Approximating the Virtual Network Embedding Problem: Theory and Practice

Transcription

1 Approximating the Virtual Network Embedding Problem: Theory and Practice 2 5 AC 3 B D rd International Symposium on Mathematical Programming 208 Bordeaux, France Matthias Rost Technische Universität Berlin, Internet Network Architectures Stefan Schmid Universität Wien, Communication Technologies

2 A Short Introduction to the Virtual Network Embedding Problem

3 Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = (V S, E S ) Data Center Network Wide-Area Network Capacities c S : G S R 0 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 3

4 Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = (V S, E S ) Data Center Network Wide-Area Network Capacities c S : G S R 0 Classic Cloud Computing requests 4 A B D C 3 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 208, Bordeaux 3

5 Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = (V S, E S ) Data Center Network Wide-Area Network Capacities c S : G S R 0 Classic Cloud Computing Goal: Virtual Networks (since 2006) requests 4 A B requests 4 A B 6 D C 3 bunch of VMs User requests virtual machines No guarantee on network performance D C 3 Virtual Network Communication requirements given Network performance will be guaranteed Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 3

6 Operators offer their Network Resources Substrate (Physical Network) Directed graph G S = (V S, E S ) Data Center Network Wide-Area Network Capacities c S : G S R 0 requests Goal: Virtual Networks (since 2006) 4 A B 6 D C 3 Virtual Network Virtual Network Request G r = (V r, E r ) demands d r : G r R 0 mapping restrictions VS i V S for i V r E i,j S E S for (i, j) E r Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 4

7 requests Goal: Virtual Networks (since 2006) 4 A B 6 D C 3 Virtual Network Virtual Network Request G r = (V r, E r ) demands d r : G r R 0 mapping restrictions VS i V S for i V r E i,j S E S for (i, j) E r Valid Mapping Virtual Network Substrate (Physical Network) A B AC B Embedding D C D Def: Valid mapping m r = (m V, m E )... m V : V r V S and m E : E r P(E S ) satisfies valid connectivity: valid node mapping: valid edge mapping: m V (i) m E (i,j) m V (i) V i S m E (i, j) E i,j S m V (j) Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 4

8 requests Goal: Virtual Networks (since 2006) 4 A B 6 D C 3 Virtual Network Virtual Network Request G r = (V r, E r ) demands d r : G r R 0 mapping restrictions VS i V S for i V r E i,j S E S for (i, j) E r 6 D C 3 Feasible Embedding Virtual Network Substrate (Physical Network) 4 A 2/3 4/5 B AC B /3 Embedding /2 /2 /2 /2 D 3/3 / Def: Valid mapping m r = (m V, m E )... m V : V r V S and m E : E r P(E S ) satisfies valid connectivity: valid node mapping: valid edge mapping: m V (i) m E (i,j) m V (i) V i S m E (i, j) E i,j S Def: Feasible embedding m r is valid and respects capacities. m V (j) Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 4

9 requests Goal: Virtual Networks (since 2006) 4 A B 6 D C 3 Virtual Network Virtual Network Request G r = (V r, E r ) demands d r : G r R 0 mapping restrictions VS i V S for i V r E i,j S E S for (i, j) E r 6 D C 3 Feasible Embedding Virtual Network Substrate (Physical Network) 4 A 2/3 4/5 B AC B /3 Embedding /2 /2 /2 /2 D 3/3 / Def: Feasible embedding m r 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 208, Bordeaux 4

10 Related Work & Overview of Contributions

11 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] for chains Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

12 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] 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 208, Bordeaux 6

13 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing N P-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 208, Bordeaux 6

14 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing N P-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 208, Bordeaux 6

15 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing N P-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 208, Bordeaux 6

16 Computational Complexity Andersen [2002] N P-hardness (argument) Amaldi et al. [206] N P-hardness and inapproximability for offline VNEP (profit) Related Work Heuristics & Exact Algorithms Generally 00 works, e.g.... Chowdhury et al. [2009] Heuristics based on Linear Programming; hoped for approximations... Approximations None for general graphs! Bansal et al. [20] for trees Even et al. [206] for chains Idea of this Talk: Give Overview on Our Results Complexity results showing N P-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 208, Bordeaux 6

