinatorially isomorphic to the d dimensional cub e The question is interesting also for abstract
ob jective functions
Arrangements
We would like to close this section with the following remark Consider an arrangement of n
d
hyp erplanes in general p osition in R and a generic linear ob jective function This arrangement
d
d
divides R into simple d p olyhedra The average value of h P over all these p olyhedra is
k
k
d
To see this just note that every vertex v in the arrangement b elongs to d p olyhedra and has
d
degree k in of these p olyhedra Similarly the average h vector over r dimensional faces of the
k
arrangement is the h vector of the r dimensional cub e
The Hirsch conjecture and sub exp onential randomized pivot
rules for the simplex algorithm
In this section we describ e recent developments concerning the simplex algorithm We describ e
sub exp onential randomized pivot rules and recent upp er b ounds for the diameter of graphs of
p olytop es The algorithms we consider should b e regarded in the general context of LP algorithms
discovered by Megiddo Clarkson Seidel Dyer Dyer and Frieze and many others
But we will not attempt giving this general picture here For the use of randomized algorithms in
computational geometry the reader is referred to Mulmuly s b o ok Another word of warning
is that the language we use is quite di erent than the usual LP terminology and we leave it to the
interested reader to make the translation
The complexity of linear programming
Given a linear program max b x sub ject to Ax c with n inequalities in d variables we denote
by L the total input size of the problem when the co e cients are describ ed in binary We denote
by C d n L the numb er of arithmetic op erations needed in the worst case by an algorithm
A
A to solve a linear programming problem with d variables n inequalities and input size L The
worst case complexity of linear programming is roughly the function C d n L which describ es
for every value of d n L the smallest p ossible value of C d n L over all p ossible algorithms
A
Khachiyan s breakthrough result was that the complexity of the ellipsoid metho d E is a
p olynomial function of d n and L namely C d n L p d n L Other algorithms which improve
E
on Khachiyan s original b ound and also had immense practical impact on the sub ject were found
by Karmarkar and many others
By considering solutions to all subsets of d from the n inequalities we can easily see that
C d n L f d n i e linear programming can b e solved by a numb er of arithmetic op erations
which is a function of d and n and indep endent of the input size L It is an outstanding op en problem
to nd a strongly polynomial algorithm for linear programming that is to nd an algorithm which
requires a p olynomial numb er in d and n of arithmetic op erations which is indep endent from L
Klee and Minty and subsequently others have shown that several common pivot rules for
the simplex algorithm are exp onential in the worst case
Explaining the excellent p erformance of the simplex algorithm in practice esp ecially in view
of the exp onential worst case b ehavior of various pivot rules is a ma jor challenge The results on
the average case b ehavior of the simplex algorithm provide one such explanation See Borgwardt s
b o ok for a description of his work and for references to other works or The fact that the
complexity of linear programming is p olynomial by Khachiyan s result even if not via the simplex
algorithm provides another partial explanation
Of course nding a pivot rule which requires a p olynomial numb er of steps in the worst case
or even proving that there are always a p olynomial numb er of pivot steps leading to the optimal
vertex without prescribing an algorithm to nd these steps are very desirable
Using randomness for pivot rules
We will consider now randomized algorithms Namely algorithms which dep end on internal random
R
choices Given such a randomized algorithm A we denote by C d n the expected numb er of
A
arithmetic op eration needed in the worst case by A on a LP problem with d variables and n
R R
d n over all p ossible algorithms A Clearly inequalities C d n will b e the minimal value of C
A
R
C d n C d n Note We are interested in a worst case analysis of the average running time
where the randomization is internal to the algorithm This is in contrast with average case analysis
where the LP problem itself is random
Perhaps the simplest random pivot rule is to cho ose at each step at random with equal proba
bilities a neighb oring vertex with a higher value of the ob jective function Unfortunately it seems
very di cult to analyze this rule for general problems Recently G artner Henk and Ziegler
managed to analyze the b ehavior of random pivoting on the Klee Minty cub e
The Hirsch conjecture
Let d n denotes the maximal diameter of the graphs of d p olyhedra P with n facets and d n
b
denotes the maximal diameter of the graphs of d p olytop es with n vertices
Given a d p olyhedron P a linear ob jective function which is b ounded from ab ove on P and a
vertex v of P denote by m v the minimal length of a monotone path in G P from v to a vertex
of P on which attains its maximum Let H d n b e the maximum of m v over all d p olyhedra
d
P with n facets all linear functionals on R and all vertices v of P A monotone path is a path
in G P on which is increasing
Let M d n b e the maximal numb er of vertices in a monotone path in G P over all d p olyhedra
d
P with n facets and all linear functionals on R
Clearly
d n H d n M d n
H d n can b e regarded as the numb er of pivot steps needed by the simplex algorithm when
the pivots are chosen by an oracle in the b est p ossible way M d n can b e regarded as the numb er
