Intra-domain weight optimization using column generation Bernard Fortz and Hakan Umit Université Catholique de Louvain Institut d'Administration et de Gestion Louvain-la-Neuve Belgium 10th Aussois Workshop on Combinatorial Optimization, Aussois 2006
Challenges Rapid growth of networks Meeting user demands Quality of service under service level agreements; less delay, promised throughput etc. 2
Intra-domain routing protocols Interior Gateway Protocol OSPF, IS-IS Routing information is distributed between routers belonging to a single Autonomous System. Traffic is routed through shortest paths wrt link weights Weights are set and can be altered by network operators Suggestion of Cisco: weight=1/capacity 3
Packet routing in OSPF 1 4 2 3 4
IGP weight optimization problem Find the best set of link metric (weights) that yields routing of a given traffic (demands between routers) with minimum congestion (load over the links). Constraint: A flow arriving at a router (node) is sent to its destination by evenly splitting the flow between the links that are on the shortest paths to the destination. 5
Evenly balancing flows NP-Hard! 6
Existing approaches and tools Weight optimization using local search heuristic [Fortz and Thorup, 2004] Tabu search implementation IGP-WO: open source software, [Fortz, Cerav and Umit, 2004] Open source software funded by Walloon government Three research groups from UCL and Univ. Liege 7
About the toolbox Unified algorithms for intra-domain and inter- domain traffic engineering purposes Project URL : http://totem.info.ucl.ac.be 8
Results so far Results are within 5% gap of General routing problem 9
Objectives Provide a lower bound Generate possible link weights for IGP routing by using column generation 10
Problem input and variables with capacitated arcs, G = ( N , A ) c a k � K commodities F demand for each and o k � N d k � N k Set of directed paths P k Objective: Minimize total cost of flows: � Decision variables: p � P f : Flow on path k p : Load on arc a � A l a 11
Piecewise linear cost function 12
Multi commodity network flow problem – path based form Minimize � � a a A � l c a A , i I , � � � � � ( 1 ) � � a i a i a � l f � a p a A , ( 2 ) � p P : a p � � k � k K , f F ( 3 ) � � p k p P � k k K , p P , ( 4 ) � � f 0 k � p 13
Column generation procedure Restricted Master Multi commodity flow Problem (RMP) problem- path based Added variables Initial variables Constraints Variables that were never considered paths 14
Solving master and restricted master problem Let be a subset of P S k k Solve restricted master problem for paths in S k Add more columns as needed until optimum solution is attained 15
Optimality Conditions Dual variables w for each arc a for each commodity � k l c a A , i I , � � � � � ( 1 ) � � a i a i a l � f � a p a A , ( 2 ) � p P : a p � � k � f F k K , ( 3 ) � � p k p P � k k K , p P , ( 4 ) � � f 0 k � p 16
Interpretation of dual variables Dual Variables and are the optimal w � k a weight for arc a and shortest path distance for commodity k , respectively. In column generation procedure dual w variables are input to check optimality: a � min w < � a k p P � k a p � Current path Newly generated path 17
Generation of new columns Dynamic shortest path computation [Buriol, Resende and Thorup, 2003] Given a graph , a shortest G = ( N , A ) path graph and a vector W G SP = ( N , A ' ) w with a weight associated with each a arc a . Update without recomputing it G SP from scratch. Gain up to factor of 20 for a 100 node graph. 18
Use of output Solving RMP until optimality, i.e. until no more shortest path exists, can provide a lower bound for IGP routing w of the optimum solution can be a used as a heuristic weight setting 19
Future research Numerical results Addition of a cut that will split the flows evenly 20
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