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Approximating the Virtual Network Embedding Problem: Theory and Practice
2 5 3 1 3 2 2 2 2
AC B D
Approximating the Virtual Network Embedding Problem: Theory and - - PowerPoint PPT Presentation
Approximating the Virtual Network Embedding Problem: Theory and - - PowerPoint PPT Presentation
Approximating the Virtual Network Embedding Problem: Theory and Practice 2 5 3 AC B 2 2 2 2 D 0 3 1 23rd International Symposium on Mathematical Programming 2018 Bordeaux, France Matthias Rost Technische Universitt Berlin,
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Operators offer their Network Resources
Data Center Network Wide-Area Network
Substrate (Physical Network) Directed graph GS = (VS, ES) Capacities cS : GS → R≥0
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 3
SLIDE 4
Operators offer their Network Resources
Data Center Network Wide-Area Network
Substrate (Physical Network) Directed graph GS = (VS, ES) Capacities cS : GS → R≥0 ‘Classic’ Cloud Computing
A B C D
1 4 3 1
requests
bunch of VMs
User requests virtual machines No guarantee on network performance
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 3
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Operators offer their Network Resources
Data Center Network Wide-Area Network
Substrate (Physical Network) Directed graph GS = (VS, ES) Capacities cS : GS → R≥0 ‘Classic’ Cloud Computing
A B C D
1 4 3 1
requests
bunch of VMs
User requests virtual machines No guarantee on network performance Goal: Virtual Networks (since ≈ 2006)
A B C D
1 4 3 1 1 1 1 1 6
Virtual Network requests
Communication requirements given Network performance will be guaranteed
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 3
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Operators offer their Network Resources
Data Center Network Wide-Area Network
Substrate (Physical Network) Directed graph GS = (VS, ES) Capacities cS : GS → R≥0 Goal: Virtual Networks (since ≈ 2006)
A B C D
1 4 3 1 1 1 1 1 6
Virtual Network requests
Virtual Network Request Gr = (Vr, Er) demands dr : Gr → R≥0 mapping restrictions
V i
S ⊆ VS for i ∈ Vr
E i,j
S ⊆ ES for (i, j) ∈ Er
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
SLIDE 7
Goal: Virtual Networks (since ≈ 2006)
A B C D
1 4 3 1 1 1 1 1 6
Virtual Network requests
Virtual Network Request Gr = (Vr, Er) demands dr : Gr → R≥0 mapping restrictions
V i
S ⊆ VS for i ∈ Vr
E i,j
S ⊆ ES for (i, j) ∈ Er
Valid Mapping
A B C D AC B D
Substrate (Physical Network) Virtual Network Embedding
Def: Valid mapping mr = (mV , mE) . . .
mV : Vr → VS and mE : Er → P(ES) satisfies valid connectivity: mV (i)
mE (i,j)
- mV (j)
valid node mapping: mV (i) ∈ V i
S
valid edge mapping: mE(i, j) ⊆ E i,j
S
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
SLIDE 8
Goal: Virtual Networks (since ≈ 2006)
A B C D
1 4 3 1 1 1 1 1 6
Virtual Network requests
Virtual Network Request Gr = (Vr, Er) demands dr : Gr → R≥0 mapping restrictions
V i
S ⊆ VS for i ∈ Vr
E i,j
S ⊆ ES for (i, j) ∈ Er
Feasible Embedding
A B C D AC B D
1 1 1 1 6
Substrate (Physical Network) Virtual Network Embedding
1/2 1/2 1/2 1/2 2/3 1/3 1 4 3 1 2/2 4/5 0/0 1/1 3/3
Def: Valid mapping mr = (mV , mE) . . .
mV : Vr → VS and mE : Er → P(ES) satisfies valid connectivity: mV (i)
mE (i,j)
- mV (j)
valid node mapping: mV (i) ∈ V i
S
valid edge mapping: mE(i, j) ⊆ E i,j
S
Def: Feasible embedding mr . . . . . . is valid and respects capacities.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
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Goal: Virtual Networks (since ≈ 2006)
A B C D
1 4 3 1 1 1 1 1 6
Virtual Network requests
Virtual Network Request Gr = (Vr, Er) demands dr : Gr → R≥0 mapping restrictions
V i
S ⊆ VS for i ∈ Vr
E i,j
S ⊆ ES for (i, j) ∈ Er
Feasible Embedding
A B C D AC B D
1 1 1 1 6
Substrate (Physical Network) Virtual Network Embedding
1/2 1/2 1/2 1/2 2/3 1/3 1 4 3 1 2/2 4/5 0/0 1/1 3/3
Def: Feasible embedding mr . . . . . . is valid and respects capacities. Virtual Network Embedding Problem Setting Online vs. Offline Objectives resource minimization, profit maximization, energy minimization, . . .
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
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Related Work & Overview of Contributions
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Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
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Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains VNEP is of crucial importance, yet is hardly understood!
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
SLIDE 13
Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing NP-completeness and inapproximability. (FPT-)Linear Programs for computing convex combinations of valid mappings. (FPT-)Approximations for offline VNEP based on randomized rounding. Computational evaluation of derived heuristics for offline profit VNEP.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
SLIDE 14
Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing NP-completeness and inapproximability. (FPT-)Linear Programs for computing convex combinations of valid mappings. (FPT-)Approximations for offline VNEP based on randomized rounding. Computational evaluation of derived heuristics for offline profit VNEP.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
SLIDE 15
Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing NP-completeness and inapproximability. (FPT-)Linear Programs for computing convex combinations of valid mappings. (FPT-)Approximations for offline VNEP based on randomized rounding. Computational evaluation of derived heuristics for offline profit VNEP.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
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Related Work
Computational Complexity
Andersen [2002] NP-hardness (argument) Amaldi et al. [2016] NP-hardness and inapproximability for offline VNEP (profit)
Heuristics & Exact Algorithms
Generally ≫ 100 works, e.g. . . . Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations...
