Virtual Network Embedding Approximations: Leveraging Randomized Rounding

Matthias Rost Stefan Schmid TU Berlin, Germany University of Vienna, Austria Email: [email protected] Email: stefan [email protected]

VM1 Abstract—The Virtual Network Embedding Problem (VNEP) LB1 Cache LB2 NAT captures the essence of many resource allocation problems VM2 VM5 of today’s infrastructure providers, which offer their physical Customer FW Internet VM VM computation and networking resources to customers. Customers 3 4 request resources in the form of Virtual Networks, i.e. as Fig. 1. Examples for virtual networks ‘in the wild’. The left graph shows a directed graph which specifies computational requirements a service chain for mobile operators [4]: load-balancers route (parts of at the nodes and communication requirements on the edges. the) traffic through a cache. Furthermore, a firewall and a network-address An embedding of a Virtual Network on the shared physical translation are used. The right graph depicts the Virtual Cluster abstraction for infrastructure is the joint mapping of (virtual) nodes to physical provisioning virtual machines (VMs) in data centers. The abstraction provides servers together with the mapping of (virtual) edges onto paths connectivity guarantees via a logical switch in the center [3]. in the physical network connecting the respective servers. This work initiates the study of approximation algorithms We study the offline setting with admission control: given for the VNEP. Concretely, we study the offline setting with admission control: given multiple request graphs the task is to multiple requests the task is to embed the most profitable embed the most profitable subset while not exceeding resource subset while not exceeding resource capacities. capacities. Our approximation is based on the randomized rounding of Linear Programming (LP) solutions. Interestingly, A. Formal Problem Statement we uncover that the standard LP formulation for the VNEP In the light of the recent interest in Service Chaining [2], exhibits an inherent structural deficit when considering general we extend the VNEP’s general definition [5] by consider- virtual network topologies: its solutions cannot be decomposed into valid embeddings. In turn, focusing on the class of cactus ing different types of computational nodes. We refer to the request graphs, we devise a novel LP formulation, whose solutions physical network as the substrate network. The substrate can be decomposed into convex combinations of valid embedding. GS = (VS,ES) is offering a set T of computational types. Proving performance guarantees of our rounding scheme, we This set of types may contain, e.g., ‘FW’ (firewall), ‘x86 obtain the first approximation algorithm for the VNEP in the server’, etc. For a type τ ∈ T , the set V τ ⊆ V denotes the resource augmentation model. S S We propose two types of rounding heuristics and evaluate their substrate nodes that can host functionality of type τ. Denoting V τ performance in an extensive computational study. Our results the node resources by RS = {(τ, u) |τ ∈ T , u ∈ VS } and all V indicate that randomized rounding can yield good solutions (even substrate resources by RS = RS ∪ ES, the capacity of nodes without augmentations). Specifically, heuristic rounding achieves and edges is denoted by dS(x, y) > 0 for (x, y) ∈ RS. 73.8% of the baseline’s profit, while not exceeding capacities. For each request r ∈ R, a directed graph Gr = (Vr,Er) I.INTRODUCTION together with a profit br > 0 is given. We refer to the Cloud applications usually consist of multiple distributed respective nodes as virtual or request nodes and similarly refer components (e.g., virtual machines, containers), which results to the respective edges as virtual or request edges. The types of virtual nodes are indicated by the function τ V → T . arXiv:1803.03622v2 [cs.NI] 4 May 2018 in substantial communication requirements. If the provider r : r Based on policies of the customer or the provider, the map- fails to ensure that these communication requirements are met, r,i τr (i) the performance can suffer dramatically [1]. Consequently, ping of virtual node i ∈ Vr is restricted to a set VS ⊆ VS , while the mapping of virtual edge (i, j) is restricted to a over the last years, several proposals have been made to jointly r,i,j provision the computational functionality together with ap- subset of substrate edges ES ⊆ ES. Each virtual node propriate network resources. The Virtual Network Embedding i ∈ Vr and each edge (i, j) ∈ Er is attributed with a resource Problem (VNEP) captures the core of this problem: given a demand dr(i) ≥ 0 and dr(i, j) ≥ 0, respectively. Virtual nodes directed graph specifying computational requirements at the and edges can only be mapped on substrate nodes and edges r,i τr (i) of sufficient capacity, i.e. VS ⊆ {u ∈ VS |dS(u) ≥ dr(i)} nodes and bandwidth