17 Idea of this Talk: Give Overview on Our Results Complexity results showing N P-completeness and inapproximability a. (FPT-)Linear Programs for computing convex combinations of valid mappings b,c. (FPT-)Approximations for offline VNEP based on randomized rounding b,c. Computational evaluation of derived heuristics for offline profit VNEP b. a Matthias Rost and Stefan Schmid. Charting the Complexity Landscape of Virtual Network Embeddings. In Proc. IFIP Networking, 208c b Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. In Proc. IFIP Networking, 208d c Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding Problem. Technical report, March 208a. URL Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

18 Complexity of the VNEP Matthias Rost and Stefan Schmid. Charting the Complexity Landscape of Virtual Network Embeddings. In Proc. IFIP Networking, 208c

19 Reminder: 3-SAT and N P-Completeness 3-SAT-Formula φ φ = C i C φ C i with C i C φ being disjunctions of at most 3 (possible negated) literals. Example 3-SAT formula φ over literals L φ = {x, x 2, x 3, x 4 } φ = (x x 2 x 3 ) ( x }{{} x 2 x 4 ) (x }{{} 2 x 3 x 4 ) }{{} C C 2 Definition of 3-SAT Decide whether satisfying assignment a : L φ {F, T} exists for formula φ. Output: Yes/No. Theorem: Karp [972] 3-SAT is N P-complete. C 3 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 8

20 Methodology: Proving N P-completeness Proving N P-completeness of the VNEP VNEP lies in N P (answer can be checked in polynomial time). 2 Reduction from 3-SAT to VNEP. Outline of Reduction Framework 3-SAT instance φ VNEP instance (G r(φ), G S(φ), restrictions) φ satisfiable? feasible embedding of G r(φ) on G S(φ) under restrictions? Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 9

21 Our Reduction Framework Proving N P-completeness of the VNEP VNEP lies in N P (answer can be checked in polynomial time). 2 Reduction from 3-SAT to VNEP. Outline of Reduction Framework 3-SAT instance φ VNEP instance (G r(φ), G S(φ), restrictions) φ satisfiable? feasible embedding of G r(φ) on G S(φ) under restrictions? Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 0

22 Our Reduction Framework Input: 3-SAT formula φ = (x x 2 x 3 ) ( x x 2 x 4 ) (x 2 x 3 x 4 ) Request G r(φ) V r(φ) = {v i C i C φ } v v 2 v 3 E r(φ) = { (v i, v j ) C i introduces literal used by C j } Substrate G S(φ) one node per clause and per satisfying assignment edges as for the requests, if assignments do not contradict x, x 2, x 3 :TTT x, x 2, x 3 :TTF x, x 2, x 3 :TFT x, x 2, x 4 : TTT x, x 2, x 4 : TTF x, x 2, x 4 : TFT x 2, x 3, x 4 : TTT x 2, x 3, x 4 : TTF x 2, x 3, x 4 : TFT Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 0

23 Complete Picture φ: (x x 2 x 3) ( x x 2 x 4) (x 2 x 3 x 4) G r(φ) : v v 2 v 3 x, x 2, x 3 : TTT x, x 2, x 4 : TTT x 2, x 3, x 4 : TTT x, x 2, x 3 : TTF x, x 2, x 4 : TTF x 2, x 3, x 4 : TTF x, x 2, x 3 : TFT x, x 2, x 4 : TFT x 2, x 3, x 4 : TFT G S(φ) : x, x 2, x 3 : TFF x, x 2, x 3 : FTT x, x 2, x 4 : FTT x 2, x 3, x 4 : TFF x 2, x 3, x 4 : FTT x, x 2, x 3 : FTF x, x 2, x 4 : FTF x, x 2, x 3 : FFT x, x 2, x 4 : FFT x 2, x 3, x 4 : FFT x, x 2, x 4 : FFF x 2, x 3, x 4 : FFF Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux

