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
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] r t1 t3 t2 t4
Phase 1 r t1 t3 t2 t4
Phase 2
re-fires

5. (cost of resulting N.E.)/OPT = O(√n polylog(n))
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 ≤  after phase ≤  after phase ≤ 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. s(j)≤ d(j,i)+s(i)-b(i)/2
Gap revealing LP
s(i): cost of i’s path on arrival of i b(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) r t1 t2 t4 t3
Left subtree
Right subtree
Path to root

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  d(j,parent(j))=O(log2 n)*OPT
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 r