Approximations
None for general graphs! Bansal et al. [2011] for trees Even et al. [2016] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing NP-completeness and inapproximability. (FPT-)Linear Programs for computing convex combinations of valid mappings. (FPT-)Approximations for offline VNEP based on randomized rounding. Computational evaluation of derived heuristics for offline profit VNEP.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
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Idea of this Talk: Give Overview on Our Results Complexity results showing NP-completeness and inapproximabilitya. (FPT-)Linear Programs for computing convex combinations of valid mappingsb,c. (FPT-)Approximations for offline VNEP based on randomized roundingb,c. Computational evaluation of derived heuristics for offline profit VNEPb.
a
Matthias Rost and Stefan Schmid. Charting the Complexity Landscape of Virtual Network
- Embeddings. In Proc. IFIP Networking, 2018c
b Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging
Randomized Rounding. In Proc. IFIP Networking, 2018d
c Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network
Embedding Problem. Technical report, March 2018a. URL http://arxiv.org/abs/1803.04452
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
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Complexity of the VNEP1
1Matthias Rost and Stefan Schmid. Charting the Complexity Landscape of Virtual Network Embeddings.
In Proc. IFIP Networking, 2018c
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Reminder: 3-SAT and NP-Completeness
3-SAT-Formula φ φ =
Ci∈Cφ Ci with Ci ∈ Cφ being disjunctions of at most 3 (possible negated) literals.
Example 3-SAT formula φ over literals Lφ = {x1, x2, x3, x4} φ = (x1 ∨ x2 ∨ x3)
- C1
∧ (¯ x1 ∨ x2 ∨ x4)
- C2
∧ (x2 ∨ ¯ x3 ∨ x4)
- C3
Definition of 3-SAT Decide whether satisfying assignment a : Lφ → {F, T} exists for formula φ. Output: Yes/No. Theorem: Karp [1972] 3-SAT is NP-complete.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 8
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Methodology: Proving NP-completeness
Proving NP-completeness of the VNEP
1 VNEP lies in NP (answer can be checked in polynomial time). 2 Reduction from 3-SAT to VNEP.
Outline of Reduction Framework
3-SAT instance φ VNEP instance (Gr(φ), GS(φ), restrictions) φ satisfiable? feasible embedding of Gr(φ) on GS(φ) under restrictions?
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 9
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Our Reduction Framework
Proving NP-completeness of the VNEP
1 VNEP lies in NP (answer can be checked in polynomial time). 2 Reduction from 3-SAT to VNEP.
Outline of Reduction Framework
3-SAT instance φ VNEP instance (Gr(φ), GS(φ), restrictions) φ satisfiable? feasible embedding of Gr(φ) on GS(φ) under restrictions?
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 10
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Our Reduction Framework
Input: 3-SAT formula φ = (x1 ∨ x2 ∨ x3) ∧ (¯ x1 ∨ x2 ∨ x4) ∧ (x2 ∨ ¯ x3 ∨ x4) Request Gr(φ) Vr(φ) = {vi | Ci ∈ Cφ}
v1 v3 v2
Er(φ) = { (vi, vj) | Ci introduces literal used by Cj } Substrate GS(φ)
- ne node per clause and
per satisfying assignment edges as for the requests, if assignments do not contradict
x1, x2, x3 :TTT x1, x2, x3 :TTF x1, x2, x3 :TFT x1, x2, x4 : TTT x1, x2, x4 : TTF x2, x3, x4 : TTT x2, x3, x4 : TTF x2, x3, x4 : TFT x1, x2, x4 : TFT
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 10
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Complete Picture
φ:
(x1 ∨ x2 ∨ x3) ∧ (¯ x1 ∨ x2 ∨ x4) ∧ (x2 ∨ ¯ x3 ∨ x4)
GS(φ):
x1, x2, x3 : TTT x1, x2, x3 : TTF x1, x2, x3 : TFT x1, x2, x3 : TFF x1, x2, x3 : FTT x1, x2, x3 : FTF x1, x2, x3 : FFT x1, x2, x4 : TTT x1, x2, x4 : TTF x1, x2, x4 : FTT x1, x2, x4 : FTF x1, x2, x4 : FFT x1, x2, x4 : FFF x2, x3, x4 : TTT x2, x3, x4 : TTF x2, x3, x4 : TFT x2, x3, x4 : TFF x2, x3, x4 : FTT x2, x3, x4 : FFT x2, x3, x4 : FFF v1 v3 v2
Gr(φ):
x1, x2, x4 : TFT Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 11
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Our Reduction Framework
Outline of Reduction Framework
3-SAT instance φ VNEP instance (Gr(φ), GS(φ), restrictions) φ satisfiable? feasible embedding of Gr(φ) on GS(φ) under restrictions?
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 12
SLIDE 25
Our Reduction Framework
Outline of Reduction Framework
3-SAT instance φ VNEP instance (Gr(φ), GS(φ), restrictions) φ satisfiable? feasible embedding of Gr(φ) on GS(φ) under restrictions?
