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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

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SLIDE 1

Approximate Graph Embeddings in the Cloud

2 5 3 1 3 2 2 2 2

AC B D

Highlights of Algorithms 2018 Matthias Rost

Technische Universität Berlin, Internet Network Architectures

Stefan Schmid

Universität Wien, Communication Technologies

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SLIDE 2

Cloud Providers Offer Data Center Resources

Cloud Data Center (Amazon, Google, . . . ) Customers

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 2

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SLIDE 3

Cloud Providers Offer Data Center Resources

Cloud Data Center (Amazon, Google, . . . ) Customers

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 3

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SLIDE 4

Cloud Providers Offer Data Center Resources

Cloud Data Center (Amazon, Google, . . . ) Customers

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Goal: Virtual Networks (since ≈ 2006)

A B C D

1 4 3 1 1 1 1 1 6

◮ Additionally:

communication requirements given

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 4

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SLIDE 5

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Goal: Virtual Networks (since ≈ 2006)

A B C D

1 4 3 1 1 1 1 1 6

◮ Additionally:

communication requirements given The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

2 2 2 2 3 3 1 4 3 1 2 5 1 3 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 5

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SLIDE 6

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Goal: Virtual Networks (since ≈ 2006)

A B C D

1 4 3 1 1 1 1 1 6

◮ Additionally:

communication requirements given The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

2 2 2 2 3 3 1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 6

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SLIDE 7

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Goal: Virtual Networks (since ≈ 2006)

A B C D

1 4 3 1 1 1 1 1 6

◮ Additionally:

communication requirements given The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 1/2 1/2 1/2 1/2 2/3 1/3 Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 7

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SLIDE 8

‘Classic’ Cloud Computing

◮ Customer specifies

number and ‘size’ of Virtual Machines

◮ Communication between

VMs not modeled

A B C D

1 4 3 1

Goal: Virtual Networks (since ≈ 2006)

A B C D

1 4 3 1 1 1 1 1 6

◮ Additionally:

communication requirements given The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 1/2 1/2 1/2 1/2 2/3 1/3

Embedding

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 8

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SLIDE 9

The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 1/2 1/2 1/2 1/2 2/3 1/3

Embedding

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

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SLIDE 10

The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 1/2 1/2 1/2 1/2 2/3 1/3

Embedding

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

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SLIDE 11

The Virtual Network Embedding Problem (VNEP)

◮ Map virtual nodes to substrate nodes ◮ Map virtual edges to paths in the substrate ◮ Respecting capacities & mapping restrictions

A B C D

1 1 1 1 6

Substrate (Physical Network) Virtual Network

1 4 3 1

AC B D

2/2 4/5 0/0 1/1 3/3 1/2 1/2 1/2 1/2 2/3 1/3

Embedding

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

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SLIDE 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

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SLIDE 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

◮ Compute opt. ‘convex combinations’ of mappings: Dr = {(

f k

r

  • weight ≥0

, mk

r

  • mapping

)}k for request r

◮ Probabilistically select mapping mk 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

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SLIDE 14

Approach: Randomized Rounding à la Raghavan & Thompson

◮ Compute opt. ‘convex combinations’ of mappings: Dr = {(

f k

r

  • weight ≥0

, mk

r

  • mapping

)}k for request r

◮ Probabilistically select mapping mk 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)

Request i j k Substrate

1 2i 1 2j 1 2k 1 2i 1 2k 1 2j 1 2j

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

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SLIDE 15

Approach: Randomized Rounding à la Raghavan & Thompson

◮ Compute opt. ‘convex combinations’ of mappings: Dr = {(

f k

r

  • weight ≥0

, mk

r

  • mapping

)}k for request r

◮ Probabilistically select mapping mk 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)

Request i j k Substrate

1 2i 1 2j 1 2k 1 2i 1 2k 1 2j 1 2j

Classic LP Solution Extraction Order i j k

◮ 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

slide-16
SLIDE 16

Approach: Randomized Rounding à la Raghavan & Thompson

◮ Compute opt. ‘convex combinations’ of mappings: Dr = {(

f k

r

  • weight ≥0

, mk

r

  • mapping

)}k for request r

◮ Probabilistically select mapping mk 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)

Request i j k Substrate

1 2i 1 2j 1 2k 1 2i 1 2k 1 2j 1 2j

Classic LP Solution Extraction Order i j k

◮ 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 16

slide-17
SLIDE 17

Approach: Randomized Rounding à la Raghavan & Thompson

◮ Compute opt. ‘convex combinations’ of mappings: Dr = {(

f k

r

  • weight ≥0

, mk

r

  • mapping

)}k for request r

◮ Probabilistically select mapping mk 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)

Request i j k Substrate

1 2i 1 2j 1 2k 1 2i 1 2k 1 2j 1 2j

Classic LP Solution Extraction Order i j k

◮ 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 17

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SLIDE 18

Main Challenge: Computing (Convex Combinations) of Valid Mappings

◮ Classic LP Formulation yields no meaningful solutions (→ unbounded integrality gap) ◮ Observation: Need to fix confluence targets a priori.