24 Our Reduction Framework Outline of Reduction Framework 3-SAT instance φ VNEP instance (G r(φ), G S(φ), restrictions) φ satisfiable? feasible embedding of G r(φ) on G S(φ) under restrictions? Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 2

25 Our Reduction Framework Outline of Reduction Framework 3-SAT instance φ VNEP instance (G r(φ), G S(φ), restrictions) φ satisfiable? feasible embedding of G r(φ) on G S(φ) under restrictions? Base Lemma Formula φ is satisfiable if and only if there exists a mapping of G r(φ) on G S(φ), s.t. () each virtual node v i 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 208, Bordeaux 2

26 Our Reduction Framework Base Lemma Formula φ is satisfiable if and only if there exists a mapping of G r(φ) on G S(φ), s.t. () each virtual node v i 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 = (x x 2 x 3 ) ( x x 2 x 4 ) (x 2 x 3 x 4 ) Request Assignment for φ x T x 2 T x 3 F x 4 F v v 2 v 3 Embedding satisfying conditions () and (2) x, x 2, x 3 : TTF x, x 2, x 3 : TFT x, x 2, x 4 : TTF x, x 2, x 4 : TFT x 2, x 3, x 4 : TTF x 2, x 3, x 4 : TFT x, x 2, x 3 : TFF x 2, x 3, x 4 : TFF Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 2

27 Our Reduction Framework Base Lemma Formula φ is satisfiable if and only if there exists a mapping of G r(φ) on G S(φ), s.t. () each virtual node v i 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 N P-complete under mapping restrictions Node placement restrictions enforce () Routing restrictions enforce (2) v v2 v3 v v2 v3 x, x2, x3 : TTT x, x2, x4 : TTT x2, x3, x4 : TTT x, x2, x3 : TTT x, x2, x4 : TTT x2, x3, x4 : TTT x, x2, x3 : TTF x, x2, x4 : TTF x2, x3, x4 : TTF x, x2, x3 : TTF x, x2, x4 : TTF x2, x3, x4 : TTF x, x2, x3 : TFT x, x2, x4 : TFT x2, x3, x4 : TFT x, x2, x3 : TFT x, x2, x4 : TFT x2, x3, x4 : TFT x, x2, x3 : TFF x2, x3, x4 : TFF x, x2, x3 : TFF x2, x3, x4 : TFF Matthias x, Rost x2, x3 (TU : FTT Berlin) x, Approximating x2, x4 : FTT the Virtual x2, x3, Network x4 : FTT Embedding Problem: x, x2, x3 Theory : FTT and Practice x, x2, x4 : FTT ISMP 208, x2, x3, Bordeaux x4 : FTT 2

28 Our Reduction Framework Base Lemma Formula φ is satisfiable if and only if there exists a mapping of G r(φ) on G S(φ), s.t. () each virtual node v i 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 N P-complete for degree-bounded, planar request graphs Reduction from planar 3-SAT variant using literals max. 4 times (see Kratochvíl [994]): each planar formula φ leads to a planar request graph G r(φ) each node of G r(φ) has degree at most 2 planar graph G φ planar graph G r(φ) u u2 u3 u4 u u2 u3 u4 u u2 u3 u4 v v2 v3 v v2 v3 v v2 v3 v v2 v3 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 2

29 (FPT-)Linear Programs for Computing Convex Combinations of Valid Mappings 2,3 2 Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. In Proc. IFIP Networking, 208d 3 Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding Problem. Technical report, March 208a. URL