Base Lemma Formula φ is satisfiable if and only if there exists a mapping of Gr(φ) on GS(φ), s.t. (1) each virtual node vi is mapped to a ‘satisfying assignment node’ of the i-th clause, and (2) all virtual edges are mapped on exactly one substrate edge.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 12
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Our Reduction Framework
Base Lemma Formula φ is satisfiable if and only if there exists a mapping of Gr(φ) on GS(φ), s.t. (1) each virtual node vi is mapped to a ‘satisfying assignment node’ of the i-th clause, and (2) all virtual edges are mapped on exactly one substrate edge. Example φ = (x1 ∨ x2 ∨ x3) ∧ (¯ x1 ∨ x2 ∨ x4) ∧ (x2 ∨ ¯ x3 ∨ x4)
Assignment for φ x1 →T x2 →T x3 →F x4 →F Request
v1 v3 v2
Embedding satisfying conditions (1) and (2)
x1, x2, x3 : TTF x1, x2, x3 : TFT x1, x2, x3 : TFF x1, x2, x4 : TTF x2, x3, x4 : TTF x2, x3, x4 : TFT x2, x3, x4 : TFF x1, x2, x4 : TFT
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 12
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Our Reduction Framework
Base Lemma Formula φ is satisfiable if and only if there exists a mapping of Gr(φ) on GS(φ), s.t. (1) each virtual node vi is mapped to a ‘satisfying assignment node’ of the i-th clause, and (2) all virtual edges are mapped on exactly one substrate edge. Decision VNEP is NP-complete under mapping restrictions Node placement restrictions enforce (1)
x1, x2, x3 : TTT x1, x2, x3 : TTF x1, x2, x3 : TFT x1, x2, x3 : TFF x1, x2, x4 : TTT x1, x2, x4 : TTF x2, x3, x4 : TTT x2, x3, x4 : TTF x2, x3, x4 : TFT x2, x3, x4 : TFF v1 v3 v2 x1, x2, x4 : TFT
Routing restrictions enforce (2)
x1, x2, x3 : TTT x1, x2, x3 : TTF x1, x2, x3 : TFT x1, x2, x3 : TFF x1, x2, x4 : TTT x1, x2, x4 : TTF x2, x3, x4 : TTT x2, x3, x4 : TTF x2, x3, x4 : TFT x2, x3, x4 : TFF v1 v3 v2 x1, x2, x4 : TFT Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 12
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Our Reduction Framework
Base Lemma Formula φ is satisfiable if and only if there exists a mapping of Gr(φ) on GS(φ), s.t. (1) each virtual node vi is mapped to a ‘satisfying assignment node’ of the i-th clause, and (2) all virtual edges are mapped on exactly one substrate edge. Decision VNEP is NP-complete for degree-bounded, planar request graphs Reduction from planar 3-SAT variant using literals max. 4 times (see Kratochvíl [1994]):
each planar formula φ leads to a planar request graph Gr(φ) each node of Gr(φ) has degree at most 12
v1 v3 v2 u1 u2 u4 u3
planar graph Gφ
u1 u2 u4 u3 v1 v3 v2 u1 u2 u4 u3 v1 v3 v2
planar graph Gr(φ)
v1 v3 v2 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 12
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(FPT-)Linear Programs for Computing Convex Combinations of Valid Mappings2,3
2
Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized
- Rounding. In Proc. IFIP Networking, 2018d
3
Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding
- Problem. Technical report, March 2018a. URL http://arxiv.org/abs/1803.04452
SLIDE 30
Linear Programming: Classic MCF Formulation and its Limits
Classic LP Formulation
Formulation 1: Classic MCF Formulation for the VNEP
- u∈V i
S
yu
r,i= xr
∀r ∈ R, i ∈ Vr (1)
- u∈VS\V i
S
yu
r,i= 0
∀r ∈ R, i ∈ Vr (2)
- (u,v)∈δ+(u)
zu,v
r,i,j
−
- (v,u)∈δ−(u)
zv,u
r,i,j
=
- yu
r,i
−yu
r,j
- ∀
- r ∈ R, (i, j) ∈ Er,
u ∈ VS
- (3)
zu,v
r,i,j= 0
∀
- r ∈ R, (i, j) ∈ Er,
(u, v) ∈ ES \ E i,j
S
- (4)
- i∈Vr,τr(i)=τ
dr(i) · yu
r,i= aτ,u r
∀r ∈ R, (τ, u) ∈ RV
S
(5)
- (i,j)∈Er
dr(i, j) · zu,v
r,i,j= au,v r
∀r ∈ R, (u, v) ∈ ES (6)
- r∈R
ax,y
r
≤ cS(x, y) ∀(x, y) ∈ RS (7)
Main Building Block: Multi-Commodity Flows yu
r,i ∈ [0, 1]: maps node i ∈ Vr on VS
zu,v
r,i,j ∈ [0, 1]: maps (i, j) ∈ Er on (u, v) ∈ ES
- (u,v)∈δ+(u)
zu,v
r,i,j
−
- (v,u)∈δ−(u)
zv,u
r,i,j = yu r,i − yu r,j
(3) Local Connectivity Property Given a (fractional) mapping of i ∈ Vr to u ∈ VS, a ‘valid’ mapping can be recovered for edges incident to i and their respective endpoints.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 14
SLIDE 31
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Solution
Main Building Block: Multi-Commodity Flows yu
r,i ∈ [0, 1]: maps node i ∈ Vr on VS
zu,v
r,i,j ∈ [0, 1]: maps (i, j) ∈ Er on (u, v) ∈ ES
- (u,v)∈δ+(u)
zu,v
r,i,j
−
- (v,u)∈δ−(u)
zv,u
r,i,j = yu r,i − yu r,j
(3) Local Connectivity Property Given a (fractional) mapping of i ∈ Vr to u ∈ VS, a ‘valid’ mapping can be recovered for edges incident to i and their respective endpoints.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 32
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Extraction i j k Order Solution
Main Building Block: Multi-Commodity Flows yu
r,i ∈ [0, 1]: maps node i ∈ Vr on VS
zu,v
r,i,j ∈ [0, 1]: maps (i, j) ∈ Er on (u, v) ∈ ES
- (u,v)∈δ+(u)
zu,v
r,i,j
−
- (v,u)∈δ−(u)
zv,u
r,i,j = yu r,i − yu r,j
(3) Local Connectivity Property Given a (fractional) mapping of i ∈ Vr to u ∈ VS, a ‘valid’ mapping can be recovered for edges incident to i and their respective endpoints.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 33
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Extraction i j k Order Solution
Local Connectivity Property Given a (fractional) mapping of i ∈ Vr to u ∈ VS, a ‘valid’ mapping can be recovered for edges incident to i and their respective endpoints. Main Issue Targets of confluences pose problems! In the example: target k of confluence (i, k), (i, j), (j, k).