requirements on the edges, an embedding r,i,j of this Virtual Network in the physical network has to be and ES ⊆ {(u, v) ∈ ES|dS(u, v) ≥ dr(i, j)} holds. d r, x, y found, such that both the computational and the network We denote by max( ) the maximal demand that a request r may impose on a resource (x, y) ∈ R : requirements are met. Figure 1 illustrates two incarnations of S r,i virtual networks: service chains [2] and virtual clusters [3]. dmax(r, τ, u) = max({0} ∪ {dr(i)|i ∈ Vr : τ(i) = τ ∧ u ∈ VS }) r,i,j A short version of this work appeared in IFIP Networking 2018 [6]. dmax(r, u, v) = max({0} ∪ {dr(i, j)|(i, j) ∈ Er :(u, v) ∈ ES }) In the following the notions of valid mappings (respecting have been studied recently by considering restricted graph mapping constraints) and feasible embeddings (respecting classes for the virtual networks and the substrate graph. For resource constraints) are introduced to formalize the VNEP. example, virtual clusters with uniform demands are studied in [3], [12], line requests are studied in [13], [14], [15] and tree Definition 1 (Valid Mapping). A valid mapping m of re- r requests were studied in [14], [16]. quest r ∈ R is a tuple (mV , mE) of functions mV : V → V r r r r S Considering approximation algorithms, Even et al. em- and mE : E → P(E ), such that the following holds: r r S ployed randomized rounding in [14] to obtain a constant ap- • Virtual nodes are mapped to allowed substrate nodes: V r,i proximation for embedding line requests on arbitrary substrate mr (i) ∈ VS holds for all i ∈ Vr. E graphs under strong assumptions on both the benefits and • The mapping m i, j of virtual edge i, j ∈ E is r ( ) ( ) r the capacities. In their interesting work, Bansal et al. [16] an edge-path connecting mV (i) to mV (j) only using r r give an nO(d) time O(d2 log (nd))-approximation algorithm allowed edges, i.e. mE i, j ⊆ P Er,i,j holds. r ( ) ( S ) for minimizing the load of embedding d-depth trees based We denote by Mr the set of valid mappings of request r ∈ R. on a n-node substrate. Their result is based on a strong LP Definition 2 (Allocations of Valid Mappings). We denote by relaxation inspired by the Sherali-Adams hierarchy. A(mr, x, y) the cumulative allocation induced by the valid To the best of our knowledge, no approximation algorithms mapping mr ∈ Mr on resource (x, y) ∈ RS: are known for arbitrary substrate graphs and classes of virtual X networks containing cyclic substructures. A(m , τ, u) = d (i) ∀(τ, u) ∈ RV r V r S Bibliographic Note: This technical report is an extended i∈Vr ,τ(i)=τ,mr (i)=u X version of our paper presented at IFIP Networking 2018 [6]. A m , u, v d i, j ∀ u, v ∈ E ( r ) = E r( ) ( ) S (i,j)∈Er ,(u,v)∈mr (i,j) In our preliminary technical report [17] similar results were The maximal allocation that a valid mapping of request r ∈ R presented. The current work presents a significantly simpler LP formulation and also provides an extensive computational may impose on a substrate resource (x, y) ∈ RS is denoted evaluation. Additionally, in our recent technical report [18], by Amax(r, x, y) = maxm ∈M A(mr, x, y). r r the approximation approach presented in this work is extended Definition 3 (Feasible Embedding). A feasible embedding beyond cactus request graphs. However, approximating more 0 of a subset of requests R ⊆ R is a collection of valid general request graphs comes at the price of non-polynomial mappings {mr}r∈R0 , such that the cumulative allocations on runtimes. nodes and edges does not exceed the substrate capacities, i.e. P C. Outline of Randomized Rounding for the VNEP r∈R0 A(mr, x, y) ≤ dS(x, y) holds for (x, y) ∈ RS. Definition 4 (Virtual Network Embedding Problem). The We shortly revisit the concept of randomized rounding [19]. Given an Integer Program for a certain problem, randomized VNEP asks for a feasible embedding {mr}r∈R0 of a subset of 0 P rounding works by (i) computing a solution to its Linear requests R ⊆ R maximizing the profit 0 br. r∈R Program relaxation, (ii) decomposing this solution into convex B. Related Work combinations of elementary solutions, and (iii) probabilisti- In the last decade, the VNEP has attracted much attention cally selecting elementary solutions based on their weight. due to its many applications and the survey [5] from 2013 Accordingly, for applying randomized rounding for already lists more than 80 different algorithms for its many the VNEP, a convex combination of valid mappings k k k k variations [5]. The VNEP is known to be NP-hard and Dr = {(fr , mr )|mr ∈ Mr, fr > 0} must be recovered from inapproximable in general (unless P = NP) [7]. Based on the the Linear Programming solution for each request r ∈ R, hardness of the VNEP, most works consider heuristics without such that (i) the profit of these convex combinations equals the any performance guarantee [5], [8]. Other works proposed profit achieved by the Linear Program