30 Linear Programming: Classic MCF Formulation and its Limits Classic LP Formulation Formulation : Classic MCF Formulation for the VNEP yr,i= u x r r R, i V r () u V i S u V S \VS i z u,v r,i,j (u,v) δ + (u) z v,u r,i,j (v,u) δ (u) yr,i= u 0 r R, i V r (2) = z u,v r,i,j = 0 d r (i) yr,i= u a τ,u i V r,τ r (i)=τ (i,j) E r [ ] [ yr,i u r R, (i, j) E r, yr,j u u V S [ ] (3) r R, (i, j) E r, (u, v) E S \ E i,j (4) S r r R, (τ, u) RS V (5) d r (i, j) z u,v r,i,j = au,v r r R, (u, v) E S (6) ar x,y c S (x, y) (x, y) R S (7) r R ] Main Building Block: Multi-Commodity Flows yr,i u [0, ]: maps node i V r on V S z u,v r,i,j [0, ]: maps (i, j) E r on (u, v) E S z u,v r,i,j (u,v) δ + (u) (v,u) δ (u) z v,u r,i,j = yu r,i yu r,j (3) Local Connectivity Property Given a (fractional) mapping of i V r to u V S, 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 208, Bordeaux 4

31 Linear Programming: Classic MCF Formulation and its Limits Substrate 2 k 2 j 2 i Example 2 i 2 j LP Solution 2 k Request i k j Main Building Block: Multi-Commodity Flows yr,i u [0, ]: maps node i V r on V S z u,v r,i,j [0, ]: maps (i, j) E r on (u, v) E S z u,v r,i,j (u,v) δ + (u) (v,u) δ (u) z v,u r,i,j = yu r,i yu r,j (3) Local Connectivity Property Given a (fractional) mapping of i V r to u V S, 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 208, Bordeaux 5

32 Linear Programming: Classic MCF Formulation and its Limits Example Substrate Request i 2 k 2 j 2 i 2 i 2 j LP Solution 2 k k k i j j Extraction Order Main Building Block: Multi-Commodity Flows yr,i u [0, ]: maps node i V r on V S z u,v r,i,j [0, ]: maps (i, j) E r on (u, v) E S z u,v r,i,j (u,v) δ + (u) (v,u) δ (u) z v,u r,i,j = yu r,i yu r,j (3) Local Connectivity Property Given a (fractional) mapping of i V r to u V S, 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 208, Bordeaux 5

33 Linear Programming: Classic MCF Formulation and its Limits Example Substrate Request i 2 k 2 j 2 i 2 i 2 j LP Solution 2 k k k i j j Extraction Order Local Connectivity Property Given a (fractional) mapping of i V r to u V S, 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 208, Bordeaux 5

34 Linear Programming: Classic MCF Formulation and its Limits Example Substrate Request i 2 k 2 j 2 i 2 i 2 j LP Solution 2 k k k i j j Extraction Order 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 208, Bordeaux 5

35 Linear Programming: Classic MCF Formulation and its Limits Example Substrate Request i 2 k 2 j 2 i 2 i 2 j LP Solution 2 k k k i j j Extraction Order 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 208, Bordeaux 5

36 Linear Programming: Classic MCF Formulation and its Limits Example Substrate Request i 2 k 2 j 2 i 2 i 2 j LP Solution 2 k k k i j j Extraction Order 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 208, Bordeaux 5

37 Outline of Novel Decomposable LP Formulations Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X 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 208, Bordeaux 6

38 Outline of Novel Decomposable LP Formulations Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! Pre-Processing Extraction Orders 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. If edge e lies on confluence Ci,j X, 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 a {a} {a, b} {a, b, c} {k} {c} k c Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

39 Outline of Novel Decomposable LP Formulations Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! a Pre-Processing Extraction Orders 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. If edge e lies on confluence Ci,j X, 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 {a} c {a, b} {a, b, c} {c} {k} k {i, l} i {l} {j, l} {j} l j {k} k Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

40 Matthias Rost (TU Berlin) Approximating the Virtual NetworkEdge Embedding Bags Problem: Theory and Practice ISMP 208, Bordeaux 6 Outline of Novel Decomposable LP Formulations Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! a Pre-Processing Extraction Orders 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. If edge e lies on confluence Ci,j X, 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 {a} c {a, b} {a, b, c} {c} {k} k {i, l} i {l} {j, l} {j} l j {k} k

