Algorithmic Problems Related To The Internet Christos H. Papadimitriou UC Berkeley www.cs.berkeley.edu/~christos

MITACS, May 7 2001

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 7 2001

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 7 2001

TCP congestion control [Jacobson 1987]: “increase by one, decrease by half’’ traffic available bandwidth B total MITACS, May 7 2001 time

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 2001

? 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 7 2001

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 7 2001

Restrict B B  [ a, b ] Optimum ratio: log (b / a) Optimum strategy: Choose x at random with probability ~ 1 / x t + 1 MITACS, May 7 2001

Restrict B (cont.) B  [ B   , B + ] Optimum ratio:  Optimum strategy: increase by  / (3 + ) decrease by  / (3 + ) t + 1 t t MITACS, May 7 2001

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 7 2001

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 7 2001

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 7 2001

   = 0.99999 documents Idea: random walk Problems: hyperlinks 1. asymmetric  2. uneven degree 3. 2nd eigenvalue?  = 0.99999 MITACS, May 7 2001

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 7 2001

Q: Is the web a random graph? Indegrees/outdegrees obey “power laws” Many K3,3’s (“communities”) MITACS, May 7 2001

Structure of the web graph (cont.) “Bowtie” structure 40% 25% 25% 10% MITACS, May 7 2001

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 7 2001

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 7 2001

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 7 2001

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 7 2001

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 7 2001

Bottom-up phase W =  u +  W  c, if > 0 0 otherwise c W W W i j 1 3 W 2 MITACS, May 7 2001

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 7 2001

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 7 2001

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 7 2001