and (ii) the (fractional) exact methods as integer or constraint programming, coming cumulative allocations do not violate substrate capacities. at the cost of an exponential runtime [9], [10], [11]. Rounding a solution is then done as follows. For each request k k A column generation approach was proposed by Jarray et al. r ∈ R, the mapping mr is selected with probability fr , P k in [10] to efficiently compute solutions to the VNEP by gen- rejecting r with probability 1 − k fr . erating feasible mappings ‘on-the-fly subject to a specific cost measure. In particular, Jarray et al. compute feasible mappings D. Results and Organization by relying on heuristics and, if no feasible heuristical solution This paper initiates the study of approximation algorithms was found, rely on Mixed-Integer Programming to compute a for the VNEP on general substrates and general virtual net- cost optimal feasible embedding in non-polynomial time. We works. Specifically, we employ randomized rounding to obtain believe that our work can bridge the gap between the heuristic the first approximation algorithm for the non-trivial class of generation of feasible mappings and the optimal generation of cactus graph requests in the resource augmentation model. feasible mappings as our approach can be used to compute Studying the classic multi-commodity flow (MCF) for- approximate mappings. mulation for the VNEP in Section II, we show that its Acknowledging the hardness of the general VNEP and the solutions can only be decomposed for tree requests: request diversity of applications, several subproblems of the VNEP graphs containing cycles can in general not be decomposed into valid mappings. This result has ramifications beyond The variable xr ∈ {0, 1} indicates whether request r ∈ R is u the inability to apply randomized rounding: we prove that embedded or not. The variable yr,i ∈ {0, 1} indicates whether the MCF formulation exhibits an unbounded integrality gap. virtual node i ∈ Vr was mapped on substrate node u ∈ VS. u,v Investigating the root cause for this surprising result, we Similarly, the flow variable zr,i,j ∈ {0, 1} indicates whether devise a novel decomposable Linear Programming formulation the substrate edge (u, v) ∈ ES is used to realize the virtual x,y in Section III for the class of cactus graph requests. We edge (i, j) ∈ Er. The variable ar ≥ 0 denotes the cumulative then present our randomized rounding algorithm and prove allocations of request r ∈ R induced on resource (x, y) ∈ RS. its performance guarantees in Section IV, obtaining the first By Constraint 2, the virtual node i ∈ Vr of request r ∈ R r,i approximation algorithm for the Virtual Network Embedding must be placed on any of the suitable substrate nodes of VS Problem. Section V presents a synthetic computational study, iff. xr = 1 holds and Constraint 3 forbids the mapping on in which two types of rounding heuristics are evaluated. Our nodes which may not host node i. Constraint 4 induces an results indicate that high-quality solutions can be obtained unsplittable unit flow for each virtual edge (i, j) ∈ Er from even without resource augmentations. Indeed, our heuristic the substrate location to which i was mapped to the substrate rounding algorithm achieves 73.8% of the baseline’s profit on location to which j was mapped. By Constraint 5 virtual edges average, while not exceeding resource capacities. may only be mapped on allowed substrate edges. Constraints 6 and 7 compute the cumulative allocations and Constraint 8 II.THE CLASSIC MULTI-COMMODITY FORMULATION guarantees that the substrate resource capacities are respected. FORTHE VNEP ANDITS LIMITATIONS The following lemma states the connectivity property enforced In this section, we study the relaxation of the standard multi- by Formulation 1. commodity flow (MCF) formulation for the VNEP (cf. [2], [8]). We first show the positive result that the formulation is Lemma 5 (Local Connectivity Property of Formulation 1). sufficiently strong to decompose virtual networks being trees For any virtual edge (i, j) ∈ Er and any substrate node r,i u u,v into convex combinations of valid mappings. Subsequently, we u ∈ VS with yr,i > 0, there exists a path Pr,i,j in GS from u r,j v show that the formulation fails to allow for the decomposition to v ∈ VS with yr,j > 0, such that the flow along any edge u,v ·,· of cyclic requests. This not only impacts its applicability for of Pr,i,j with respect to the variables zr,i,j is greater 0. u,v randomized rounding but renders the formulation useless for The path Pr,i,j can be computed in polynomial time. approximations in general: we show that the formulation’s P u P v Proof. First, note that r,i yr,i = r,j yr,i holds by integrality gap is unbounded. u∈VS v∈VS Constraint 2 and hence both virtual nodes i and j must be A. The Classic Multi-Commodity Formulation mapped to an equal extent on suitable substrate