41 Outline of Novel Decomposable LP Formulations Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! Pre-Processing Extraction Orders 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. If edge e lies on confluence Ci,j X, 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 a {a} {a, b} a {a, b, c} {k} {c} k c c Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

42 Outline of Novel Decomposable LP Formulations a Pre-Processing Extraction Orders If edge e lies on confluence Ci,j X, 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 {a} c {a, b} {a, b, c} {c} {k} k a b c {a, b, c} {k} k b a {a, b} c {c} Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 6

43 Outline of Novel Decomposable LP Formulation Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! Pre-Processing Extraction Orders If edge e Er X lies on confluence Ci,j X, then it is labeled with the confluence s target j. Outgoing edges are partitioned into edge bags not sharing labels. Each label has unique root node at which the mapping of the label must be fixed. 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. Generation of Linear Program If e E r is labeled with L X r,e, then V S LX r,e many commodities are considered for e. For each edge bag variables are introduced to enumerate all potential label mappings. Root nodes of labels decide on the confluence s mapping. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 7

44 Outline of Novel Decomposable LP Formulation Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! Pre-Processing Extraction Orders If edge e Er X lies on confluence Ci,j X, then it is labeled with the confluence s target j. Outgoing edges are partitioned into edge bags not sharing labels. Each label has unique root node at which the mapping of the label must be fixed. 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. Generation of Linear Program If e E r is labeled with L X r,e, then V S LX r,e many commodities are considered for e. For each edge bag variables are introduced to enumerate all potential label mappings. Root nodes of labels decide on the confluence s mapping. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 7

45 Outline of Novel Decomposable LP Formulation Extraction Order Gr X Rooted acyclic reorientation of the original request graph G r. Gr X is not unique! Pre-Processing Extraction Orders If edge e Er X lies on confluence Ci,j X, then it is labeled with the confluence s target j. Outgoing edges are partitioned into edge bags not sharing labels. Each label has unique root node at which the mapping of the label must be fixed. 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. Generation of Linear Program If e E r is labeled with L X r,e, then V S LX r,e many commodities are considered for e. For each edge bag variables are introduced to enumerate all potential label mappings. Root nodes of labels decide on the confluence s mapping. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 7

46 Outline of Novel Decomposable LP Formulation Pre-Processing Extraction Orders If edge e Er X lies on confluence Ci,j X, then it is labeled with the confluence s target j. Outgoing edges are partitioned into edge bags not sharing labels. Each label has unique root node at which the mapping of the label must be fixed. Generation of Linear Program If e E r is labeled with L X r,e, then V S LX r,e many commodities are considered for e. For each edge bag variables are introduced to enumerate all potential label mappings. Root nodes of labels decide on the confluence s mapping. Stitching Flow Variables via Node Mapping Variables y i r,i (i, c), [j j] a {j} {i, l} i {j, l} c f y i r,i (i, a), [i i, l l] y i r,f (i, a), [i i, l l2] γ i r,i,,[j j,l l] γ i r,i,,[j j,l l2] y i r,i (i, c), [j j2] y i r,i (f, i), [j j, l l] y i r,i (f, i), [j j, l l2] y i r,i (i, a), [i i2, l l] γ i r,i,,[j j2,l l] y i r,i (f, i), [j j2, l l] y i r,i (i, a), [i i2, l l2] γ i r,i,,[j j2,l l2] y i r,i (f, i), [j j2, l l2] Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 7

47 Outline of Novel Decomposable LP Formulation Pre-Processing Extraction Orders If edge e Er X lies on confluence Ci,j X, then it is labeled with the confluence s target j. Outgoing edges are partitioned into edge bags not sharing labels. Each label has unique root node at which the mapping of the label must be fixed. Def. Extraction Width ew X (G X r )... is the size of the largest edge bag plus one of the extraction order ew X (G X r ). Generation of Linear Program If e E r is labeled with L X r,e, then V S LX r,e many commodities are considered for e. For each edge bag variables are introduced to enumerate all potential label mappings. Root nodes of labels decide on the confluence s mapping. Proof of Decomposability... via decomposition algorithm. Overall runtime O(poly( G S ew X (G X r ) G r )). Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 7