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 34
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Extraction i j k Order Solution
Main Issue Targets of confluences pose problems! In the example: target k of confluence (i, k), (i, j), (j, k). Theorem Decomposing solutions to the MCF LP is not possible in general.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 35
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Extraction i j k Order Solution
Main Issue Targets of confluences pose problems! In the example: target k of confluence (i, k), (i, j), (j, k). Theorem Decomposing solutions to the MCF LP is not possible in general. Theorem MCF LP Formulation has infinite integrality gap.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 36
Linear Programming: Classic MCF Formulation and its Limits
Example
Request i j k Substrate
1 2i 1 2j 1 2k
LP
1 2i 1 2k 1 2j 1 2j
Extraction i j k Order Solution
Main Issue Targets of confluences pose problems! In the example: target k of confluence (i, k), (i, j), (j, k). Key Insight If we fix confluence target nodes valid mappings can always be extracted, when following the extraction order. In the example: Consider one sub-LP formulation per potential mapping location of k.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 15
SLIDE 37
Outline of Novel Decomposable LP Formulations
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 38
Outline of Novel Decomposable LP Formulations
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders If edge e lies on confluence C X
i,j, then it is labeled with the confluence’s target j.
Labeling can be computed in polynomial-time (by applying Menger’s theorem). Each label has unique root node at which the mapping of the label must be fixed. Outgoing edges are partitioned into edge bags not sharing labels.
b c a k {c} {a} {k}
{a, b, c}
{a, b}
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 39
Outline of Novel Decomposable LP Formulations
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders If edge e lies on confluence C X
i,j, then it is labeled with the confluence’s target j.
Labeling can be computed in polynomial-time (by applying Menger’s theorem). Each label has unique root node at which the mapping of the label must be fixed. Outgoing edges are partitioned into edge bags not sharing labels.
b c a k {c} {a} {k}
{a, b, c}
{a, b} {j, l} i l j k {j} {i, l} {k} {l}
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 40
Outline of Novel Decomposable LP Formulations
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders If edge e lies on confluence C X
i,j, then it is labeled with the confluence’s target j.
Labeling can be computed in polynomial-time (by applying Menger’s theorem). Each label has unique root node at which the mapping of the label must be fixed. Outgoing edges are partitioned into edge bags not sharing labels.
b c a k {c} {a} {k}
{a, b, c}
{a, b} {j, l} i l j k {j} {i, l} {k} {l}
Edge Bags
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 41
Outline of Novel Decomposable LP Formulations
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders If edge e lies on confluence C X
i,j, then it is labeled with the confluence’s target j.
Labeling can be computed in polynomial-time (by applying Menger’s theorem). Each label has unique root node at which the mapping of the label must be fixed. Outgoing edges are partitioned into edge bags not sharing labels.
b c a k {c} {a} {k}
{a, b, c}
{a, b}
a
c
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 42
Outline of Novel Decomposable LP Formulations
Pre-Processing Extraction Orders If edge e lies on confluence C X
i,j, then it is labeled with the confluence’s target j.
Labeling can be computed in polynomial-time (by applying Menger’s theorem). Each label has unique root node at which the mapping of the label must be fixed. Outgoing edges are partitioned into edge bags not sharing labels.
b c a k {c} {a} {k}
{a, b, c}
{a, b} b c a k {k}
{a, b, c}
b c a {c} {a, b}
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 16
SLIDE 43
Outline of Novel Decomposable LP Formulation
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders Generation of Linear Program If edge e ∈ E X
r
lies on confluence C X
i,j, then
it is labeled with the confluence’s target j. If e ∈ Er is labeled with LX
r,e, then |VS||LX
r,e|
many commodities are considered for e. Outgoing edges are partitioned into edge bags not sharing labels. For each edge bag variables are introduced to enumerate all potential label mappings. Each label has unique root node at which the mapping of the label must be fixed. Root nodes of labels ‘decide’ on the confluence’s mapping.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 17
SLIDE 44
Outline of Novel Decomposable LP Formulation
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders Generation of Linear Program If edge e ∈ E X
r
lies on confluence C X
i,j, then
it is labeled with the confluence’s target j. If e ∈ Er is labeled with LX
r,e, then |VS||LX
r,e|
many commodities are considered for e. Outgoing edges are partitioned into edge bags not sharing labels. For each edge bag variables are introduced to enumerate all potential label mappings. Each label has unique root node at which the mapping of the label must be fixed. Root nodes of labels ‘decide’ on the confluence’s mapping.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 17
SLIDE 45
Outline of Novel Decomposable LP Formulation
Extraction Order G X
r
Rooted acyclic reorientation of the original request graph Gr. G X
r
is not unique! Confluence C X
i,j
A confluence C X
i,j from i to j is a pair of
(node-)disjoint paths connecting i to j in G X
r .