nodes. Fix any r,i u The classic MCF formulation for the VNEP is presented as substrate node u ∈ VS with yr,i > 0. If j is also partially u Formulation 1 . We first describe its (mixed-)integer variant, mapped on u, i.e. if yr,j > 0 holds, then the result follows which computes a single valid mapping for each request by directly, as u connects to u using (and allowing) the empty u,u u using binary variables. The Linear Programming variant is path Pr,i,j = hi. If, on the other hand, yr,j = 0 holds, then u obtained by relaxing the binary variables’ domain to [0, 1]. Constraint 4 induces a flow of value yr,i at u in the commodity ·,· zr,i,j. As the right hand side of Constraint 4 may only attain r,j v negative values at nodes v ∈ VS for which yr,j > 0 holds, the Formulation 1: Classic MCF Formulation for the VNEP r,j X flow emitted at node u must eventually reach a node v ∈ VS max brxr (1) v u,v with yr,j > 0 and hence the result follows. The path Pr,i,j r∈R X u can be constructed in polynomial-time by a simple breadth- yr,i= xr ∀r ∈ R, i ∈ Vr (2) 0 0 first search that only considers edges (u , v ) ∈ ES for which r,i 0 0 u∈VS u ,v X u zr,i,j > 0 holds. yr,i= 0 ∀r ∈ R, i ∈ Vr (3) r,i u∈VS \VS B. Decomposing Solutions for Tree Requests  P zu,v  r,i,j " u # " # + y r ∈ R, (i, j) ∈ E , Given Lemma 5, we now present Algorithm 1 to decompose (u,v)∈δ (u) = r,i ∀ r (4)  P v,u  u solutions to the LP Formulation 1 into convex combinations of − z −y u ∈ VS r,i,j r,j k k k k (v,u)∈δ−(u) valid mappings D = {(f , m )|m ∈ M , f > 0} (cf. Sec- " # r r r r r r r ∈ R, (i, j) ∈ E , tion I-C), if the request’s underlying undirected graph is a zu,v ∀ r r,i,j= 0 r,i,j (5) tree. Recall that in the LP formulation the binary variables are (u, v) ∈ ES \ ES X u τ,u V relaxed to take any value in the interval [0, 1]. dr(i) · y = a ∀r ∈ R, (τ, u) ∈ R (6) r,i r S Given a request r ∈ R, the algorithm processes all vir- i∈Vr ,τr (i)=τ X u,v u,v tual edges according to an arbitrary acyclic representation dr(i, j) · zr,i,j= ar ∀r ∈ R, (u, v) ∈ ES (7) A A Gr = (Vr,Er , rr) of the undirected interpretation of Gr (i,j)∈Er X being rooted at r ∈ V . Concretely, the edge set EA is ax,y≤ d (x, y) ∀(x, y) ∈ R (8) r r r r S S obtained from E by reorienting (some of the) edges, such r∈R r that any node i ∈ Vr can be reached from rr. Considering tree Request G Substrate G LP Solution Decomposition Attempt Algorithm 1: Decompositioning MCF solutions for Tree Requests r S u1 0.5i 0.5i Input : Tree request r ∈ R, solution (xr, ~yr, ~zr,~ar) to LP i A A Formulation 1, acyclic reorientation Gr = (Vr,Er , rr) u6 u2 0.5k 0.5j 0.5k 0.5j k k Output: Convex combination Dr = {(f , m )}k r r k j u5 u3 0.5j 0.5k 0.5k 1 set Dr ← ∅ and k ← 1 u4 0.5i 2 while xr > 0 do k V E 3 set mr = (mr , mr ) ← (∅, ∅) Fig. 2. Example showing that solutions to the LP Formulation 1 can in general 4 set Q = {rr} not be decomposed into convex combinations of valid mappings. Request r r,rr u V 5 choose u ∈ V with y > 0 and set m (rr) ← u is a simple cyclic graph which shall be mapped on the substrate graph GS . S r,rr r r,i r,j 6 while |Q| > 0 do We assume following node mapping restrictions VS = {u1, u4}, VS = r,k 7 choose i ∈ Q and set Q ← Q \ {i} {u2, u5}, VS = {u3, u6}. The LP solution with xr = 1 is depicted A 8 foreach (i, j) ∈ E do as follows. Substrate nodes are annotated with the mapping of virtual nodes. r u1 Hence, 0.5i at node u1 indicates y = 1/2, i.e. that virtual node i is mapped 9 if (i, j) ∈ Er then r,i −→u,v V r,j with 0.5 on substrate node u1. Substrate edges are colored according to the 10 compute P r,i,j connecting mr (i) = u to v ∈ VS color of virtual links mapped onto it. Virtual links are all mapped using flow 11 according to Lemma 5 u1,u2 values 1/2. Accordingly, for example zr,i,j = 1/2 holds. V E −→u,v set mr (j) = v and mr (i.j) = P r,i,j 12 else 0 0 0 0 ←−v ,u u ,v 0 0 13 let z r,i,j , zr,j,i for all (u , v ) ∈ ES ~ar (Lines 20 and 21), the Constraints 2 - 7 continue to hold. ←−v,u V r,j 14 compute P r,i,j connecting mr (i) = v to u ∈ VS As the decomposition process continues as long as xr > 0 15 according to Lemma 5 −→ ←− holds and in the k-th iteration at least one variable’s value is set u,v reverse v,u set P r,j,i = (P r,i,j) to 0, the algorithm terminates with a complete decomposition V E −→u,v 16 set m (i) = u and m (j, i) = P P k r r r,j,i for which k fr = xr holds. Furthermore, the algorithm has 17 set Q ← Q ∪ {j} polynomial runtime, as in each iteration at least one variable is

