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Online Multicast with Egalitarian Cost Sharing

Online Multicast with Egalitarian Cost Sharing. Moses Charikar (Princeton) Howard Karloff (AT&T) Claire Mathieu (Brown) Seffi Naor (Technion) Mike Saks (Rutgers) Talk presented by Warren Schudy (Brown ). Introduction.

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Online Multicast with Egalitarian Cost Sharing

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  1. Online Multicast with Egalitarian Cost Sharing Moses Charikar (Princeton) Howard Karloff (AT&T) Claire Mathieu (Brown) Seffi Naor (Technion) Mike Saks (Rutgers) Talk presented by Warren Schudy (Brown)

  2. Introduction • Multicast: n terminal vertices must connect to root vertex r via paths. Edges have costs c(e). Goal is to minimize total cost: OPT Steiner Tree. • Egalitarian cost sharing: an edge e used by the paths of n(e) terminals charges each terminal c(e)/n(e). Terminals are selfish, non-cooperative. Nash equilibrium (N.E.): no terminal wants to change its path if everything else stays the same. Question: how much more costly is the outcome of selfish choices? That is: bound (cost of N.E.)/ OPT?

  3. Impact of selfishness (cost of worst N.E.)/OPT = n [Koutsoupias Papadimitriou‘99]Price of anarchy (cost of best N.E.)/OPT = O(log n/ loglog n) [Anshelevich Dasgupta Kleinberg Tardos Wexler Roughgarden ‘04, Agarwal Charikarr ‘06] Price of stability Question: what about (cost of N.E.)/OPT for N.E. reachable by some process? Best response dynamics: when activated, a terminal always chooses its current cheapest path to root In which order do activations occur?

  4. Two phase model r r Phase 1 Phase 2 t4 t1 t4 t3 t2 t1 t3 t2 • Activation model [Chekuri Chuzhoy Lewin-Eytan Naor Orda ‘06] • Phase 1: Terminals are activated one by one • Phase 2: Re-activated terminals may change their path (arbitrary sequence of re-activations) Ω(log n/ loglog n)≤(cost of resulting N.E.)/OPT≤ O(√n log2 n)[CCLNO] re-fires

  5. New results • For two phase model: Ω(log n) ≤(cost of resulting N.E.)/OPT ≤ O(log3 n) • For General sequence of interleaved activations and re-activations, except that terminal arrivals (first activations) are in random order: (cost of resulting N.E.)/OPT = O(√n polylog(n)) Next 5 slides: proof sketch of O(log3 n) result

  6. Proving O(log3 n) • Potential function • cost ≤  ≤ H(n)*cost • Re-activations decrease  • So, cost after phase 2 ≤  after phase 2 ≤  after phase 1 ≤ O(log n)*cost after phase 1 • Must prove: (cost after phase 1)≤ O(log2 n)*OPT

  7. Analysis of phase 1 • Define “Gap revealing” linear program (cost after phase 1) ≤ Value(LP) • Relax the LP and write dual linear program Value(LP) ≤ Value(Dual) by linear programming duality • Define feasible dual solution… Value(Dual) ≤ Value(solution) • … of value O(log2 n) OPT Value(solution) = O(log2 n) OPT

  8. Gap revealing LP • s(i): cost of i’s path on arrival of ib(i): cost of new edges bought by i •  b(i) =(Cost after phase 1) =  b(i)’s s(i) = • If terminal j arrives after terminal i, then j could go to i and reuse i’s path with discount: s(j)≤ d(j,i)+s(i)-b(i)/2

  9. Relax, take dual Take a tree T over the terminals, such that children arrive after their parent. Relax the linear program by writing the constraint s(j)≤ d(j,i)+s(i)-b(i)/2 for j child of i in T only So, dual has one variable z(j) for each edge of T between j and parent(j) (C(i): children of i in T)

  10. How is T defined? Must have: children arrive after their parent Take Eulerian tour π of min spanning tree of terminals. Note: Cost(π) ≤ 2(OPT Steiner tree) Try to have: parent(j) is in the vicinity of j along π, and so:  d(j,parent(j))=O(log2 n)* Cost(π) Left subtree r Path to root t1 t3 t2 t4 Right subtree

  11. Random Arrivals Result O(√n polylog(n)) proof sketch • Arbitrary interleaving of arrivals and reactivations, but: assume order of arrivals is random • Analyze potential Φ • Reactivations decrease potential • Φ(k): potential right after kth terminal arrives; bound E[Φ(k+1) - Φ(k) given Φ(k)]

  12. Analysis: arrival of j • Path picked by j is difficult to analyze. Instead, • Consider again Eulerian tour π of min spanning tree of terminals. • Pick i randomly from previously arrived terminals in the vicinity of j along π. Connect j to i and follow i’s path.

  13. Conclusion and Open Problems • General theme: Bound cost of solutions reachable by best response dynamics • Obvious open question:analyze arbitrary mix of arrivals and reactivations • Random slow arrivals: • Arrivals in random order + arbitrary interleavings • When new terminal arrives, solution in equilibrium • Other problems ? • Multiple source-sink pairs: linear lower bound

  14. Properties of T r • Children arrive after their parents • Every node has at most 2 children, and nodes with 2 children are at levels integer* log n •  d(j,parent(j))=O(log2 n)*OPT

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