of pivot steps needed when the pivots are chosen by an adversary in the worst p ossible way
In Hirsch conjectured that d n n d Klee and Walkup showed that the Hirsch
conjecture is false for unb ounded p olyhedra The Hirsch conjecture for p olytop es is still op en The
sp ecial case asserting that d d d is called the d step conjecture and it was shown by Klee
b
and Walkup to imply the general case
Theorem Klee and Walkup
d n n d minf d n d g
Theorem Holt and Klee For al l d and n d
d n n d
b
Theorem Larman
d
d n n
Theorem Kalai and Kleitman
log n d
log d
d n n n
log n
Klee and Minty considered a certain geometric realization of the d cub e called now the
Klee Minty cub e to show that
d
Theorem Klee and Minty M d d
Sub exp onential randomized pivot rules
We will assume and there is no loss of generality assuming this that the LP problem is non
degenerate i e the feasible p olyhedron is simple and that a vertex v of the feasible p olyhedron is
given With a slight change of terminology all the algorithms and results we describ e apply to the
degenerate case
Several years ago the author and indep endently Matou sek Sharir and Welzl found a
randomized sub exp onential pivot rule for LP thus proving that
p
R
C d n exp K d log n
Slightly sharp er b ounds are describ ed b elow In my pap er various variants of the algorithm
were presented and we will see here two variants The rst and simplest variant is one of my original
and is equivalent in a dual setting to the Sharir Welzl algorithm on which is based The
second variant presented here is a joint work with Martin Dyer and Nimro d Megiddo It is a b etter
and simpli ed version of other variants from All these algorithms apply to abstract ob jective
functions and even more general settings See also G artner s pap er
Consider an LP problem of optimizing a linear ob jective function over a d p olyhedron P and
a vertex v of P Our aim is to reach top P which is a vertex of P on which the ob jective function
is maximal or an edge of P on which the ob jective function is unb ounded from ab ove
ALGORITHM I
Given a vertex v P cho ose a facet F containing v at random
Apply the algorithm on F until reaching w top F
rep eat the algorithm from w
Remark The algorithm terminates if v top P If v top F for some facet F containing
v in which case v has only one improving edge we cho ose F at random from the other d facets
containing F Unless v top P there is at most one such facet F
ALGORITHM II
Cho ose at random an ordering of the facets F F F
n
Phase I Apply the algorithm until you reach a vertex in F or reach top P
Phase II Apply the algorithm recursively inside F until reaching w top F
Phase III Delete the facet F from the ordering and continue to run the algorithm from
w
Phase I and phase III are p erformed w r t the initial random ordering of the n inequalities but
in phase II you have to nd again a new random ordering of the facets
Analysis of the rules
We say that a facet F of P is active w r t the vertex v if v maxf x x F g We will
study the numb er of pivot steps as a function of the numb er of variables d and the numb er of active
facets n The numb er of pivot steps will not dep end on the total numb er of facets N However we
do not assume that we know while running the algorithm which facets are active and the numb er
of arithmetic op erations p er pivot step dep ends therefore p olynomially also on N Note that in
Algorithm II only the ordering of the active facets matters
For a linear programming problem U with d variables and N inequalities and a feasible vertex
v for U such that there are n active facets w r t v we denote by f U v the exp ected numb er of
pivot steps needed by algorithm I on the problem U starting with the vertex v f d n denotes the
maximal value of f U v over all problems U and vertices v The function f d n is not decreasing
with n Similarly g d n will b e the average numb er of pivot steps in the worst case problem for
Algorithm I I
Analysis of Algorithm I
We start with a situation where there are n active facets Let F F F b e the facets
d
containing v ordered such that top F top F top F Note that unless
d
v top P at most one namely only F of these facets can b e non active The average numb er
of steps needed to reach top F from v is at most f d n
If F is active then with probability d the chosen random facet F equals F for i d
i
and then after reaching w top F there are at most n i active facets remaining and the average
numb er of steps needed to reach top P from w is at most f d n i Averaging over i we get
P
d
f d n i that the average numb er of steps needed to reach top P from w is at most d
i
If F is not active then F F with probability d for i d and by the same token
i
P
d
the average numb er of steps needed to reach top P from w is at most d f d n i
i
This is slightly higher than the previous expression by the monotonicity of f d n as a function
of n In sum
d
X
f d n f d n f d n i
d
i
p
This gives f d n exp K n log d see
Analysis of Algorithm I I
For phase II we need at most g d n steps on the average For phase III we can rep eat
the argument of the previous algorithm With probability n there are at most n i active
facets left after reaching top F for i n So the average numb er of pivot step for this
P
n
phase is at most g d i We claim now that the average numb er of pivot steps for phase
i
n
P
n
g d i I is also
i
n
To see this note
As long as we run the algorithm from v meeting only vertices in r active facets we can regard
ourself running the algorithm from v in the LP problem obtained by deleting the inequalities
corresp onding to the other active facets This LP problem has only r active facets Since the
average numb er of pivot steps needed for this problem is at most g d r we conclude that