V ! set to 0 and the number of variables for request r is bounded mr (i) {xr} ∪ {y |i ∈ Vr} 18 r,i set Vk ← u,v E by O(|Gr| · |GS|). Hence, we obtain the following: ∪ {zr,i,j|(i, j) ∈ Er, (u, v) ∈ mr (i, j)} k 19 set fr ← min Vk Lemma 6. Given a virtual network request r ∈ R, k 20 set v ← v − fr for all v ∈ Vk whose underlying undirected graph is a tree, Algorithm 1 x,y x,y k k 21 set ar ← ar − fr · A(mr , x, y) for all (x, y) ∈ RS k k k decomposes a solution (xr, ~yr, ~zr,~ar) to LP Formula- 22 add D f , m D and set k ← k r = ( r r ) to r + 1 tion 1 into valid a convex combination of valid mappings 23 return Dr k k k k Dr = {(mr , fr )|mr ∈ Mr, fr > 0}k, such that: P k • The decomposition is complete, i.e. xr = k fr holds. • The decomposition’s allocations are bounded by ~ar, i.e. GA x,y P k k requests for now, r is an arborescence and can be computed ar ≥ k fr · A(mr , x, y) holds for (x, y) ∈ RS. by a simple graph search of the underlying undirected graph starting at an arbitrary root node rr ∈ Vr. We denote by C. Limitations of the Classic MCF Formulation ←−A A E r = Er \ Er the edges whose orientations were reversed Above it was shown that LP solutions to the classic MCF A in the process of computing Gr . formulation can be decomposed into convex combinations of k k The algorithm extracts mappings mr of value fr iteratively, valid mappings if the underlying graph is a tree. This does as long as xr > 0 holds. Initially, in the k-th iteration, none of not hold anymore when considering cyclic virtual networks: the virtual nodes and edges are mapped. As xr > 0 holds, there u ∈ V r,rr yu > Theorem 7. Solutions to the standard LP Formulation 1 can must exist a node S with r,rr 0 by Constraint 2 V in general not be decomposed into convex combinations of and the algorithm accordingly sets mr (rr) = u. Given this initial fixing, the algorithm iteratively extracts nodes from the valid mappings if the virtual networks contain cycles. queue Q which have been already mapped and considers all Proof. In Figure 2 we visually depict an example of a solution A outgoing virtual edges (i, j) ∈ Er . If an outgoing edge (i, j) to the LP Formulation 1 from which not a single valid mapping is contained in Er, Lemma 5 can be readily applied to obtain can be extracted. The validity of the depicted solution follows a joint mapping of the edge (i, j) and its head j. If the edge’s from the fact that the virtual node mappings sum to 1 and ←−A orientation was reversed, i.e. if (i, j) ∈ E r holds, Lemma 5 each virtual node connects to its neighboring node with half is applied while reversing the flow’s direction (Lines 13 - 16). a unit of flow. Assume for the sake of contradiction that the First, note that by the repeated application of Lemma 5, depicted solution can be decomposed. As virtual node i ∈ Vr A the mapping of virtual nodes and edges is valid. As Gr is mapped onto substrate node u1 ∈ VS, and u2 ∈ VS is A ·,· is an arborescence, each edge and each node of Gr will the only neighboring node with respect to variables zr,i,j that k V E eventually be mapped and hence mr is a valid mapping. The hosts j ∈ Vr, there must exist a mapping (mr , mr ) with k V V V mapping value fr is computed as the minimum of the mapping mr (i) = u1 and mr (j) = u2. Similarly, mr (k) = u3 must k V variables Vk used for constructing mr . Reducing the values hold. However, for mr (i) = u1, the virtual node k must be of the mapping variables together with the allocation variables mapped to u6, as otherwise the embedding of (k, i) cannot lead to substrate node u1. Hence the virtual node k ∈ Vr must III.NOVEL DECOMPOSABLE LPFORMULATION be mapped both on u6 and u3. As this is not possible, and the same argument holds when considering the mapping of In this section, we present a novel LP formulation and its i onto u4, no valid mapping can be extracted from the LP accompanying decomposition algorithm for the class of cactus solution. request graphs, i.e. graphs for which cycles intersect in at most a single node (in its undirected interpretation). Accordingly, these graphs can be uniquely decomposed into cycles and a This non-decomposability also induces large integrality single forest (cf. Lemma 10 below). gaps. Specifically, when considering edge mapping restrictions Before delving into the details of our novel LP formula- the integrality gap of the MCF Formulation 1 is unbounded. tion, we discuss our main insight on how to overcome the Theorem 8. The integrality gap of the MCF formulation is limitations of the MCF formulation and accordingly how to unbounded. This even holds under infinite substrate capacities. derive decomposable formulations. To this end, it is instruc- tive, to revisit the non-decomposable example of Figure 2 Proof. We again consider the example of Figure 2. We intro- by applying the decomposition Algorithm 1 on the depicted duce the following restrictions for mapping the virtual links: LP solution. Concretely, we consider the acyclic reorientation r,i,j r,j,k A A A ES = {(u1, u2), (u4, u5)}, ES = {(u2, u3), (u5, u6)}, Gr = (Vr,Er , rr) with Er = {(i, k), (i, j), (j, k)}, such r,k,i ES = {(u3, u4), (u6, u1)}. Note that the LP solution that i is the root, rr = i. Assuming that i is initially mapped depicted in Figure 2 is still feasible and hence the LP will on node