Main Contributions

◮ LP Formulations for cactus request graphs → first approximation algorithmsa ◮ Derivation of heuristics & extensive computational evaluationa ◮ Extension to arbitrary virtual network topologies → FPT-approximationsb ◮ FPT required: no poly.-time algorithms for computing valid mappings for general graphsc

aMatthias Rost and Stefan Schmid. “Virtual Network Embedding Approximations: Leveraging

Randomized Rounding”. In: Proc. IFIP Networking. 2018.

bMatthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network

Embedding Problem.

  • Tech. rep. Mar. 2018. URL: http://arxiv.org/abs/1803.04452.

cMatthias Rost and Stefan Schmid. “Charting the Complexity Landscape of Virtual Network

Embeddings”. In: Proc. IFIP Networking. 2018.

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 18

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SLIDE 19

Main Challenge: Computing (Convex Combinations) of Valid Mappings

◮ Classic LP Formulation yields no meaningful solutions (→ unbounded integrality gap) ◮ Observation: Need to fix confluence targets a priori.

Main Contributions

◮ LP Formulations for cactus request graphs → first approximation algorithmsa ◮ Derivation of heuristics & extensive computational evaluationa ◮ Extension to arbitrary virtual network topologies → FPT-approximationsb ◮ FPT required: no poly.-time algorithms for computing valid mappings for general graphsc

aMatthias Rost and Stefan Schmid. “Virtual Network Embedding Approximations: Leveraging

Randomized Rounding”. In: Proc. IFIP Networking. 2018.

bMatthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network

Embedding Problem.

  • Tech. rep. Mar. 2018. URL: http://arxiv.org/abs/1803.04452.

cMatthias Rost and Stefan Schmid. “Charting the Complexity Landscape of Virtual Network

Embeddings”. In: Proc. IFIP Networking. 2018.

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 19

slide-20
SLIDE 20

Main Challenge: Computing (Convex Combinations) of Valid Mappings

◮ Classic LP Formulation yields no meaningful solutions (→ unbounded integrality gap) ◮ Observation: Need to fix confluence targets a priori.

Main Contributions

◮ LP Formulations for cactus request graphs → first approximation algorithmsa ◮ Derivation of heuristics & extensive computational evaluationa ◮ Extension to arbitrary virtual network topologies → FPT-approximationsb ◮ FPT required: no poly.-time algorithms for computing valid mappings for general graphsc

aMatthias Rost and Stefan Schmid. “Virtual Network Embedding Approximations: Leveraging

Randomized Rounding”. In: Proc. IFIP Networking. 2018.

bMatthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network

Embedding Problem.

  • Tech. rep. Mar. 2018. URL: http://arxiv.org/abs/1803.04452.

cMatthias Rost and Stefan Schmid. “Charting the Complexity Landscape of Virtual Network

Embeddings”. In: Proc. IFIP Networking. 2018.

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 20

slide-21
SLIDE 21

Main Challenge: Computing (Convex Combinations) of Valid Mappings

◮ Classic LP Formulation yields no meaningful solutions (→ unbounded integrality gap) ◮ Observation: Need to fix confluence targets a priori.

Main Contributions

◮ LP Formulations for cactus request graphs → first approximation algorithmsa ◮ Derivation of heuristics & extensive computational evaluationa ◮ Extension to arbitrary virtual network topologies → FPT-approximationsb ◮ FPT required: no poly.-time algorithms for computing valid mappings for general graphsc

aMatthias Rost and Stefan Schmid. “Virtual Network Embedding Approximations: Leveraging

Randomized Rounding”. In: Proc. IFIP Networking. 2018.

bMatthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network

Embedding Problem.

  • Tech. rep. Mar. 2018. URL: http://arxiv.org/abs/1803.04452.

cMatthias Rost and Stefan Schmid. “Charting the Complexity Landscape of Virtual Network

Embeddings”. In: Proc. IFIP Networking. 2018.

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 21

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SLIDE 22

Main Contributions

◮ LP Formulations for cactus request graphs → first approximation algorithmsa ◮ Derivation of heuristics & extensive computational evaluationa ◮ Extension to arbitrary virtual network topologies → FPT-approximationsb ◮ FPT required: no poly.-time algorithms for computing valid mappings for general graphsc

aMatthias Rost and Stefan Schmid. “Virtual Network Embedding Approximations: Leveraging

Randomized Rounding”. In: Proc. IFIP Networking. 2018.

bMatthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network

Embedding Problem.

  • Tech. rep. Mar. 2018. URL: http://arxiv.org/abs/1803.04452.

cMatthias Rost and Stefan Schmid. “Charting the Complexity Landscape of Virtual Network

Embeddings”. In: Proc. IFIP Networking. 2018.

Thanks for your attention!

Matthias Rost (TU Berlin) Approximate Graph Embeddings in the Cloud Highlights of Algorithms 2018 22