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Approximate Graph Embeddings in the Cloud 2 5 3 AC B 2 2 2 2 D 0 3 1 Highlights of Algorithms 2018 Matthias Rost Technische Universitt Berlin, Internet Network Architectures Stefan Schmid Universitt Wien, Communication

  1. Approximate Graph Embeddings in the Cloud 2 5 3 AC B 2 2 2 2 D 0 3 1 Highlights of Algorithms 2018 Matthias Rost Technische Universität Berlin, Internet Network Architectures Stefan Schmid Universität Wien, Communication Technologies

  2. Cloud Providers Offer Data Center Resources Customers Cloud Data Center (Amazon, Google, . . . ) Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 2

  3. Cloud Providers Offer Data Center Resources Customers Cloud Data Center (Amazon, Google, . . . ) ‘Classic’ Cloud Computing 1 4 ◮ Customer specifies A B number and ‘size’ of Virtual Machines ◮ Communication between D C VMs not modeled 3 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 3

  4. Cloud Providers Offer Data Center Resources Customers Cloud Data Center (Amazon, Google, . . . ) ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) ◮ Customer specifies 1 4 1 4 1 A B A B number and ‘size’ of ◮ Additionally: 6 Virtual Machines communication 1 1 ◮ Communication between requirements given D C D C 1 VMs not modeled 3 1 3 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 4

  5. ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) ◮ Customer specifies 1 4 1 4 1 A B A B number and ‘size’ of ◮ Additionally: 6 Virtual Machines communication 1 1 ◮ Communication between requirements given D C D C 1 VMs not modeled 3 1 3 1 The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 1 4 2 5 3 1 A B 3 ◮ Map virtual edges to paths in the substrate 6 1 1 2 2 ◮ Respecting capacities & mapping restrictions 2 2 D C 1 3 1 0 3 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 5

  6. ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) ◮ Customer specifies 1 4 1 4 1 A B A B number and ‘size’ of ◮ Additionally: 6 Virtual Machines communication 1 1 ◮ Communication between requirements given D C D C 1 VMs not modeled 3 1 3 1 The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) 2 / 2 4 / 5 1 4 3 ◮ Map virtual nodes to substrate nodes 1 A B AC B 3 ◮ Map virtual edges to paths in the substrate 6 1 1 2 2 ◮ Respecting capacities & mapping restrictions 2 2 D C D 1 3 / 3 3 1 0 / 0 1 / 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 6

  7. ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) 1 4 ◮ Customer specifies 1 4 1 A B A B number and ‘size’ of ◮ Additionally: Virtual Machines 6 communication 1 1 ◮ Communication between requirements given D C D C 1 VMs not modeled 3 1 3 1 The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 1 4 2 / 2 4 / 5 2 / 3 1 A B AC B 1 / 3 ◮ Map virtual edges to paths in the substrate 6 1 1 1 / 2 1 / 2 ◮ Respecting capacities & mapping restrictions 1 / 2 1 / 2 D C D 1 0 / 0 3 / 3 1 / 1 3 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 7

  8. ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) 1 4 ◮ Customer specifies 1 4 1 A B A B number and ‘size’ of ◮ Additionally: Virtual Machines 6 communication 1 1 ◮ Communication between requirements given D C D C 1 VMs not modeled 3 1 3 1 The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 1 4 2 / 2 4 / 5 2 / 3 1 A B AC B 1 / 3 ◮ Map virtual edges to paths in the substrate Embedding 6 1 1 1 / 2 1 / 2 ◮ Respecting capacities & mapping restrictions 1 / 2 1 / 2 D C D 1 0 / 0 3 / 3 1 / 1 3 1 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 8

  9. The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 2 / 2 4 / 5 1 4 2 / 3 1 A B AC B 1 / 3 ◮ Map virtual edges to paths in the substrate Embedding 6 1 1 1 / 2 1 / 2 ◮ Respecting capacities & mapping restrictions 1 / 2 1 / 2 D C D 1 3 / 3 3 1 0 / 0 1 / 1 Related Work ◮ VNEP (and related problems) studied intensively in the networking community: > 100 papers. ◮ VNEP is related to classical problems as, e.g., subgraph isomorphism, but different . . . ◮ No approximations known for arbitrary virtual networks graphs. Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 9

