Optimal Oblivious Routing in
Polynomial Time
Yossi Azar Amos Fiat Haim Kaplan
Tel-Aviv University
Edith Cohen
AT&T Labs-Research
June 10, 2003
STOC 2003
Harald Räcke
Paderborn
Routing, Demands, Flow, Congestion
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Routing: a unit s-t flow for each origindestination pair:
fab(i,j) >= 0 routing for OD pair a,b on edge (i,j)
Demands: Dab >= 0 for each OD pair a,b
Flow on edge e=(i,j) when routing D with f:
flow(e,f,D)=Sab fab(i,j) Dab
Congestion on edge e=(i,j) when routing D with f:
cong(e,f,D)=flow(e,f,d)/capacity(e)
June 10, 2003
STOC 2003
Congestion, Oblivious Routing
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Congestion of demands D with routing
f: cong(f,D)= maxe cong(e,f,D)
Optimal routing for D: min possible
congestion: opt(D) = minf cong(f,D)
Oblivious ratio of f:
obliv(f)= maxD cong(f,D)/opt(D)
Optimal Oblivious Ratio of G:
obliv-opt(G)=minf obliv(f)
June 10, 2003
STOC 2003
Example
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Routing f: Route each OD pair on direct edge
Demands D: unit demand for all pairs
cong(e,f,D)=2 for all edges
Thus, cong(f,D)=2
(f is optimal for D)
June 10, 2003
STOC 2003
Example
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Routing f: Route each OD pair on direct edge
Demands D: unit demand for ONE pair
cong(e,f,D)=1 for used edge, 0 otherwise.
Thus, cong(f,D)=1
(f is NOT optimal for D)
June 10, 2003
STOC 2003
Example
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Routing f: Route each OD pair on the 3 1,2 hop paths
Demands D: unit demand for one pair
cong(e,f,D)=1/3 for used edges
cong(f,D)=1/3
“direct” routing has oblivious ratio >= 3
June 10, 2003
STOC 2003
Example
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Routing f: Route each OD pair on the 3 1,2 hop paths
Demands D: unit demand for all pairs
cong(e,f,D)=10/3 for all edges (10 pairs use each edge)
cong(f,D)=10/3 (f is NOT optimal for D)
2-hop routing has oblivious ratio >= 5/3
June 10, 2003
STOC 2003
Optimal oblivious routing
• Balances performance across all
demand matrices.
• Why is it interesting?
– Demands are dynamic
– Changes to routing are hard
– Sometimes we don’t know the demands
June 10, 2003
STOC 2003
History
• Specific networks, VC routing
– Raghavan/Thompson 87…Aspnes et al 93
– Valiant/Brebner 81: Hypercubes
• Räcke 02:
Any undirected network has an oblivious routing
with ratio O(log^3 n)!!
• Questions:
– Poly time algorithm.
– Get an optimal routing.
– Directed networks?
June 10, 2003
STOC 2003
LP for Optimal Oblivious Ratio
• Minimize r s.t.
fab(i,j) is a routing (1-flow for every a,b)
For all demands Dab >= 0 which can be
routed with congestion 1:
For all edges e=(i,j) : (cong(e,f,D) <= r)
Sab fab(i,j) Dab/capacity(e) <= r
But… Infinite number of constraints  use Ellipsoid
June 10, 2003
STOC 2003
Separation Oracle
• Given a routing fab(i,j), find its oblivious
ratio and a demand matrix D which maximizes
the ratio (the “worst” demands for f).
For each edge e=(i,j) solve the LP (and then
take the maximum over these LPs):
• Maximize Sab fab(i,j) Dab/capacity(e)
• gab(i,j) is a flow of demand Dab >= 0
• For all edges h,
S gab(h) <= capacity(h)
** Need to insure that the numbers don’t grow too much
June 10, 2003
STOC 2003
Directed Networks
(Asymmetric link capacities)
• Our algorithm computes optimal oblivious
routing for undirected and directed
networks.
• Räcke’s O(log^3 n) bound applies only to
undirected networks.
• We show that some directed networks have
optimal oblivious ratio of W(sqrt(n)).
June 10, 2003
STOC 2003
{i,j}
k/2
i
k
2
( )
j
k
t
Any flow from {i,j} to t is split on the two possible paths.
Thus, a routing is determined by the split ratio for each {i,j}.
For any routing f, there is at least one mid-layer node i that
routes >= half the flow for >= k/2 pairs.
“Bad” demands for f: 1 on pairs {i,*} to t, 0 otherwise.
congestion is >= k/4 with f. But optimal is 1 (via alternate paths)
June 10, 2003
STOC 2003
Extensions
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Subset of OD pair demands
Ranges of demands
Node congestion
Limiting dilation
June 10, 2003
STOC 2003
Follow up/subsequent work
• Polytime construction of a Räcke-like
decomposition
(two SPAA 03 papers: Harrelson/Hildrum/Rao
Bienkowski/Korzeniowski/Räcke)
• More efficient polynomial time algorithm
(Applegate/Cohen SIGCOMM 03)
• Oblivious routing on ISP topologies
(Applegate/Cohen SIGCOMM 03)
• Online oblivious routing
(Bansal/Blum/Chawla/Meyerson SPAA 03)
June 10, 2003
STOC 2003
Open Problems
• Tighten Räcke’s bound
O(log^3 n)  W(log n)
(Currently, O(log^2 n log log n) by
Harrelson/Hildrum/Rao 03)
• Single source demands:
Is there a constant optimal oblivious
ratio ?
June 10, 2003
STOC 2003