Prot-Maximizing Multiast Priing by
Approximating Fixed Points
Aranyak Mehta
y
Sott Shenker.
Vijay V. Vazirani
Abstrat
We desribe a xed point approah for the following stohasti optimization problem: given
a multiast tree and probability distributions of user utilities, ompute pries to oer the users
in order to maximize the expeted prot of the servie provider. We show that any optimum
priing is a xed point of an eÆiently omputable map. In the language of lassial numerial
analysis, we show that the non-linear Jaobi and Gauss-Seidel methods of oordinate desent are
appliable to this problem. We provide proof of onvergene to the optimum pries for speial
ases of utility distributions and tree edge osts.
1 Introdution
We study a \stohasti optimization" problem for priing multiast transmissions. A single servie
provider is sending multiast pakets to a large number of users. For eÆieny, multiasting is
ahieved over a multiast tree [DC90, DEF 94℄, with the servie provider at the root of the tree
and one user per node of the tree. Pakets are sent down the tree and are dupliated at every
branh node. We assume that this dupliation is ahieved free of ost. However, there is a ost
involved in transmitting over eah edge of the tree. User i has a utility ui for the servie. The
servie provider does not know ui, but he does know that ui omes from a distribution Fi . The
servie provider oers to sell the servie to user i at prie pi. The user aepts i ui pi. The
servie is then provided to all the users who aept, through a subtree of the original, universal
multiast tree, at a ost equal to the ost of the subtree and a revenue equal to the sum of the
pries aepted. The goal of the servie provider is to nd pries pi so as to maximize his expeted
prot. We desribe the problem in detail in Setion 2.
This problem is motivated by the reent work on mehanism design in general [MS01, NR99, JV01℄
and mehanism design for multiasting in partiular [FPS01, GHW01℄. The standard mehanism
design problem involves a mehanism designer (in the ase of multiasting, the servie provider)
and a set of users. Eah user is assumed to be selsh, and tries to maximize some private utility
value for some servie that the designer an oer him. This utility is unknown to the designer.
The goal of the designer is (roughly) to use the selshness and rationality of the users to design a
protool that will eliit truthful responses (regarding the utilities) from the users and hene fulll
some goal of the designer himself.
The goal of the designer in mehanism design problems is usually related to soial welfare - either
maximizing the eÆieny of the protool (the dierene between the sum of the utilities of the
+
Address:
College
of
Computing,
faranyak,vaziranig.gateh.edu.
y
Georgia
Institute
of
Tehnology,
Atlanta,
GA
30332{0280,
Researh supported by NSF Grant CCR-9820896.
ICSI, 1947 Center Street, Suite 600, Berkeley, CA 94704{1198, E-mail: shenkerisi.berkeley.edu
E-mail:
users who get the servie and the ost of providing the servie to these users) or maintaining a
balaned budget (suh that the servie provider retrieves exatly the ost of providing the servie).
For multiasting these questions were investigated in [FPS01℄ (in terms of the network omplexity
of ahieving these goals). The more reent work of Fiat et.al. [FGHK02℄ asks a dierent question are there mehanisms whih maximize the prot to the servie provider? This is a prot maximizing
question as opposed to the ost sharing questions above. Fiat et.al. nd mehanisms whih are
ompetitive in terms of maximizing prot. In fat, they study the prot maximization question for
a broad lass of problems (whih inludes multiasting). They nd a onstant fator mehanism,
in the restrited ase that there is more than one user at every node of the multiast tree and no
user's bid \dominates" the bids of other users in the same node. Furthermore, the onstant fator
depends ritially on the assumption that the optimal prot is itself a onstant fator more than
the ost involved.
In this paper we also address the prot maximization question, but in a dierent ontext. We
remove all restritions on the number of agents at eah node and on the optimal prot. However,
we assume that while the individual values of the utilities are not known to the servie provider,
their distribution is. This is not the typial assumption made in the mehanism design literature,
but it might apply to settings, suh as what we onsider here, where there are a random set of
users drawn from a pool whose overall harateristis are well-known from market researh or other
means.