48 Novel Decomposable LP Formulation: Takeaways Overview of Construction Request Graph G r Extraction Order Gr X Labeling / Extraction Width ew X (Gr X ) LP of size O(poly( G S ew X (Gr X ) G r )) Decomposition Algorithm with runtime O(poly( G S ew X (Gr X ) G r )) Convex Combinations of valid mappings: D r = {(fr k, mr k ) fr k > 0, mr k M r } Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 8

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 208, Bordeaux 9

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( V r ) Minimizing extraction width is N P-hard (via reduction from Vertex-Cover) Cactus graphs (cycles intersect in at most a single node) have bounded extraction width w w 2 w 3 ew X (G X w ) = 2 w c w n w w n/2+2 w w n/2+ n/2 w 2 w 3 ew X (G X w ) V r /2 w c w n w n/2+2 w w n/2+ n/2 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 9

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( V r ) Minimizing extraction width is N P-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 208, Bordeaux 9

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( V r ) Minimizing extraction width is N P-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 Virtual Cluster VM VM 5 VM 2 VM 4 VM 3 Service Chain LB LB 2 NAT Customer Cache FW Internet Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 9

53 Extraction Width Overview of Results Extraction Width: Overview of Results Extraction width may vary by factor Ω( V r ) Minimizing extraction width is N P-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 Virtual Cluster VM VM 5 VM 2 VM 4 VM 3 Service Chain LB LB 2 NAT Customer Cache FW Internet Can we do substantially better? No! Computing valid mappings for planar graphs is N P-complete FPT algorithms are necessary. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 9

54 (FPT-)Approximations for offline VNEP based on Randomized Rounding 2,3 2 Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. In Proc. IFIP Networking, 208d 3 Matthias Rost and Stefan Schmid. (FPT-)Approximation Algorithms for the Virtual Network Embedding Problem. Technical report, March 208a. URL

55 Approximating the Offline VNEP Profit Variant A set of request R = {r, r 2,...} is given. Profit for request p r > 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 208, Bordeaux 2

56 Approximating the Offline VNEP Profit Variant A set of request R = {r, r 2,...} is given. Profit for request p r > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r, r 2,...} is given. Substrate resource costs k S : G S 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 208, Bordeaux 2

57 Approximating the Offline VNEP Focus: Profit Variant A set of request R = {r, r 2,...} is given. Profit for request p r > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r, r 2,...} is given. Substrate resource costs k S : G S 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 208, Bordeaux 2

58 Approximating the Offline VNEP Focus: Profit Variant A set of request R = {r, r 2,...} is given. Profit for request p r > 0. Task: Embed subset of requests feasibly maximizing the attained profit. Cost Variant A set of request R = {r, r 2,...} is given. Substrate resource costs k S : G S 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 208, Bordeaux 2

59 Approximating the Offline VNEP Combine Single Decomposable LP Formulations while enforcing capacity constraints and maximizing the profit. LP for offline VNEP (profit). Request 4 A B 6 D C 3 Valid Mappings M r = {m r, m 2 r, m 3 r,...} 2/3 4/5 2/3 7/5 AC B AC BD /3 2/3 /2 /2 /2 /2 D 3/3 0/3 Valid mappings do not necessarily respect capacity constraints!... Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 2

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: r R f k r {0, } r R, m k r M r (8) m k r Mr f k r r R (9) A(mr k, x) fr k c S (x) x R S (0) mr k Mr max p r fr k () r R mr k Mr Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 2

61 Randomized Rounding Revisited Example Substrate Network Request r : profit 00$ Request r 2 : profit 50$ A B E D C 3 G F 2 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

62 Substrate Network Randomized Rounding Revisited Example Request r : profit 00$ Request r 2 : profit 50$ A B E D C 3 G F 2 Example Solution to Linear Program: Profit 33$ Variables of r (profit: 00$) Variables of r 2 (profit: 50$) f = 0.5 2/3 4/5 AC B /3 f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