Pre-Processing Extraction Orders Generation of Linear Program If edge e ∈ E X
r
lies on confluence C X
i,j, then
it is labeled with the confluence’s target j. If e ∈ Er is labeled with LX
r,e, then |VS||LX
r,e|
many commodities are considered for e. Outgoing edges are partitioned into edge bags not sharing labels. For each edge bag variables are introduced to enumerate all potential label mappings. Each label has unique root node at which the mapping of the label must be fixed. Root nodes of labels ‘decide’ on the confluence’s mapping.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 17
SLIDE 46
Outline of Novel Decomposable LP Formulation
Pre-Processing Extraction Orders Generation of Linear Program If edge e ∈ E X
r
lies on confluence C X
i,j, then
it is labeled with the confluence’s target j. If e ∈ Er is labeled with LX
r,e, then |VS||LX
r,e|
many commodities are considered for e. Outgoing edges are partitioned into edge bags not sharing labels. For each edge bag variables are introduced to enumerate all potential label mappings. Each label has unique root node at which the mapping of the label must be fixed. Root nodes of labels ‘decide’ on the confluence’s mapping.
Stitching Flow Variables via Node Mapping Variables
yi1
r,i(i, a), [i → i1, l → l1]
yi1
r,f(i, a), [i → i1, l → l2]
yi1
r,i(i, a), [i → i2, l → l1]
yi1
r,i(i, a), [i → i2, l → l2]
γi1
r,i,1,[j→j1,l→l1]
{j} {i, l} {j, l} a i c f
γi1
r,i,1,[j→j1,l→l2]
γi1
r,i,1,[j→j2,l→l1]
γi1
r,i,1,[j→j2,l→l2]
yi1
r,i(i, c), [j → j1]
yi1
r,i(i, c), [j → j2]
yi1
r,i(f, i), [j → j1, l → l1]
yi1
r,i(f, i), [j → j1, l → l2]
yi1
r,i(f, i), [j → j2, l → l1]
yi1
r,i(f, i), [j → j2, l → l2]
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 17
SLIDE 47
Outline of Novel Decomposable LP Formulation
Pre-Processing Extraction Orders Generation of Linear Program If edge e ∈ E X
r
lies on confluence C X
i,j, then
it is labeled with the confluence’s target j. If e ∈ Er is labeled with LX
r,e, then |VS||LX
r,e|
many commodities are considered for e. Outgoing edges are partitioned into edge bags not sharing labels. For each edge bag variables are introduced to enumerate all potential label mappings. Each label has unique root node at which the mapping of the label must be fixed. Root nodes of labels ‘decide’ on the confluence’s mapping.
- Def. Extraction Width ewX (G X
r )
. . . is the size of the largest edge bag plus
- ne of the extraction order ewX (G X
r ).
Proof of Decomposability . . . via decomposition algorithm. Overall runtime O(poly(|GS|ewX (G X
r ) · |Gr|)). Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 17
SLIDE 48
Novel Decomposable LP Formulation: Takeaways
Overview of Construction Request Graph Gr ⇓ Extraction Order G X
r
⇓ Labeling / Extraction Width ewX (G X
r )
⇓ LP of size O(poly(|GS|ewX (G X
r ) · |Gr|))
⇓ Decomposition Algorithm with runtime O(poly(|GS|ewX (G X
r ) · |Gr|))
⇓ Convex Combinations of valid mappings: Dr = {(f k
r , mk r )|f k r > 0, mk r ∈ Mr}
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 18
SLIDE 49
Extraction Width – Overview of Results
Which graphs have bounded extraction width? How to find extraction orders of small width?
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 19
SLIDE 50
Extraction Width – Overview of Results
Which graphs have bounded extraction width? How to find extraction orders of small width? Extraction Width: Overview of Results Extraction width may vary by factor Ω(|Vr|) Minimizing extraction width is NP-hard (via reduction from Vertex-Cover) Cactus graphs (cycles intersect in at most a single node) have bounded extraction width ewX(GX
w) ≥ |Vr|/2
ewX(GX
w) = 2
w1 w2 w3 wn/2 wn wn/2+1 wn/2+2 w1 w2 w3 wn/2 wn wn/2+1 wn/2+2 wc wc
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 19
SLIDE 51
Extraction Width – Overview of Results
Which graphs have bounded extraction width? How to find extraction orders of small width? Extraction Width: Overview of Results Extraction width may vary by factor Ω(|Vr|) Minimizing extraction width is NP-hard (via reduction from Vertex-Cover) Cactus graphs (cycles intersect in at most a single node) have bounded extraction width
Vertex Cover
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 19
SLIDE 52
Extraction Width – Overview of Results
Which graphs have bounded extraction width? How to find extraction orders of small width? Extraction Width: Overview of Results Extraction width may vary by factor Ω(|Vr|) Minimizing extraction width is NP-hard (via reduction from Vertex-Cover) Cactus graphs (cycles intersect in at most a single node) have bounded extraction width Real World Cactus Graph Examples
Customer Internet LB1 LB2 Cache FW NAT VM1 VM2 VM3 VM4 VM5
Service Chain Virtual Cluster
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 19
SLIDE 53
Extraction Width – Overview of Results
Extraction Width: Overview of Results Extraction width may vary by factor Ω(|Vr|) Minimizing extraction width is NP-hard (via reduction from Vertex-Cover) Cactus graphs (cycles intersect in at most a single node) have bounded extraction width Real World Cactus Graph Examples
Customer Internet LB1 LB2 Cache FW NAT VM1 VM2 VM3 VM4 VM5
Service Chain Virtual Cluster Can we do substantially better? No! Computing valid mappings for planar graphs is NP-complete ⇒ FPT algorithms are necessary.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 19
SLIDE 54