after an average numb er of g d r pivot steps we either reach top P or reach vertices in more
than r active facets
The pivot steps taken running the algorithm while meeting vertices on r active facets do not
dep end on the ordering of the remaining active facets Therefore the identity of the active
facet to b e the next we meet which is a probability distribution on the remaining active
facets do es not dep end on the ordering of the remaining n r active facets
It follows that with probability n the facets F will b e the i th active facet to b e met
i n
So we get
n
X
g d n g d n g d n i
n
i
This relation implies the following
p
d log n K
g d n e
p
K B d
If d and n are comparable we get a b etter estimate g d B d e K B is a constant
dep ending on B
t
The following estimates are useful when t n d is small w r t n g d d t K d
t
and g d d t K log d These b ounds apply to f d d t as well
d
n for every The following estimates are useful when d is small w r t n g d n K
d
and g d n K log n n
It is p ossible to use generating function techniques to get precise asymptotic for f d n and
g d n It follows from the recursion that n g d n is b ounded ab ove by b d n the numb er of
p ermutations of f ng such that each cycle in the p ermutation considered as a pro duct of
disjoint cycles is decorated by a nonnegative integer and by a plus or minus sign such that the
P
d k
k
sum of the integers is d For b d n there is the closed formula b d n c n k where
k
c n k is the numb er of p ermutations of f ng with k cycles c n k is the absolute value of
the Stirling numb er of the rst kind However for the asymptotic facts describ ed ab ove without
getting the precise constants the simplest pro ofs are by direct estimations
Remark Matou sek found remarkable classes of abstract ob jective functions on the d
dimensional cub e for which the exp ected numb er of pivot steps for Algorithm I describ ed ab ove
p
is indeed exp C d Further understanding of similar examples may shed light on some of the
problems describ ed in this section
LP duality
LP duality allows us to move from a problem with d variables and n inequalities to the dual problem
with n d variables and n inequalities Note that the running time of the algorithms as well as
the b ounds on the diameter are not invariant under LP duality The upp er b ounds on d n as
well as on the running time for the algorithms describ ed here agree with the common wisdom that
when n is large w r t d it is b etter to move to the dual problem However note that the average
numb er of pivot steps of Algorithm II is rather small close to linear even when d is xed and n
tends to in nity
It is an interesting problem to study the relations b etween the combinatorics e g the face
numb ers h numb ers etc of the feasible p olyhedra for an LP problem and for its dual
Non deterministic analysis of the rule and application to the Hirsch problem
Now let us consider again Algorithm II but this time let us assume that the random choices are
made by a friendly oracle and that we can instruct the oracle to make as go o d as p ossible choices
Studying non deterministic p erformance of randomized algorithms is imp ortant for understanding
the algorithm but in this case this is of particular imp ortance since it is immediately related to
the Hirsch problem discussed ab ove
First we order the active facets F F so that top F top F top F
n n
The instructions for the oracle are as follows The only condition on the rst active facet F F
is that top F is ab ove the median So when you run the algorithm you declare the rst facet F
you reach with top F ab ove the median as F Of course this instruction applies recursively
for the rst and last stages as well as when you run the algorithm inside F
Let h d n is the numb er of pivot steps made with the help of our friendly oracle instructed
ab ove in phase I we need at most h d n steps Indeed as long as we met vertices only on
m active facets we can consider ourselves as running the algorithm in the p olyhedra where the
inequalities corresp ond to the other active facets are deleted and the numb er of pivot steps is at
most h d m So in h d n pivot steps we must reach either top P or vertices in more than
n active facets and hence we must reach a vertex in a facet F with top F ab ove the median
When we reach top F then the numb er of remaining active facets is smaller than d Therefore
also in step III we need at most h d n steps Thus
h d n h d n h d n
This recursion gives
log n d
log d
h d n n n
log n
How clever should the oracle b e Not so much The oracle should b e able to run LP problems
and this is p olynomial via Khachiyan By a well known result of Tardos we do not even need
to consider the ob jective function in the input size So we get
Theorem Let P be a d polytope with n facets described by a system of inequalities with input
size L Let v be a vertex of P and be a linear objective function Then there is an algorithm
which nds in T h d n steps a monotone path of length T from v to top P and each step is
performed by a polynomial number in d n and L of arithmetic operations
Conclusion
The situation concerning the Hirsch conjecture and the worst case complexity of the simplex
algorithm is rather frustrating We are short of p olynomial b ounds for the diameter and despite
the simplicity of the pro ofs for the known b ounds we cannot push them any further For n d
we cannot nd a randomized pivot rule which will require exp d pivot steps for some
even if the feasible p olytop e is combinatorially equivalent to a d dimensional cub e And we cannot
nd a deterministic pivot rule without randomization which is not exp onential We leave these
tasks for you the reader
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