63 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ Variables of r (profit: 00$) Variables of r 2 (profit: 50$) f = 0.5 2/3 4/5 AC B /3 f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

64 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ Variables of r (profit: 00$) Variables of r 2 (profit: 50$) f = 0.5 2/3 4/5 AC B /3 f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

65 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ Variables of r 2 (profit: 50$) Variables of r 2 (profit: 50$) f = 0.5 2/3 4/5 AC B /3 f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load m m2 2 50$ 200% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

66 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ f = 0.5 2/3 4/5 AC B /3 Variables of request f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 Variables of r 2 (profit: 50$) f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load m m $ 200% 2 m 3 00$ 00% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

67 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ f = 0.5 2/3 4/5 AC B /3 Variables of request f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 Variables of r 2 (profit: 50$) f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load m m $ 200% 2 m 3 00$ 00% 3 m m 2 50$ 200% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

68 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ f = 0.5 2/3 4/5 AC B /3 Variables of request f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 Variables of r 2 (profit: 50$) f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load m m $ 200% 2 m 3 00$ 00% 3 m m 2 50$ 200% 4 m 2 m 2 50$ 200% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

69 Randomized Rounding Revisited Example Solution to Linear Program: Profit 33$ f = 0.5 2/3 4/5 AC B /3 Variables of request f 2 = 0.3 2/3 7/5 AC BD 2/3 f 3 = 0.2 /3 4/5 AC B /3... f 2 = 0.5 3/3 /5 G E 0/3 Variables of r 2 (profit: 50$) f 2 2 = 0.6 3/3 2/5 EF G /3 f 3 2 = 0 5/5 0/3 EFG... /2 /2 /2 /2 /2 D 3/3 0/3 D 3/3 /2 F /3 0/3 0/3 Idea: Treat weights as probabilities! Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Rounding Outcomes Iter. Req. Req. 2 Profit max Load m m $ 200% 2 m 3 00$ 00% 3 m m 2 50$ 200% 4 m 2 m 2 50$ 200%..... Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 22

70 First (FPT-)Approximation Algorithm for VNEP Randomized Rounding Approximation Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while and rounding tries not exceeded Algorithm: RoundingProcedure Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k end return solution Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 23

71 First (FPT-)Approximation Algorithm for VNEP Randomized Rounding Approximation Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while 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 optimal 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 208, Bordeaux 23

72 First (FPT-)Approximation Algorithm for VNEP Randomized Rounding Approximation Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while and rounding tries not exceeded α =/3 Definition of Parameters β =( + ε 2 (V S ) log( V S )) γ =( + ε 2 (E S ) log( E S )) ε = max d max(r, x)/c S (x) r R,x R S (X ) = max x X r R (relative achieved profit) (max node load) (max edge load) (max demand/capacity) (A max(r, x)/d max(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 optimal 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 208, Bordeaux 23

73 First (FPT-)Approximation Algorithm for VNEP Randomized Rounding Approximation Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while and rounding tries not exceeded α =/3 Definition of Parameters β =( + ε 2 (V S ) log( V S )) γ =( + ε 2 (E S ) log( E S )) ε = max d max(r, x)/c S (x) r R,x R S (X ) = max x X r R (relative achieved profit) (max node load) (max edge load) (max demand/capacity) (A max(r, x)/d max(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 max r R V r log( V S )) and γ O(ε R max r R E r log( E S )) Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 23

74 First (FPT-)Approximation Algorithm for VNEP Randomized Rounding Approximation Algorithm: VNEP Approximation (Profit) // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while and rounding tries not exceeded α =/3 Definition of Parameters β =( + ε 2 (V S ) log( V S )) γ =( + ε 2 (E S ) log( E S )) ε = max d max(r, x)/c S (x) r R,x R S (X ) = max x X r R (relative achieved profit) (max node load) (max edge load) (max demand/capacity) (A max(r, x)/d max(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 max r R V r log( V S )) and γ O(ε R max r R E r log( E S )) Consider Heuristics Return best solution found within X iterations. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 23

