June 10, 2003STOC 2003 Optimal Oblivious Routing in Polynomial Time Harald Räcke Paderborn Edith Cohen AT&T Labs-Research Yossi Azar Amos Fiat Haim Kaplan Tel-Aviv University

June 10, 2003STOC 2003 Routing, Demands, Flow, Congestion Routing: a unit s-t flow for each origin- destination pair: f ab ( i,j )  0 routing for OD pair a,b on edge (i,j) Demands: D ab >= 0 for each OD pair a,b Flow on edge e=(i,j) when routing D with f:  flow(e,f,D)=  ab f ab ( i,j ) D ab  Congestion on edge e=(i,j) when routing D with f: cong(e,f,D)=flow(e,f,d)/capacity(e)

June 10, 2003STOC 2003 Congestion, Oblivious Routing Congestion of demands D with routing f: cong(f,D)= max e cong(e,f,D) Optimal routing for D: min possible congestion: opt(D) = min f cong(f,D) Oblivious ratio of f: obliv(f)= max D cong(f,D)/opt(D) Optimal Oblivious Ratio of G: obliv-opt(G)=min f obliv(f)

June 10, 2003STOC 2003 Example 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, 2003STOC 2003 Example 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, 2003STOC 2003 Example 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, 2003STOC 2003 Example 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, 2003STOC 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, 2003STOC 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, 2003STOC 2003 LP for Optimal Oblivious Ratio Minimize r s.t. f ab ( i,j) is a routing (1-flow for every a,b) For all demands D ab >= 0 which can be routed with congestion 1: For all edges e=(i,j) : (cong(e,f,D) <= r)  ab f ab ( i,j ) D ab /capacity(e) <= r But… Infinite number of constraints  use Ellipsoid

June 10, 2003STOC 2003 Separation Oracle Given a routing f ab ( 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  ab f ab ( i,j ) D ab /capacity(e) g ab (i,j) is a flow of demand D ab >= 0 For all edges h,  g ab (h) <= capacity(h) ** Need to insure that the numbers don’t grow too much

June 10, 2003STOC 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  (sqrt(n)).

June 10, 2003STOC 2003 t {i,j} ( ) k 2 ij k k/2 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, 2003STOC 2003 Extensions Subset of OD pair demands Ranges of demands Node congestion Limiting dilation

June 10, 2003STOC 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, 2003STOC 2003 Open Problems Tighten Räcke’s bound O(log^3 n)   (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 ?