On assuming knowledge of the distributions of the utilities, the problem hanges to a stohasti
optimization problem. Reent work in the same vein is that of Kleinberg, Rabani and Tardos
[KRT97℄ - they study the stohasti versions of minimum makespan, bin-paking and knapsak,
i.e. versions in whih the item/job sizes are not known but only the distributions from where
they are piked. However, in their ase the non-stohasti problems are well-known NP omplete
problems. Hene they look for approximation algorithms. In our ase, the non-stohasti problem
is polynomial time solvable: knowing the utility of a user, the only options are to set the prie to
the utility or to not oer the servie at all. Whih users to make that oer to an be gured out
by dynami programming (e.g. trikling a value up the tree whih says \if serving my parent then
the maximum extra prot you an get through the subtree rooted at me is so muh.")
Thus our problem is (in terms of diÆulty) between (1) knowing the utilities (easy) and (2) not
knowing anything about the utilities (hard). We do not know whether this stohasti problem is
polynomial time solvable even for \nie" distributions. A naive approah would be to assume the
utility of eah user to be his expeted utility and oer that as the prie. However, it an be shown
that this approah an give an arbitrarily large approximation fator, in fat it may result in a loss
even if the situation is protable.
The main diÆulty in the problem is that the probability that a user aepts a prie depends on
the prie, but the prie in turn depends on the probability that the servie is routed through that
user. A diret numerial method would not work beause the funtion we want to optimize takes
exponential spae to even write down. We suggest an indiret xed point approah to the problem
- we nd a map whih is easy to ompute suh that the optimum pries are a xed point of this
map. Fixed point methods for stohasti optimization problems have been used elsewhere, e.g. in
traÆ route guidane systems [BBABI98, BC01℄.
Results: We desribe a quadrati time omputable map suh that the optimal pries form a xed
point of this map, and hene a simple xed point approah may lead to the optimal solution. Fixed
point problems have been well studied, and several general purpose methods are known in the
literature (see e.g., [OR70℄). We analyze the simplest iteration method for speial distributions.
These iterations are known in the numerial analysis literature as the Jaobi or the Gauss-Seidel
method. We show that the simple iterations onverge geometrially to the optimum pries (and
hene yield a fully polynomial approximation) in speial ases for the uniform distribution and for
the exponential distribution.
This approah an be used in any multi-variate optimization problem in whih the expeted ost
of adding one variable when the rest of the variables are xed at arbitrary values an be omputed
easily.
This paper has three setions. In Setion 2 we dene the expeted prot maximization problem.
In Setion 3 we desribe a map for whih solutions to the problem are xed points. In Setion 4 we
study two partiular distributions (the exponential and the uniform) and show that for speial ases
of edge osts the simplest iterative proess does onverge to a solution of the prot maximization
problem.
2 The Expeted Prot Maximization Problem
To motivate the problem we study in this paper, onsider rst the simple situation in whih there
is a servie provider who oers a servie and a single user who requires this servie. The ost to
the servie provider to serve the user is . The user has a utility value u whih is hosen from a
(umulative) probability distribution F . The provider does not know u, but he does know F . F
has non-negative support (i.e. F (0) = 0). We also assume throughout that omputing F (x) is a
unit time operation for all x 2 R.
The servie provider has to set the prie p at whih he oers the servie to the user. If u p then
the user aepts the oer and the provider's prot is (p ); if u < p then the user rejets the
oer, the servie is not provided, and there is no prot. The goal of the provider is to maximize
his expeted prot (1 F (p))(p ). We assume throughout that the distribution F is suh that
there is preisely one value of p whih maximizes the expeted prot. Thus we have the following
denition:
Denition 1 The single user prot maximizing funtion for a distribution F is the funtion
F : R+ ! R+ suh that F () is the prie whih maximizes the expeted prot in serving a single
user at a ost of , when the user's utility omes from F .
We restrit attention to distributions F suh that F () an be omputed in polynomial time, and
assume throughout that omputing F () is a unit time operation for all 2 R. We also assume
that F is a monotonially inreasing funtion.
For example, if F is the exponential distribution E with parameter then the expeted prot is
e p (p ), and this is maximized at E () = + . If F is the uniform distribution U on [0; 1℄
then the expeted prot is (1 p)(p ) when 0 p 1 and 0 when p > 1, and this is maximized
at U () = minf1; g.
A subtle point to be noted here is that the protool must be prie xing, in the sense that if the
user rejets the oer and oers in return another prie at whih he is willing to buy the servie, the
provider should not aept this ounter oer, even if it is greater that . This is essential, otherwise
the user will laim a lower utility than the true value, and the protool may be suboptimal.