75 Derived Heuristics Randomized Rounding Approximation Algorithm: VNEP Approximation // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution ( RoundingProcedure({D r } r R ) ) solution not (α, β, γ)-approximate while and rounding tries not exceeded Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 24

76 Derived Heuristics Heuristic Idea: Fixed #Iterations Algorithm: Heuristic Adaptation // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution RoundingProcedure({D r } r R ) while rounding tries not exceeded return best solution Vanilla Rounding: RR MinLoad 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 208, Bordeaux 24

77 Heuristic Idea: Fixed #Iterations Derived Heuristics Algorithm: Heuristic Adaptation // perform preprocessing compute optimal LP solution compute {D r } r R from LP solution do solution RoundingProcedure({D r } r R ) while rounding tries not exceeded return best solution Vanilla Rounding: RR MinLoad still may exceed capacities return solution with least resource violations (among those: highest profit) Algorithm: RoundingProcedure (Heuristic) Input : Optimal convex combinations {D r } r R foreach r R do choose mr k with probability fr k discard mapping if capacity violated end return solution Heuristic Rounding: RR Heuristic 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 208, Bordeaux 24

78 Computational Evaluation 4,5 4 Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. In Proc. IFIP Networking, 208d 5 Matthias Rost and Stefan Schmid. Virtual Network Embedding Approximations: Leveraging Randomized Rounding. Technical report, March 208b. URL

79 Computational Evaluation: Setup Generation Parameters for,500 instances Number of requests: 40, 60, 80, 00 Substrate: GEANT Node-Resource Factor (NRF): 0.2, 0.4, 0.6, 0.8,.0 Edge-Resource Factor (ERF): 0.25, 0.5,.0, 2.0, 4.0 Instances per combination: 5 Requests: Synthetic Cactus Requests Code available: evaluation-ifip-networking-208 ECDF number of nodes: Vr 0.2 number of edges: Er 0. number of cycles: Er Vr Profit: minimum embedding resource costs Node mapping restriction: /4 substrate nodes Demands: exp. dist. according to NRF/ERF Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 26

80 Computational Evaluation: Setup Baseline Algorithm MIP MCF : solve classic MIP Formulation for upto 3 hours Acceptance Ratio Avg. Node Load Avg. Edge Load Edge Resource Factor Number of Requests Node Resource Factor Number of Requests Edge Resource Factor Number of Requests Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 26

81 Computational Evaluation: Heuristic Performance Vanilla Rounding Performance Max Load (RRMinLoad) [%] ERF Profit(RR MinLoad )/Profit(MIP MCF ) [%] Relative profit 80-20% Resource augmentations mostly < 200% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 27

82 Computational Evaluation: Heuristic Performance Vanilla Rounding Performance Heuristic Rounding (w/o augmentations) Max Load (RRMinLoad) [%] ERF Profit(RR MinLoad )/Profit(MIP MCF ) [%] Edge Resource Factor Number of Requests Relative profit 80-20% Resource augmentations mostly < 200% Relative profit 65-90% min: 22.5% / mean: 73.8% / max: 0% Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 27

83 Computational Evaluation: Runtimes Runtime MIP MCF (Gurobi 7.5.) Runtime LP novel (Gurobi 7.5.) Edge Resource Factor Node Resource Factor Edge Resource Factor Number of Requests Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 28

84 Computational Evaluation: Formulation Strengths ECDF Bound(MIP MCF ) initial final #Requests Bound(MIP MCF ) / Bound(LP novel ) 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.3. Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 29

85 Conclusion

86 Conclusion Summary Complexity: Computing valid mappings is N P-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 208, Bordeaux 3

87 Conclusion Summary Complexity: Computing valid mappings is N P-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. [20]). Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 3

88 Thank You! Questions? Matthias Rost (TU Berlin) Approximating the Virtual Network Embedding Problem: Theory and Practice ISMP 208, Bordeaux 32