Transcript 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
 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?
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?
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)
Phase 1
r
Phase 2
t4
t1
t4
t1
t3
t3
re-fires
t2
t2
2
Ω(log n/ loglog n)≤ (cost of resulting N.E.)/OPT≤ O(√n log n)[CCLNO]
r
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
Proving O(log3 n)
 Potential function
 cost ≤  ≤ H(n)*cost
 Re-activations decrease 
 So, cost after phase 2
after phase 2
phase 1
after phase 1
≤
≤  after
≤ O(log n)*cost
 Must prove: (cost after phase 1)≤ O(log2 n)*OPT
Analysis of phase 1
1. Define “Gap revealing” linear program
(cost
after phase 1) ≤ Value(LP)
2. Relax the LP and write dual linear program
Value(LP) ≤ Value(Dual) by linear programming duality
3. Define feasible dual solution…
Value(Dual) ≤
Value(solution)
4. … of value O(log2 n) OPT
O(log2 n) OPT
Value(solution) =
Gap revealing LP
 s(i): cost of i’s path on arrival of i
cost of new edges bought by i
b(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
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)
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
t2
t4
Right subtree
t3
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)]
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.
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
Properties of T
1. Children arrive after their parents
2. Every node has at most 2 children, and nodes with 2
children are at levels integer* log n
r
3.  d(j,parent(j))=O(log2 n)*OPT