Now onsider the situation in whih there are several users, eah with some utility, independent of
the other users. If these users have to be provided the servie independent of eah other (say by
1
1+
2
diret links, one to eah user) then the maximum expeted prot is omputed by simply omputing
the values for eah user. Of ourse, servie is never delivered in this manner.
For our problem we assume that we are given a xed multiast tree. The servie provider is at
the root of the tree and there are n other nodes. Eah node (other than the root) orresponds
to a user, and we use the terms node i and user i interhangeably. The edges of the tree have
speied edge osts, whih ould be dierent for dierent edges. The servie an be provided to
the nodes only through the edges of the tree. We assume that there is no ost of dupliating the
servie at the branhing nodes of the tree, whenever suh dupliation is required. Eah node has a
utility value whih determines how muh it is willing to pay for the servie. Node i's utility ui is
distributed aording to the (umulative) distribution Fi , independent of the utilities of the other
nodes. These distributions are known to the servie provider. The provider has to name the pries
at whih he is willing to provide the servie to the users. The pries need not be the same for all
users. Again, the goal of the provider is to set pries so as to maximize his expeted prot. Let
p = (p ; p ; :::; pn ) be the oered prie vetor (node i is oered prie pi ). Node i aepts the oer
if ui pi and rejets otherwise. Let A be the (random) set of nodes that aept the oer.
1
2
Denition 2 The aepting subtree for a given prie vetor p is the smallest subtree of the multiast tree that ontains all the nodes in A.
The servie is routed through the aepting subtree to the nodes in A. Nodes whih are not in the
aepting subtree have rejeted the oer and are not provided the servie. Again, we insist that a
node whih is not in A but is in the aepting subtree is also not provided the servie, even though
the servie is routed through it. This is essential, otherwise the users may laim a utility lower
than the true value and hope to end up in the aepting subtree and get the servie at lesser prie.
Denition 3 The ost of providing the servie is the ost of the aepting subtree. The revenue is
the sum of the pries whih were aepted.
Prot = revenue { ost.
Every prie vetor p results in some expeted prot (the expetation is over the outomes of the
utility values drawn from the distributions).
The Expeted Prot Maximization Problem: Given the multiast tree with edge osts and
given the distributions from whih the utilities of the users are piked, nd a prie vetor p whih
maximizes the expeted prot.
Optimum pries exist due to the extreme value theorem for ontinuous funtions (we an restrit
the funtion to a bounded region, outside whih the prot is always negative). The prot funtion
is random, as it depends on whih nodes aept. To write down the expeted prot itself would take
exponential number of terms. Hene it is not possible to use numerial methods on the expeted
prot funtion itself { we need to use more indiret methods.
We approximate the optimum pries oered to the nodes, rather than the optimum prot. In
fat, it is not lear how we an alulate the expeted prot from the pries, as the funtion is
exponentially big.
3 A Fixed Point Method
In this setion we suggest a solution to the Expeted Prot Maximization Problem using xed
points of a ertain map. In general, nding xed points of maps is not known to be in polynomial
time. We suggest a very simple iterative method whose orretness and eÆieny we analyze for
speial ases in the next setion.
3.1
Relatively Optimum Priing
Given a prie vetor p = (p ; p ; :::; pn ) we denote by p i the vetor (p ; p ; :::pi ; pi ; :::pn ), i.e.
i is a prie vetor without the ith oordinate. A prie vetor p i is interpreted as xing the
pries for all the nodes exept node i. For p i xed, dierent pries oered to node i will result in
dierent prots.
1
p
2
1
2
1
+1
Denition 4 Given a vetor p i , the relatively optimum prie for node i (or the optimum prie
for node i relative to p i ) is the prie pi for node i whih maximizes the expeted prot onditional
on the fat that pries for the other nodes are xed at p i .
The denition assumes that the relatively optimum prie is unique. We show next that this is
indeed the ase, and that pi an be omputed in O(n) time.
Fix p i, i.e. pries for all nodes exept node i. Ignoring node i, onsider all possible outomes for
the other (n 1) nodes. The probability spae is the set of all possible 2n events: whether node
j aepts or rejets the oer pj , for j 6= i. For an ! 2 there is a orresponding aepting subtree.
This is a subtree of the multiast tree whih may or may not ontain i depending on whether some
desendant of i in the multiast tree aepted the oer made in p i or not.
Let X be the random variable whih denotes the ost of onneting node i to the aepting subtree.