  10. The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 2 / 2 4 / 5 1 4 2 / 3 1 A B AC B 1 / 3 ◮ Map virtual edges to paths in the substrate Embedding 6 1 1 1 / 2 1 / 2 ◮ Respecting capacities & mapping restrictions 1 / 2 1 / 2 D C D 1 3 / 3 3 1 0 / 0 1 / 1 Related Work ◮ VNEP (and related problems) studied intensively in the networking community: > 100 papers. ◮ VNEP is related to classical problems as, e.g., subgraph isomorphism, but different . . . ◮ No approximations known for arbitrary virtual networks graphs. Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 10

  11. The Virtual Network Embedding Problem (VNEP) Virtual Network Substrate (Physical Network) ◮ Map virtual nodes to substrate nodes 2 / 2 4 / 5 1 4 2 / 3 1 A B AC B 1 / 3 ◮ Map virtual edges to paths in the substrate Embedding 6 1 1 1 / 2 1 / 2 ◮ Respecting capacities & mapping restrictions 1 / 2 1 / 2 D C D 1 3 / 3 3 1 0 / 0 1 / 1 Related Work ◮ VNEP (and related problems) studied intensively in the networking community: > 100 papers. ◮ VNEP is related to classical problems as, e.g., subgraph isomorphism, but different . . . ◮ No approximations known for arbitrary virtual networks graphs. Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 11

  12. Related Work ◮ VNEP (and related problems) studied intensively in the networking community: > 100 papers. ◮ VNEP is related to classical problems as, e.g., subgraph isomorphism, but different . . . ◮ No approximations known for arbitrary virtual networks graphs. Focus: Offline Variant Setting Multiple Virtual Network requests are given Objectives Maximize profit (admission control) or minimize ‘cost’ s.t. capacity constraints . Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 12

  13. Related Work ◮ VNEP (and related problems) studied intensively in the networking community: > 100 papers. ◮ VNEP is related to classical problems as, e.g., subgraph isomorphism, but different . . . ◮ No approximations known for arbitrary virtual networks graphs. Focus: Offline Variant Setting Multiple Virtual Network requests are given Objectives Maximize profit (admission control) or minimize ‘cost’ s.t. capacity constraints . Approach: Randomized Rounding à la Raghavan & Thompson f k m k ◮ Compute opt. ‘convex combinations’ of mappings: D r = { ( ) } k for request r , r r ���� ���� weight ≥ 0 mapping ◮ Probabilistically select mapping m k r according to weight f k r for each request r ◮ Yields: approximate solutions of bounded resource augmentations with high probability Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 13

  14. Approach: Randomized Rounding à la Raghavan & Thompson f k m k ◮ Compute opt. ‘convex combinations’ of mappings: D r = { ( ) } k for request r , r r ���� ���� mapping weight ≥ 0 ◮ Probabilistically select mapping m k r according to weight f k r for each request r ◮ Yields: approximate solutions of bounded resource augmentations with high probability Main Challenge: Computing (Convex Combinations) of Valid Mappings ◮ Classic LP Formulation yields no meaningful solutions ( → unbounded integrality gap) 1 2 i i 1 1 2 k 2 j k j 1 1 1 2 j 2 j 2 k 1 2 i Substrate Request Classic LP Solution ◮ Observation: Need to fix confluence targets (here: node k ) a priori. Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 14

  15. Approach: Randomized Rounding à la Raghavan & Thompson f k m k ◮ Compute opt. ‘convex combinations’ of mappings: D r = { ( ) } k for request r , r r ���� ���� mapping weight ≥ 0 ◮ Probabilistically select mapping m k r according to weight f k r for each request r ◮ Yields: approximate solutions of bounded resource augmentations with high probability Main Challenge: Computing (Convex Combinations) of Valid Mappings ◮ Classic LP Formulation yields no meaningful solutions ( → unbounded integrality gap) 1 2 i i i 1 1 2 k 2 j k j k j 1 1 1 2 j 2 j 2 k 1 2 i Substrate Request Classic LP Solution Extraction Order ◮ Observation: Need to fix confluence targets (here: node k ) a priori. Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 15

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