(FPT-)Approximations for offline VNEP based on Randomized Rounding2,3
2
Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized
- Rounding. In Proc. IFIP Networking, 2018d
3
Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding
- Problem. Technical report, March 2018a. URL http://arxiv.org/abs/1803.04452
SLIDE 55
Approximating the Offline VNEP
Profit Variant A set of request R = {r1, r2, . . .} is given. Profit for request pr > 0. Task: Embed subset of requests feasibly maximizing the attained profit.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 56
Approximating the Offline VNEP
Profit Variant A set of request R = {r1, r2, . . .} is given. Profit for request pr > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r1, r2, . . .} is given. Substrate resource costs kS : GS → R≥0. Task: Find feasible embeddings for all requests minimizing cost.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 57
Approximating the Offline VNEP
Focus: Profit Variant A set of request R = {r1, r2, . . .} is given. Profit for request pr > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r1, r2, . . .} is given. Substrate resource costs kS : GS → R≥0. Task: Find feasible embeddings for all requests minimizing cost.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 58
Approximating the Offline VNEP
Focus: Profit Variant A set of request R = {r1, r2, . . .} is given. Profit for request pr > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r1, r2, . . .} is given. Substrate resource costs kS : GS → R≥0. Task: Find feasible embeddings for all requests minimizing cost. Combine Single Decomposable LP Formulations while . . . . . . enforcing capacity constraints and maximizing the profit. ⇒ LP for offline VNEP (profit).
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 59
Approximating the Offline VNEP
Combine Single Decomposable LP Formulations while . . . . . . enforcing capacity constraints and maximizing the profit. ⇒ LP for offline VNEP (profit). Request A B C D
1 4 3 1 1 1 1 1 6
Valid Mappings Mr = {m1
r , m2 r , m3 r , . . .}
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
. . . Valid mappings do not necessarily respect capacity constraints!
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 60
Approximating the Offline VNEP
Combine Single Decomposable LP Formulations while . . . . . . enforcing capacity constraints and maximizing the profit. ⇒ LP for offline VNEP (profit). Decomposable LP Formulation allows us to solve Fractional VNEP
Is k-th mapping of request r chosen? Select at most one mapping: Enforce capacity for each resource x: Maximize the profit: f k
r ∈ {0, 1}
∀r ∈ R, mk
r ∈ Mr
(8)
- mk
r ∈Mr
f k
r ≤ 1
∀r ∈ R (9)
- r∈R
- mk
r ∈Mr
A(mk
r , x) · f k r ≤ cS(x)
∀x ∈ RS (10) max
- r∈R
- mk
r ∈Mr
prf k
r
(11)
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 21
SLIDE 61
Randomized Rounding Revisited
Example Substrate Network
2 5 3 1 3 2 2 2 2
Request r1: profit 100$
A B C D
1 4 3 1 1 1 1 1 6
Request r2: profit 50$
E F G
1 2 1 3 1 2 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 62
Randomized Rounding Revisited
Example Substrate Network
2 5 3 1 3 2 2 2 2
Request r1: profit 100$
A B C D
1 4 3 1 1 1 1 1 6
Request r2: profit 50$
E F G
1 2 1 3 1 2
Example Solution to Linear Program: Profit 133$ Variables of r1 (profit: 100$) f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . .
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 63
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of r1 (profit: 100$) f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 64
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of r1 (profit: 100$) f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 65
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of r2 (profit: 50$) f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load 1 m1
1
m2
2
150$ 200%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 66
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of request 1 f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load 1 m1
1
m2
2
150$ 200% 2 m3
1
∅ 100$ 100%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 67
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of request 1 f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load 1 m1
1
m2
2
150$ 200% 2 m3
1
∅ 100$ 100% 3 m1
1
m1
2
150$ 200%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 68
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of request 1 f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load 1 m1
1
m2
2
150$ 200% 2 m3
1
∅ 100$ 100% 3 m1
1
m1
2
150$ 200% 4 m2
1
m1
2
150$ 200%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 69
Randomized Rounding Revisited
Example Solution to Linear Program: Profit 133$ Variables of request 1 f 1
1 = 0.5
AC B D
1/2 1/2 1/2 1/2 2/3 1/3 2/2 4/5 0/0 3/3 0/0
f 2
1 = 0.3
AC
2/2 7/5 0/0 0/3 2/3 2/3 0/0 0/0 0/0 0/0 0/0
BD
f 3
1 = 0.2
AC B D
2/2 2/2 1/3 1/3 2/2 4/5 0/0 3/3 0/0 0/0 0/0
. . . Variables of r2 (profit: 50$) f 1
2 = 0.5
1/2 3/3 0/3 2/2 1/5 0/0 1/3 0/0
G E F
1/2 2/2 2/2
f 2
2 = 0.16
EF G
0/2 3/3 1/3 2/2 2/5 0/0 0/0 0/3 0/2 0/2 0/2
f 3
2 = 0
EFG
0/2 0/3 0/2 5/5 0/0 0/0 0/3 0/2 0/2 0/2
. . . Idea: Treat weights as probabilities!
Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution Rounding Outcomes Iter.
- Req. 1
- Req. 2
Profit max Load 1 m1
1
m2
2
150$ 200% 2 m3
1
∅ 100$ 100% 3 m1
1
m1
2
150$ 200% 4 m2
1
m1
2
150$ 200% . . . . . . . . . . . . . . .