If some desendant of node i in the original multiast tree aepts the oer made in p i then X = 0,
otherwise X is the distane of node i to the losest anestor in the multiast tree whih aepts
the oer in p i . Reall the denition of the single user prot maximizing funtion F ().
1
Lemma 1 pi = Fi (E [X ℄).
Proof : For an outome ! 2 , let r(!) be the revenue in that outome, i.e. the sum of the
pries pj ; j 6= i that are aepted. Let (!) be the ost of the aepting subtree. The expeted
value of the prot as a funtion of pi is:
(1 Fi (pi ))
X P [!℄ (r(!)
!2
(!) + pi
X (!)) + Fi (pi )
X P [!℄ (r(!)
!2
(!))
Here the rst summation is the expeted prot given that node i aepts, whih happens with
probability (1 Fi (pi)); and the seond summation is the expeted prot given that node i rejets,
whih happens with probability Fi(pi ). Splitting the rst summation, and then simplifying, we get:
X P [!℄ (r(!) (!)) + (1 F (p )) X P [!℄ (p
2
2
X
X
F (p ))
P [!℄ (p X (!)) +
P [!℄ (r(!) (!))
(1 Fi (pi ))
= (1
= (1
i i
i i
! !2
Fi (pi ))(pi
i
E [X ℄) + (p i )
! i
X (!)) + Fi (pi )
X P [!℄ (r(!)
!2
(!))
!2
where (p i ) is the expeted prot in the absene of node i. Sine (p i) is independent of pi,
it follows (from the denition of the single user prot maximizing funtion) that the value for pi
whih maximizes the above funtion is pi = Fi (E [X ℄).
2
Lemma 2 For any given p
i
we an ompute the relatively optimum prie pi in O(n) time.
By Lemma 1 it suÆes to prove that E [X ℄ an be omputed in O(n) time. If node
i is at a distane of j edges from the root, then X takes j + 1 distint values: X equals zero if
some desendant of node i aepts the oer, X equals the ost of the edge onneting node i to its
parent if no desendant of i aepts and the parent of i aepts, and so on. For any given p i the
probabilities of all of these events an be omputed in O(n) time, hene also the relatively optimum
prie.
2
Proof :
3.2
A Fixed Point Map
Lemma 1 suggests the following subroutine. It takes as input a prie vetor p and an index i
between 1 and n, and it modies the prie vetor.
RELATIVE-OPT(p; i):
Obtain p i from p by deleting the ith oordinate.
Compute the optimum prie pi for node i relative to p i.
Replae the ith oordinate of p by pi .
This motivates the following map whih takes a prie vetor to another prie vetor.
REFINE(p):
for i = 1 to n
RELATIVE-OPT(p; i)
We know from Lemma 2 that RELATIVE-OPT is a linear time subroutine, and so REFINE
an be implemented in O(n ) time.
2
Denition 5 A prie vetor p is a xed point of the map REFINE if p = REFINE(p).
Thus p is a xed point of REFINE if for every i, pi is the optimum prie for node i relative to
p i . It is lear that any optimal prie vetor is a xed point of REFINE. Consider a very simple
iterative sheme:
SIMPLE-ITERATIONS:
Initialize p to all oordinates at innity.
Repeat:
REFINE(p)
The next lemma proves that in these simple iterations the prie for eah node is dereasing monotonially. Suppose that p i and q i are prie vetors suh that p i is greater than or equal to
q i in every oordinate. Let pi be the optimum prie for node i relative to p i and let qi be the
optimum prie for node i relative to q i.
Lemma 3 pi qi .
It an be shown that E [X ℄ for p i is greater than or equal to E [X ℄ for q i. Sine Fi
is an inreasing funtion, Lemma 1 implies that pi qi.
2
Proof :
While the SIMPLE-ITERATIONS ause the pries to derease monotonially (or inrease monotonially if we initialize all pries at 0), it is not lear whether they will onverge to an optimum
priing:
Question 1: Do SIMPLE-ITERATIONS onverge to an optimum prie? If so, do they onverge
rapidly?
SIMPLE-ITERATIONS are essentially Jaobi or Gauss-Seidel iterations, whih have been well
studied in numerial analysis. However, there are no general onvergene theorems that seem to
diretly apply to this problem. Note that we have to also exlude the ase that the iterations
onverge to some point whih is not an optimum prie, but is a sub-optimal xed point. Again
there are general theorems that guarantee this, but they require onvexity of the funtion being
maximized, and that may not be true for our funtion for general utility distributions.
