1. Algorithmic Problems Related To The Internet
Christos H. Papadimitriou
UC Berkeley

2. MITACS, May

3. The Internet huge, growing, open, anarchic
built, operated and used by a multitude of diverse economic interests
as information repository: open, huge, available, unstructured
foundational understanding urgently needed
MITACS, May

4. congestion control [with Karp, Koutsoupias, Shenker, FOCS 2000]
this talk
congestion control [with Karp, Koutsoupias, Shenker, FOCS 2000]
the price of anarchy [with Koutsoupias, 1998; also, Roughgarden and Tardos, FOCS 2000]
multicast and pricing [with Feigenbaum and Shenker, STOC 2000]
the Internet as graph [Bar Yosef et al. VLDB 2000]
information retrieval [Kleinberg 1998, LSI]
MITACS, May

5. TCP congestion control [Jacobson 1987]:
“increase by one, decrease by half’’
traffic
available bandwidth B
total
MITACS, May
time

6. To which question is TCP the answer?
Increasingly realistic models:
B fixed (variants of binary search)
B varying (on-line algorithms)
B depends on other agents (game theory)
MITACS, May

7. ? x the bridge problem: bridge what is the best strategy?
What does this mean?
actual cost
ideal cost
min max cost(S, x) / | x | S x
state of the
world
strategy
(answer: 9)
MITACS, May

8. time-varying bandwidth
state of the world: B = (B1 ,B2 ,…, Bt , … )
sequence of trials: X = ( x1, x2, …, x t , …)
 t gain ( Bt , x t )
= ?
max min
 B S B t t
ideal gain
MITACS, May

9. Restrict B B  [ a, b ] Optimum ratio: log (b / a)
Optimum strategy: Choose x at random with probability ~ 1 / x
t + 1
MITACS, May

10. Restrict B (cont.) B  [ B   , B + ] Optimum ratio: 
Optimum strategy: increase by  / (3 + ) decrease by  / (3 + )
t + 1 t t
MITACS, May

11. Restrict B (cont.) B  [ B (1 ) , B (1 + )] Optimum ratio:  + 1
Optimum strategy: increase by  1 +  decrease by : 1 + 
t + 1 t t
MITACS, May

12. A related question: the price of anarchy
cost of worst Nash equilibrium
“socially optimum” cost s t
3/2 [Koutsoupias and P, 1998]
general
multicommodity
network
2 [Roughgarden and Tardos, 2000]
MITACS, May

13. The web as a graph cf: [Kleinberg 98, Google 98]
how do you sample the web?
[Bar-Yossef, Berg, Chien, Fakcharoenphol, Weitz, VLDB 2000]
e.g.: 42% of web documents are in html. How do you find that?
What is a “random” web document?
MITACS, May

14.    = 0.99999 documents Idea: random walk Problems: hyperlinks
1. asymmetric 
2. uneven degree
3. 2nd eigenvalue?
 =
MITACS, May

15. the web walker: results
[Jerrum-Sinclair 1989]:
mixing time is ~log N/(1-)
WW mixing time: 3,000,000
actual WW mixing time: 100
.com 49%, .jp 9%,
.edu 7%, .cn 0.8%
MITACS, May

16. Q: Is the web a random graph?
Indegrees/outdegrees obey “power laws”
Many K3,3’s (“communities”)
MITACS, May

17. Structure of the web graph (cont.)
“Bowtie” structure
40%
25%
25%
10%
MITACS, May

18. utilities of agents in the node
multicast tree 40 30
costs {} 20 24 17 4
{11, 10, 9, 9}
{14, 8}
{9, 5, 5, 3} 13
{23, 17, 14, 9}
{17, 10}
utilities of agents in the node
(u = the intrinsic value of the information
to agent i, known only to agent i) i
MITACS, May

19. We wish to design a protocol that will result in the computation of:
x (will i get it?)
v (how much will i pay?)
mechanism design
protocol must obey a set of desiderata: i i
MITACS, May

20. Shapley mechanism 0  v  u lim x = 1
strategy proofness: (w = u  x  v )
w (u …u …u )  w (u … u'…u )
welfare maximization
 u  x - c(T [x]) = max
marginal cost mechanism i i i
u  i
def i i i i i 1 i n i 1 i n
budget balance
 v = c ( T [x])
Shapley mechanism i i
MITACS, May

21. our contribution:
In the context of the Internet, there is another desideratum:
Tractability: the protocol should require few
(constant? logarithmic?) messages per link.
This new requirement changes drastically the space
of available solutions.
MITACS, May

22. Shapley mechanism 0  v  u lim x = 1
strategy proofness: (w = u  x  v )
w (u …u …u )  w (u … u'…u )
welfare maximization
 w = max
marginal cost mechanism i i i
u  i
def i i i i i 1 i n i 1 i n
budget balance
 v = c ( T [x])
Shapley mechanism i i
MITACS, May

23. Bottom-up phase W =  u +  W  c, if > 0 0 otherwise c W W W i j 13 W 2
MITACS, May

24. Top-down phase Theorem: The marginal cost mechanism is tractable. A c
D = min {A, W} D
v = max {0, u  D} i i
Theorem: The marginal cost mechanism is tractable.
MITACS, May

25. Theorem: “The Shapley value mechanism is intractable.”
Model: Nodes are linear decision trees, and they exchange
messages that are linear combinations of the u’s and c’s c 1
It reduces to checking whether Au > Bc
by two sites, one of which knows u and
the other c, where A, B are nonsingular c 2
agents drop out one-by-one c n
{u < u < … < u } 1 2 n
MITACS, May

26. Open Questions Of which game is TCP/IP the Nash equilibrium?
The price of the Internet architecture (stateless, best-effort, ex post, …)
Realistic protocols for content auctions
The structure of the web graph
MITACS, May