u1, Algorithm 1 will map edges (i, k) and (i, j) V V attain an objective of br. first, setting mr (k) = u6 and mr (j) = u2 However, when On the other hand, there does not exist a valid mapping the edge (j, k) is processed, k must be mapped on substrate V of request r: assume i to be mapped on u1, then j must be node u3 =6 mr (k) and the algorithm hence fails to produce mapped on u2 and k must be mapped on u3 due to the node a valid mapping. Accordingly, to avoid such diverging node and edge mapping. However, the edge mapping restrictions mappings, our key idea is to decide the mapping location of for (k, i) do not allow the establishment of a path from u3 nodes with more than one incoming edge (with respect to the A to u1. By the same argument, the mapping of i on u4 is not request’s acyclic reorientation Gr ) a priori. feasible. Accordingly, as the optimal solution achieves a profit By considering only cactus request graphs, the above can of 0 while the LP solution yields a profit of br, the integrality be implemented rather easily as exactly one node of each gap of the MCF LP Formulation 1 is unbounded. cycle has more than one incoming edge: one only needs to ensure compatibility of node mappings for this node. To resolve potential conflicts for the mapping of this unique cycle As the above proof strongly relied on the fact that edge map- target, our formulation employs multiple copies of the MCF ping restrictions are employed, we also derive the following formulation for the respective cyclic subgraph. Specifically, result when only node mapping restrictions are considered. considering a cycle with virtual target node k, we instantiate r,k Theorem 9. The integrality gap of the MCF formulation lies one MCF formulation per substrate node w ∈ VS onto which k can be mapped. Accordingly, this yields at most |V | many in Ω(|VS|), when only considering node mapping restrictions. S copies and for each of these copies k is fixed to one specific Proof. Consider the following instance. The substrate is a (substrate) mapping location. Accordingly, as the mapping cycle of an even number of n nodes ui, with 1 ≤ i ≤ n location of k is fixed to a specific node, valid mappings for the and edges {(u1, u2), (u2, u3),..., (un−1, un), (un, u1)}. We respective cycles can always be extracted from such a MCF consider unit edge capacities, i.e. we set dS(e) = 1 for copy: the mappings of k cannot possibly diverge. e ∈ ES. Consider now a request r with Vr = {i, j} and edges E { i, j , j, i } and unit bandwidth demands d i, j r ( ) ( ) r( ) = A. Cactus Request Graph Decomposition and Notation dr(j, i) = 1. Assume that i can only be mapped on substrate nodes having an uneven index and that j can only be mapped Based on the assumed cactus graph nature of request graphs, on substrate nodes of even index. Clearly, any valid mapping of we consider the following decomposition of request graphs. this request will use all edge resources. Consider now the fol- Lemma 10. Consider a cactus request graph Gr and an lowing MCF solution with xr = 1: nodes i and j are, together A A with the respective edges, mapped in an alternating fashion: acyclic reorientation Gr of Gr. The graph Gr can be A,C1 A,Cn u v r,i uniquely partitioned into subgraphs {Gr ,...,Gr } t yr,i = yr,j = 2/n holds for all u ∈ VS = {u1, u3,... } A,F r,j Gr , such that the following holds: and all v ∈ V = {u2, u4,... }, respectively. Similarly, 0 S e e 0 A,C1 A,Cn zr,i,j = zr,j,i = 2/n is set for all edges e and e originating 1) The subgraphs {Gr ,...,Gr } correspond to the A,F at uneven and even nodes, respectively. Hence, the allocation (undirected) cycles of Gr and Gr is the forest remain- equals 2/n on each edge. Hence, n/2 ∈ Θ(|VS|) many copies ing after removing the cyclic subgraphs. We denote the of this request may be embedded in the MCF solution, while index set of the cycles by Cr = {C1,...,Cn}. A the optimal solution may only embed a single request and the 2) The subgraphs partition the edges of Er : an edge A integrality gap therefore lies in Ω(|VS|). (i, j) ∈ Er is contained in exactly one of the subgraphs. A,Ck 3) The edge set Er of each cycle Ck ∈ Cr can itself be node mapping variables (cf. Constraint 2 of Formulation 1). Ck Ck partitioned into two branches B1 and B2 , such that By Constraints 13 and 14, the node mappings of the sub- Ck A,Ck Ck A,Ck both lead from sr ∈ Vr to tr ∈ Vr . LPs for mapping the subgraphs must agree with the global node mapping variables. With respect to cyclic subgraphs, we Proof. We prove that the lemma holds independently of A note that Constraint 14 allows for distributing the global node the chosen acyclic reorientation. Accordingly, let Gr be Ck mappings to any of the |VS,t | formulations: only the sum of an arbitrary acyclic reorientation of Gr. We first show that − the node mapping variables must agree with the global node |δ (i)| ≤ 2 holds for each virtual