If the answer to Question 1 is in the negative then the immediate question is whether we an use
other methods like Newton's Methods or Sarf's algorithm [Sa67℄ to nd the optimum priing fast.
Question 2: Are there other numerial methods whih nd rapidly the optimum pries as a xed
point of REFINE?
4 Analysis for partiular distributions
In this setion we provide a partial answer to Question 1 (from Setion 3.2). We onsider two
distributions for the utilities - the exponential and the uniform. With ertain assumptions on
the parameters of the distributions (or alternatively, on the ost of the tree edges) we prove that
REFINE is a ontrative map. This means that the iterations onverge geometrially to a unique
xed point (whih is the optimal priing). Suh ontration results are well known to be diÆult
to prove. However, we believe that the simple iterations are a good tehnique to solve the general
problem, and may be applied to other maximization problems of a similar nature. In ase they do
not onverge fast, more ompliated xed point methods an be used. These may run muh longer
but may guarantee a good result.
4.1
The Exponential Distribution
In this setion we assume that the utility of eah node omes independently from the exponential
distribution E with parameter .
As noted in Setion 2, the single user prot maximizing funtion is E () = + .
The probability that node i aepts prie pi is e pi . Let qi = 1 e pi , the probability that node
i rejets. Using Lemma 1 we an obtain expressions for the relatively optimum prie for eah node.
For example, if the tree is simply n nodes in a line with all edge osts , we have the following
expressions (the nodes are in order with node 1 losest to the root and node n the farthest). To see
the equation Si observe that the ost of joining i to the aepting subtree of the rest of the nodes
is i if no node aepts, (i 1) if only node 1 aepts, (i 2) if node 2 aepts and no node j
(j > 2) aepts, et. The ost is zero if any node j > i aepts.
1
S1 : p1
S2 : p2
= E ( q q q ::: qn)
= E (2 q q q ::: qn + (1
2
3
1
4
3
4
q1 ) q3 q4 ::: qn )
Si : pi
...
= E (i q q ::: qi qi ::: qn + (i 1) (1 q ) q q ::: qi qi ::: qn +
(i 2) (1 q ) q q ::: qi qi ::: qn + ::: + (1 qi ) qi :::: qn )
...
= E (n q q ::: qn + (n 1)(1 q ) q q ::: qn + (n 2)(1 q ) q
+(1 qn ))
1
2
1
2
Sn : pn
1
3
2
+1
4
1
1
2
3
+1
1
1
1
1
2
3
+1
+1
1
2
3
q4 ::: qn + :::
1
Eah equation an be onsidered to be a surfae (the surfae Si indiates the optimum prie for
node i relative to xed pries p i for the other nodes). A prie vetor p is a xed point of the map
REFINE i it is a point of intersetion of all n surfaes.
For this paper a funtion g(p ; p ; :::; pi ; pi ; :::; pn ) is alled a multiexponential funtion if it is
multilinear in the qj = 1 e pi , j 6= i.
For a general tree the system of equations obtained from Lemma 1 is similar to the partiular
example above:
1
2
1
+1
Si : pi = E (hi (p1 ; p2 ; :::; pi 1 ; pi+1 ; :::; pn ))
i = 1; :::; n
(1)
where the hi are some multiexponential funtions with positive oeÆients.
Convergene
As noted before, we are interested in nding the following (l1) approximation to the optimum
priing
Denition 6 For > 0, a vetor p is said to be an -approximation of the xed point pf if
jpi
pfi j ,
8 i = 1; :::; n.
We use a general method based on the norm of the partial derivatives matrix to prove that REFINE
is ontrative in a restrited ase. This will imply that the optimum priing is the unique xed
point, and that the iterations onverge geometrially to it. A sketh of the proof is given in the
appendix.
Theorem 4 For any multiast tree with eah edge of ost , if the utilities of the nodes are taken
independently from the exponential distribution with parameter , and if the ostliest path has ost
less than 1 , then SIMPLE-ITERATIONS onverge geometrially to the optimum prie vetor,
i.e. we reah an -approximation of the prot maximizing prie vetor in poly(n; log 1 ) time.
4.2
The Uniform Distribution
In this setion we assume that the utility of eah node omes independently from the uniform
distribution U on [0; 1℄. Sine the utilities ome from the distribution U , we restrit our attention
to prie vetors p 2 [0; 1℄n .