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 22
SLIDE 70
First (FPT-)Approximation Algorithm for VNEP
Randomized Rounding Approximation
Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Algorithm: RoundingProcedure
Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
end return solution
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 23
SLIDE 71
First (FPT-)Approximation Algorithm for VNEP
Randomized Rounding Approximation
Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Main Theorem: (FPT-)Approximation for the Virtual Network Embedding Problem
The Algorithm returns (α, β, γ)-approximate solutions for the of at least an α fraction of the
- ptimal profit, and allocations on nodes and edges within factors of β and γ of the original
capacities, respectively, with high probability.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 23
SLIDE 72
First (FPT-)Approximation Algorithm for VNEP
Randomized Rounding Approximation
Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Definition of Parameters
α =1/3 (relative achieved profit) β =(1 + ε ·
- 2 · ∆(VS) · log(|VS|))
(max node load) γ =(1 + ε ·
- 2 · ∆(ES) · log(|ES|))
(max edge load) ε = max
r∈R,x∈RS
dmax(r, x)/cS(x) ≤ 1 (max demand/capacity) ∆(X) = max
x∈X
- r∈R
(Amax(r, x)/dmax(r, x))2 sum over R of squared max (total / single) alloc
- Main Theorem: (FPT-)Approximation for the Virtual Network Embedding Problem
The Algorithm returns (α, β, γ)-approximate solutions for the of at least an α fraction of the
- ptimal profit, and allocations on nodes and edges within factors of β and γ of the original
capacities, respectively, with high probability.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 23
SLIDE 73
First (FPT-)Approximation Algorithm for VNEP
Randomized Rounding Approximation
Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Definition of Parameters
α =1/3 (relative achieved profit) β =(1 + ε ·
- 2 · ∆(VS) · log(|VS|))
(max node load) γ =(1 + ε ·
- 2 · ∆(ES) · log(|ES|))
(max edge load) ε = max
r∈R,x∈RS
dmax(r, x)/cS(x) ≤ 1 (max demand/capacity) ∆(X) = max
x∈X
- r∈R
(Amax(r, x)/dmax(r, x))2 sum over R of squared max (total / single) alloc
- Applicability in Practice: Computing β and γ is hard . . .
Computing β and γ requires enumerating all valid mappings. β ∈ O(ε ·
- |R| · maxr∈R |Vr| · log(|VS|)) and γ ∈ O(ε ·
- |R| · maxr∈R |Er| · log(|ES|))
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 23
SLIDE 74
First (FPT-)Approximation Algorithm for VNEP
Randomized Rounding Approximation
Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Definition of Parameters
α =1/3 (relative achieved profit) β =(1 + ε ·
- 2 · ∆(VS) · log(|VS|))
(max node load) γ =(1 + ε ·
- 2 · ∆(ES) · log(|ES|))
(max edge load) ε = max
r∈R,x∈RS
dmax(r, x)/cS(x) ≤ 1 (max demand/capacity) ∆(X) = max
x∈X
- r∈R
(Amax(r, x)/dmax(r, x))2 sum over R of squared max (total / single) alloc
- Applicability in Practice: Computing β and γ is hard . . .
Computing β and γ requires enumerating all valid mappings. β ∈ O(ε ·
- |R| · maxr∈R |Vr| · log(|VS|)) and γ ∈ O(ε ·
- |R| · maxr∈R |Er| · log(|ES|))
Consider Heuristics Return best solution found within X iterations.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 23
SLIDE 75
Derived Heuristics
Randomized Rounding Approximation
Algorithm: VNEP Approximation // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while solution not (α, β, γ)-approximate and rounding tries not exceeded
- Matthias Rost (TU Berlin)
Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 24
SLIDE 76
Derived Heuristics
Heuristic Idea: Fixed #Iterations
Algorithm: Heuristic Adaptation // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while rounding tries not exceeded return best solution
Vanilla Rounding: RRMinLoad still may exceed capacities return solution with least resource violations (among those: highest profit)
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 24
SLIDE 77
Derived Heuristics
Heuristic Idea: Fixed #Iterations
Algorithm: Heuristic Adaptation // perform preprocessing compute optimal LP solution compute {Dr}r∈R from LP solution do solution ← RoundingProcedure({Dr}r∈R) while rounding tries not exceeded return best solution Algorithm: RoundingProcedure (Heuristic) Input : Optimal convex combinations {Dr}r∈R foreach r ∈ R do choose mk
r with probability f k r
discard mapping if capacity violated end return solution
Vanilla Rounding: RRMinLoad still may exceed capacities return solution with least resource violations (among those: highest profit) Heuristic Rounding: RRHeuristic RoundingProcedure: discard chosen mappings exceeding capacities always yields feasible solutions return solution with highest profit
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 24
SLIDE 78
Computational Evaluation4,5
4
Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized
- Rounding. In Proc. IFIP Networking, 2018d
5
Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized
- Rounding. Technical report, March 2018b. URL http://arxiv.org/abs/1803.03622
SLIDE 79
Computational Evaluation: Setup
Substrate: GEANT Code available:
https://github.com/vnep-approx/ evaluation-ifip-networking-2018
Generation Parameters for 1,500 instances
Number of requests: 40, 60, 80, 100 Node-Resource Factor (NRF): 0.2, 0.4, 0.6, 0.8, 1.0 Edge-Resource Factor (ERF): 0.25, 0.5, 1.0, 2.0, 4.0 Instances per combination: 15
Requests: Synthetic Cactus Requests
3 6 9 12 15 18 21 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ECDF number of nodes: |Vr| number of edges: |Er| number of cycles: |Er| |Vr| + 1