node i ∈ Vr with respect A mapping variable. Constraint 15 is of crucial importance for to the edge set Er , i.e. any virtual node has at most 2 A − the decomposability: considering the sub-LP for cycle Ck and incoming edges in G . If |δ (i)| > 2 held, then by the C r target node w ∈ V k , it enforces that the target node tCk of the definition of the acyclic reorientation there must exist at least S,t r cycle C must be mapped on w. Thus, in the sub-LP [C , w] 3 paths P , P , P from the root r to i. Let p , p ∈ V k k 1 2 3 r 1,2 2,3 r both branches BCk and BCk of cycle C are pre-determined be the last common nodes lying on both P and P and 1 2 k 1 2 to lead to the node w. Lastly, for computing node allocations P and P , respectively. Clearly, from p , there exist two 2 3 1,2 the global node mapping variables are used (cf. Constraint 16) (otherwise) node-disjoint paths to i and from p there exist 2,3 and for computing edge allocations the sub-LP formulations’ two (otherwise) node-disjoint paths towards i. Accordingly, allocations are considered (cf. Constraint 17). there exist at least two cycles intersecting either in i and p1,2 or i and p2,3, which contradicts the assumption on Gr being C. Decomposing Solutions to the Novel LP Formulation a cactus graph. Accordingly, |δ−(i)| ≤ 2 holds for all virtual We now show how to adapt the decomposition Algorithm 1 nodes i ∈ V according to EA. r r to decompose solutions to Formulation 2. Now, consider any node i ∈ V with |δ−(i)| ≤ 2. By r To decompose the LP solution for a request r, the acyclic performing a graph-search in the opposite direction of edges reorientation GA, which was also used for constructing the in EA, a unique common ancestor node i0 can be determined, r r LP, must be handed over to the decomposition algorithm. such that there exist exactly two paths B and B (branches) 1 2 As the novel LP formulation does not contain (global) from i0 to i. The union of these branches represents a single edge mapping variables, the edge mapping variables used ‘cycle’ (cf. Statement 3). Removing the identified cycle from in Lines 10 and 14 of Algorithm 1 must be substituted by the graph, the cactus graph property still holds, as removing edge mapping variables of the respective sub-LP formulations. edges can never refute it. Accordingly, ‘cycles’ in the acyclic Concretely, as each edge of the request graph G is covered reorientation GA,C1 ,...,GA,Cn can be uniquely identified r r r exactly once, it is clear whether a virtual edge (i, j) ∈ E (decomposed) and after repeated removal only the forest GA,F r r is part of GF or a cyclic subgraph GCk . If (i, j) ∈ GF remains. Hence, the first statement of the lemma follows. r r r Lastly, the second statement of the lemma holds trivially as each edge is either contained in any of the cycles or is part of the remaining forest. Formulation 2: Novel LP for Cactus Requests X Ck max brxr (9) Besides the above introduced notation, we denote by Gr F r∈R and Gr the subgraphs that agree with Gr on the edge F Ck Cons. (2) - (7) for Gr on V Ck V r,tr ∀r ∈ R (10) orientations and use S,t = S to denote the substrate variables (xr, ~yr, ~zr,~ar)[Fr] Ck nodes on which tr can be mapped. Ck Cons. (2) - (7) for Gr on Ck ∀r ∈ R,Ck ∈ Cr, w ∈ VS,t (11) B. Novel LP Formulation for Cactus Requests variables (xr, ~yr, ~zr,~ar)[Ck, w] A X u Our novel Formulation 2 uses the a priori partition of Gr xr= yr,i ∀r ∈ R, i ∈ Vr (12) A,Ck A,F u∈V r,i into cycles Gr and the forest Gr to construct MCF S formulations for the respective subgraphs: for the subgraph yu = yu [F] ∀r ∈ R, i ∈ V F , u ∈ V r,i (13) F r,i r,i r S Gr a single copy is used (cf. Constraint 10) while for " r,i # the cyclic subgraphs a single MCF formulation is employed u X u r ∈ R, i ∈ Vr, u ∈ VS , y = y [Ck, w] ∀ (14) Ck r,i r,i C Ck ∈ Cr : i ∈ V k per potential target location contained in the set VS,t (cf. Ck r w∈tr Constraint 11). We index the variables of these sub-LPs by " Ck # u r ∈ R,Ck ∈ Cr, w ∈ VS,t , employing square brackets. 0= y Ck [Ck, w] ∀ C (15) r,tr u ∈ V k \{w} To bind together these (at first) independent MCF formu- S,t τ,u X u V lations, we reuse the variables ~x,~y, and ~a introduced already ar = dr(i) · yr,i ∀r ∈ R, (τ, u) ∈ RS (16) for the MCF formulation. We refer to these variables, which i∈Vr ,τr (i)=τ are defined outside of the sub-LP formulations, as global u,v u,v X u,v ar = ar [F] + ar [Ck, w] ∀r ∈ R, (u, v) ∈ ES (17) variables and do not index these. As we only consider the Ck Ck∈Cr ,w∈VS,t LP formulation, all variables are continuous. X ax,y ≤ d x, y ∀ x, y ∈ R The different sub-formulations are linked as follows. We r S( ) ( ) S (18) r∈R employ Constraint 12 to enforce the setting of the (global) ·,· holds, then the edge mapping variables zr,i,j[Fr] are used. Synopsis of our Approximation Algorithm: The algorithm If on the other hand the edge (i, j) ∈ Er is covered in the first performs