As noted in Setion 2, the single user prot maximizing funtion is U () = minf1; g.
(1+ )
2
The probability that node i aepts prie pi is (1 pi). Using Lemma 1 we an obtain expressions
for the relatively optimum prie for eah node.
For a general tree the system of equations obtained from Lemma 1 looks similar to that for the
exponential distribution ase:
Si : pi = U (hi (p ; p ; :::; pi ; pi ; :::; pn ))
i = 1; :::; n
(2)
where the hi are some multilinear funtions.
Eah equation an be onsidered to be a surfae (the surfae Si indiates the optimum prie for
node i relative to xed pries p i for the other nodes). A prie vetor p is a xed point of the map
REFINE i it is a point of intersetion of all n surfaes.
1
2
1
+1
Convergene
Consider an n node multiast tree with the utilities of the nodes drawn independently from the
uniform distribution on [0; 1℄. Suppose the edge osts are suh that for some onstant K < 1
Si (1; 1; ::::1) K < 1; i = 1; :::; n
(3)
Let the optimum prie vetor be pf = (pf ; pf ; :::; pfn ).
We prove that under assumption 3, REFINE is a ontrative map and hene the iterations onverge
geometrially to the optimum priing. A sketh of the proof is given in the appendix.
1
2
Theorem 5 For any multiast tree if the utilities of the nodes are drawn independently from the
uniform distribution on [0; 1℄, and the edge osts are suh that Si(1; 1; :::; 1) K < 1; i = 1; ::; n, for
some onstant K, then, for any given > 0, SIMPLE-ITERATIONS reah an -approximation
of the prot maximizing prie vetor in time poly(n; log(1=)).
5 Disussion
Prot maximization in multiasting attempts to tread the ne line between over-harging, whih
ould lead to loss of ustomers and under-harging, whih results in lower revenues. A naive
approah of assuming the utility of eah user to be his expeted utility and oering that as the
prie, an give an arbitrarily large approximation fator.
The main ontribution of this paper is to suggest a xed point approah for this optimization
problem. Although we an prove onvergene to an optimum priing only in very speial ases,
simulations show that in fat simple iterations onverge extremely fast to optimum priing even
for ases in whih we have no proofs. Fixed point iterations have been studied extensively in
the numerial analysis literature. In ases where we annot prove fast onvergene of the simple
iterations, we an resort to more powerful tehniques for nding xed points of REFINE, e.g.
Newton's methods or the algorithm in [Sa67℄.
The xed point approah suggested here for multiast trees, an be used for any multivariate
optimization problem in whih we an ompute the expeted additional ost of adding one variable
when the rest are xed at any xed values (i.e. when some result like Lemma 1 holds). Of ourse,
onvergene of the sheme will need a proof. However, omputing the expeted additional ost does
not seem to be easy for multiasting on general DAGs.
A dierent approah to the problem on multiast trees is to disretize the pries and use dynami
programming. However, even in that approah it is not lear that we get an FPAS for reasonable
distributions.
Finally we should point out that while the expeted prot is an important quantity to maximize,
other quantities have been investigated : e.g. minimizing the probability that the quantity we want
to maximize is less than a ertain value ([KRT97℄ use the expeted value for the makespan problem,
but have overow probability as a parameter for bin-paking and knapsak). Other measures of
stohasti optimization an be found in [Pin95℄.
It would be interesting to get simple proofs of onvergene for a large lass of distributions. Prot
maximization for other problems an also be studied using the approah proposed here. Two
problems that seem partiularly attrative are minimum spanning tree and faility loation.
Aknowledgments: We wish to thank Saugata Basu for help on uniqueness of the xed point.
We also wish to thank Milena Mihail, Christos Papadimitriou and Craig Tovey for disussions on
understanding the omplexity of our problem, as also Parikshit Gopalan and Amin Saberi for a
areful reading and for several important omments on the results.
Referenes
[BBABI98℄ J. Bottom, M.E. Ben-Akiva, M. Bierlaire, and Chabini I. Generation of onsistent
antiipatory route guidane. Presented at the Tristan III Conferene, San Juan, Puerto
Rio, 1998.
[BC01℄
M. Bierlaire and F. Crittin. Newalgorithmi approahes for the antiipatory route
guidane generation problem. Proeedings of the 1st Swiss Transportation Researh
Conferene, 2001.