Profit: minimum embedding resource costs Node mapping restriction: 1/4 substrate nodes Demands: exp. dist. according to NRF/ERF
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 26
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Computational Evaluation: Setup
Baseline Algorithm – MIPMCF: solve classic MIP Formulation for upto 3 hours Acceptance Ratio
40 60 80 100 Number of Requests 0.25 0.5 1.0 2.0 4.0 Edge Resource Factor 20 40 60 80 100
- Avg. Node Load
40 60 80 100 Number of Requests 0.2 0.4 0.6 0.8 1.0 Node Resource Factor 10 20 30 40 50 60
- Avg. Edge Load
40 60 80 100 Number of Requests 0.25 0.5 1.0 2.0 4.0 Edge Resource Factor 6 12 18 24 30 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 26
SLIDE 81
Computational Evaluation: Heuristic Performance
Vanilla Rounding Performance
60 80 100 120 140 Profit(RRMinLoad)/Profit(MIPMCF) [%] 100 125 150 175 200 225 Max Load (RRMinLoad) [%]
ERF 0.25 0.5 1.0 2.0 4.0
Relative profit ≈ 80 - 120% Resource augmentations mostly < 200%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 27
SLIDE 82
Computational Evaluation: Heuristic Performance
Vanilla Rounding Performance
60 80 100 120 140 Profit(RRMinLoad)/Profit(MIPMCF) [%] 100 125 150 175 200 225 Max Load (RRMinLoad) [%]
ERF 0.25 0.5 1.0 2.0 4.0
Relative profit ≈ 80 - 120% Resource augmentations mostly < 200%
Heuristic Rounding (w/o augmentations)
40 60 80 100 Number of Requests 0.25 0.5 1.0 2.0 4.0 Edge Resource Factor 60 70 80 90 100
Relative profit ≈ 65 - 90% min: 22.5% / mean: 73.8% / max: 101%
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 27
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Computational Evaluation: Runtimes
Runtime MIPMCF (Gurobi 7.5.1)
0.2 0.4 0.6 0.8 1.0 Node Resource Factor 0.25 0.5 1.0 2.0 4.0 Edge Resource Factor 36 72 108 144 180
Runtime LPnovel (Gurobi 7.5.1)
40 60 80 100 Number of Requests 0.25 0.5 1.0 2.0 4.0 Edge Resource Factor 2 4 6 8 10
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 28
SLIDE 84
Computational Evaluation: Formulation Strengths
0.75 1.00 1.50 2.00 2.50 3.00 3.50 Bound(MIPMCF) / Bound(LPnovel) 0.0 0.2 0.4 0.6 0.8 1.0 ECDF Bound(MIPMCF)
initial final
#Requests
40 60 80 100
Root relaxation values upto 3.5 times better than when using the MCF LP. Final MIP bounds improve novel LP bounds by at most a factor of 1.3.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 29
SLIDE 85
Conclusion
SLIDE 86
Conclusion
Summary Complexity: Computing valid mappings is NP-complete for planar graphs. (FPT-)Linear Programs: Valid mappings can be computed in FPT using novel LP. (FPT-)Approximations: For offline VNEP (profit & cost) based on randomized rounding. Evaluation: Solutions quite good even without resource augmentations. Novel formulation is much stronger. Runtime becomes an issue.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 31
SLIDE 87
Conclusion
Summary Complexity: Computing valid mappings is NP-complete for planar graphs. (FPT-)Linear Programs: Valid mappings can be computed in FPT using novel LP. (FPT-)Approximations: For offline VNEP (profit & cost) based on randomized rounding. Evaluation: Solutions quite good even without resource augmentations. Novel formulation is much stronger. Runtime becomes an issue. Future Work Runtime: Column generation could be readily applied, need to try it. Heuristics: Many possibilities, also for online problem. Extraction width: Can improve the formulation further (→ tree-width). Online Approximation: Need to improve rounding scheme (using e.g. Bansal et al. [2011]).
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 31
SLIDE 88
Thank You!
Questions?
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 32
SLIDE 89
References I
Edoardo Amaldi, Stefano Coniglio, Arie M.C.A. Koster, and Martin Tieves. On the computational complexity of the virtual network embedding problem. Electronic Notes in Discrete Mathematics, 52:213 – 220, 2016. David G. Andersen. Theoretical approaches to node assignment. [Online]. Available: http://repository.cmu.edu/compsci/86/, December 2002. Nikhil Bansal, Kang-Won Lee, Viswanath Nagarajan, and Murtaza Zafer. Minimum congestion mapping in a cloud. In Proc. ACM PODC, 2011.
- N. Chowdhury, M.R. Rahman, and R. Boutaba. Virtual network embedding with coordinated
node and link mapping. In Proc. IEEE INFOCOM, 2009. Guy Even, Matthias Rost, and Stefan Schmid. An approximation algorithm for path computation and function placement in sdns. In Jukka Suomela, editor, Structural Information and Communication Complexity. Springer, 2016.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 33
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References II
Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85–103. Springer, 1972. Jan Kratochvíl. A special planar satisfiability problem and a consequence of its np-completeness. Discrete Applied Mathematics, 52(3), 1994. Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding Problem. Technical report, March 2018a. URL http://arxiv.org/abs/1803.04452. Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. Technical report, March 2018b. URL http://arxiv.org/abs/1803.03622. Matthias Rost and Stefan Schmid. Charting the Complexity Landscape of Virtual Network
- Embeddings. In Proc. IFIP Networking, 2018c.
Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. In Proc. IFIP Networking, 2018d.
Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 34