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 Universität Berlin, Internet Network Architectures Stefan Schmid Universität Wien, Communication Technologies
A Short Introduction to the Virtual Network Embedding Problem
Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = ( V S , E S ) Capacities c S : G S → R ≥ 0 Data Center Network Wide-Area Network Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 3
Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = ( V S , E S ) Capacities c S : G S → R ≥ 0 Data Center Network Wide-Area Network ‘Classic’ Cloud Computing 1 4 requests A B D C 3 1 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
Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = ( V S , E S ) Capacities c S : G S → R ≥ 0 Data Center Network Wide-Area Network ‘Classic’ Cloud Computing Goal: Virtual Networks (since ≈ 2006) 1 4 1 4 requests 1 requests A B A B 6 1 1 D C D C 1 3 1 3 1 bunch of VMs Virtual Network User requests virtual machines Communication requirements given No guarantee on network performance Network performance will be guaranteed Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 3
Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = ( V S , E S ) Capacities c S : G S → R ≥ 0 Data Center Network Wide-Area Network Goal: Virtual Networks (since ≈ 2006) Virtual Network Request G r = ( V r , E r ) 1 4 requests 1 A B demands d r : G r → R ≥ 0 6 1 1 mapping restrictions D C 1 V i S ⊆ V S for i ∈ V r 3 1 E i , j Virtual Network S ⊆ E S for ( i , j ) ∈ E r Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
Goal: Virtual Networks (since ≈ 2006) Virtual Network Request G r = ( V r , E r ) 1 4 requests 1 A B demands d r : G r → R ≥ 0 6 1 1 mapping restrictions D C 1 V i S ⊆ V S for i ∈ V r 3 1 E i , j Virtual Network S ⊆ E S for ( i , j ) ∈ E r Valid Mapping Def: Valid mapping m r = ( m V , m E ) . . . Virtual Network Substrate (Physical Network) m V : V r → V S and m E : E r → P ( E S ) satisfies m E ( i , j ) A B AC B valid connectivity: m V ( i ) m V ( j ) � Embedding m V ( i ) ∈ V i valid node mapping: S m E ( i , j ) ⊆ E i , j D C D valid edge mapping: S Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
Goal: Virtual Networks (since ≈ 2006) Virtual Network Request G r = ( V r , E r ) 1 4 requests 1 A B demands d r : G r → R ≥ 0 6 1 1 mapping restrictions D C 1 V i S ⊆ V S for i ∈ V r 3 1 E i , j Virtual Network S ⊆ E S for ( i , j ) ∈ E r Def: Valid mapping m r = ( m V , m E ) . . . Feasible Embedding m V : V r → V S and m E : E r → P ( E S ) satisfies Virtual Network Substrate (Physical Network) m E ( i , j ) valid connectivity: m V ( i ) m V ( j ) � 1 4 2 / 2 4 / 5 2 / 3 m V ( i ) ∈ V i 1 valid node mapping: A B AC B S 1 / 3 6 m E ( i , j ) ⊆ E i , j Embedding 1 1 valid edge mapping: 1 / 2 1 / 2 S 1 / 2 1 / 2 D C D 1 Def: Feasible embedding m r . . . 3 1 0 / 0 3 / 3 1 / 1 . . . is valid and respects capacities . Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
Goal: Virtual Networks (since ≈ 2006) Virtual Network Request G r = ( V r , E r ) 1 4 requests 1 A B demands d r : G r → R ≥ 0 6 1 1 mapping restrictions D C 1 V i S ⊆ V S for i ∈ V r 3 1 E i , j Virtual Network S ⊆ E S for ( i , j ) ∈ E r Def: Feasible embedding m r . . . Feasible Embedding . . . is valid and respects capacities . Virtual Network Substrate (Physical Network) Virtual Network Embedding Problem 1 4 2 / 2 4 / 5 2 / 3 1 A B AC B 1 / 3 Setting Online vs. Offline 6 Embedding 1 1 1 / 2 1 / 2 Objectives resource minimization, 1 / 2 1 / 2 D C D profit maximization, 1 3 1 0 / 0 3 / 3 1 / 1 energy minimization, . . . Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 4
Related Work & Overview of Contributions
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 2018, Bordeaux 6
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... 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
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... 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
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... 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
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... 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
Related Work Computational Complexity Heuristics & Exact Algorithms Approximations Andersen [2002] Generally None for general graphs! NP -hardness (argument) ≫ 100 works, e.g. . . . Bansal et al. [2011] for Amaldi et al. [2016] Chowdhury et al. [2009] trees NP -hardness and Heuristics based on Linear inapproximability for offline Programming ; hoped for Even et al. [2016] for chains VNEP (profit) approximations ... 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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