[DC90℄
Stephen E. Deering and David R. Cheriton. Multiast routing in datagram internetworks and extended LANs. ACM Transations on Computer Systems, 8(2):85{110,
May 1990.
[DEF 94℄ S. Deering, D. Estrin, D. Farinai, V. Jaobson, C.-G. Liu, and L. Wei. An arhiteture
for wide-area multiast routing. 24(4), Otober 1994. SIGCOMM '94 Symposium.
[EWK90℄ Lars Elden. and Linde Wittmeyer-Koh. Numerial Analysis - An Introdution. Aademi Press, 1990.
[FGHK02℄ A. Fiat, A. Goldberg, J. Hartline, and A. Karlin. Competitive generalized autions.
STOC, 2002.
[FPS01℄ J. Feigenbaum, C. Papadimitriou, and S. Shenker. Sharing the ost of multiast transmissions. J. Computer an System Sienes, 63, 2001.
[GHW01℄ A. Goldberg, J. Hartline, and A. Wright. Competitive autions and digital goods.
SODA, 2001.
[JV01℄
Kamal Jain and Vijay V. Vazirani. Appliations of approximation algorithms to ooperative games. In ACM Symposium on Theory of Computing, 2001.
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[KRT97℄
J. Kleinberg, Y. Rabani, and E. Tardos. Alloating bandwidth for bursty onnetions.
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vs. eÆieny. Eonomi Theory, 18, 2001.
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STOC,
[MS01℄
[NR99℄
[OR70℄
[Pin95℄
[Sa67℄
A Sketh of Proof of Theorem 4
We use the following standard result (see e.g. [EWK90℄) to prove geometri onvergene to an
optimum priing under ertain assumptions on the edge osts.
Let : Rn ! Rn be a map suh that (p) = ( (p); (p); ::::; n (p)). Let pf be a xed point
of , i.e. (pf ) = pf . Consider iterations p1; p2; :::; pk ; ::: of starting at some initial point p1.
i
Assume that the partial derivatives mij = exist in a region R around pf in whih the iterations
pj
run. Let M be the n n matrix (mij ).
We say that the iterations onverge geometrially to the xed point if jpk 1 pf j jpk pf j, for
some < 1. Geometri onvergene implies that we reah an -approximation of the xed point in
O(log ) iterations.
+
1
+
2
+
1
Lemma 6 If for some matrix norm, kM k m < 1 for all points in R, then the iterations onverge
geometrially to the xed point pf in the orresponding vetor norm.
The innity norm of the matrix M is
j1 = max
kM k1 = maxv jMv
in
jvj
1
1
0
1
X
jm jA
n
ij
j =1
Suppose that we are given a multiast tree with eah edge weight equal to . Note that for any i
if we set all pries in p i to 1, then the relatively optimum prie for node i will be less than or
equal to D + , where D is the depth of the tree. On the other hand the iterations will never
set the prie for a node to less than . Thus we an take R to be the region fpj pi D + g
and start our iterations with pi = D + , i = 1; ::; n.
By bounding eah of the pries by from below and D + from above, and assuming that D < ,
we an show that kM k1 m < 1, for some onstant m. Hene the map is ontrative, and the
iterations onverge to the optimum prie vetor.
2(Theorem 4)
1
1
1
1
1
1
1
1
B Sketh of proof of Theorem 5
By the hange of oordinates p^i = pi pfi ; i = 1; :::; n we an make the equation of eah surfae
have onstant term zero, while remaining multilinear. In the new oordinates, the iterations start
with the initial prie vetor (1 pf ; 1 pf ; :::::; 1 pfn ) and the xed point is at the origin. We wish
to prove that the iterations onverge geometrially to the origin.
Due to ondition (3), at the end of the rst iteration p^i K pfi ; i = 1; :::; n, i.e. eah p^i drops
f
by a multipliative fator of at least K ppfi K .
i
Sine eah of the equations is multilinear with onstant term zero, at the end of the seond iteration
eah variable drops further by a fator of at least K . It is easy to see that every iteration will result
in a drop by a fator of at least K , and so eah variable onverges geometrially to zero, i.e. the
iterations onverge geometrially to the (unique) xed point.
Every iteration takes O(n ) time, and we need log =K (1=) iterations to reah an -approximation
of the xed point. This proves the theorem.
2(Theorem 5)
1
2
1
2
1