a preprocessing in Lines 1 - 3 by removing Ck Ck cyclic subgraph Gr , then there exist |VS,t | many sub-LPs to all requests which cannot be fully (fractionally) embedded choose the respective edge mapping variables from. To ensure in the absence of other requests, as these can never be part the decomposability, we proceed as follows. of any feasible solution. In Lines 4 - 6 an optimal solution A Ck If the edge (i, j) ∈ Er is the first edge of Gr to be mapped to the novel LP Formulation 2 is computed and afterwards ·,· in the k-th iteration, the mapping variables zr,i,j[Ck, w] be- decomposed into convex combinations. Then, in Lines 7 - 9, mV (i) the rounding is performed: for each request r ∈ R a mapping longing to an arbitrary target node w, with y r [C , w] > 0, r,i k mk f k are used. Such a node w exists by Constraint 14. r is selected with probability r . Importantly, the summed probabilities may not sum to 1, i.e. with probability 1−P f k If another edge (i0, j0) of the same cycle was already k r the request r is not embedded. mapped in the k-th iteration, the same sub-LP before is The rounding procedure is iterated as long as the constructed considered. Accordingly, the mapping of cycle target nodes solution is not of sufficient quality or until the number of cannot conflict, and as these are the only nodes with potential maximal rounding tries is exceeded. Concretely, we seek mapping conflicts, the returned mappings are always valid. (α, β, γ)-approximate solutions which achieve at least a factor To successfully iterate the extraction process, the steps of α ≤ 1 times the optimal (LP) profit and exceed node and taken in Lines 18 - 21 of Algorithm 1 must be adapted to edge capacities by at most factors of β ≥ 1 and γ ≥ 1, consider the sub-LP variables. Again, as in each iteration at respectively. In the following we derive parameters α, β, and least a single variable of the LP is set to 0 and as the novel γ for which solutions can be found with high probability. Formulation 2 contains at most O(|V |) times more variables S Note that Algorithm 2 is indeed a polynomial-time algo- than the MCF Formulation 1, the decomposition algorithm still rithm, as the size of the novel LP Formulation 2 is polynomi- runs in polynomial-time. Hence, we conclude that the result ally bounded and can hence be solved in polynomial-time. of Lemma 6 carries over to the novel LP Formulation 2 for cactus request graphs and state the following theorem. A. Probabilistic Guarantee for the Profit

Theorem 11. Given a solution (xr, ~yr, ~zr,~ar) to the novel LP For bounding the profit achieved by the randomized round- Formulation 2 for a cactus request graph Gr, the solution can ing scheme, we recast the profit achieved in terms of random be decomposed into a convex combination of valid mappings variables. The discrete random variable Yr ∈ {0, br} models k k k k Dr = {(mr , fr )|mr ∈ Mr, fr > 0}k, such that: the profit achieved by the rounding of request r ∈ R. Accord- P k k ing to our rounding scheme, we have (Y = b ) = f • P P r r k r The decomposition is complete, i.e. xr = k fr holds. P k and P(Yr = 0) = 1 − k fr . We denote the overall profit by • The decomposition’s allocations are bounded by ~ar, i.e. P P P k x,y P k k B = r∈R Yr with E(B) = r∈R br · k fr . Denoting the ar ≥ k fr · A(mr , x, y) holds for (x, y) ∈ RS. profit of an optimal LP solution by BLP, we have BLP = E(B) due to the decomposition’s completeness (cf. Theorem 11). IV. APPROXIMATION VIA RANDOMIZED ROUNDING By preprocessing the requests and confirming that each request can be fully embedded, the LP will attain at least the Above we have shown how convex combinations for the maximal profit of any of the considered requests: VNEP can be computed for cactus requests. Given these convex combinations, we now present our approximation Lemma 12. E(B) = BLP ≥ maxr∈R br holds. algorithm (see Algorithm 2) and analyze its probabilistic We employ the following Chernoff bound over continuous guarantees in Sections IV-A to IV-C. Afterwards, we propose variables to bound the probability of achieving a small profit. several rounding heuristics to apply our approximation in Pn practice (see Section IV-D). Theorem 13 (Chernoff Bound [20]). Let X = i=1 Xi, Xi ∈ [0, 1], be a sum of n independent random variables. For any 0 < ε < 1, the following holds:
2 Algorithm 2: Randomized Rounding for the VNEP P X ≤ (1 − ε) · E(X) ≤ exp(−ε · E(X)/2) 1 foreach r ∈ R do // preprocess requests 2 compute LP Formulation 2 for request r maximizing x r Theorem 14. Let BIP denote the profit of an optimal solution. 3 if xr < 1 then remove request r from the set R Then P(B < 1/3 · BIP) ≤ exp(−2/9) ≈ 0.8007 holds. P 4 compute LP Formulation 2 for R maximizing br · xr r∈R Proof. ˆb b 5 foreach r ∈ R do // perform decomposition Let = maxr∈R r be the maximum bene- k k fit among the preprocessed requests. We consider ran- 6 compute Dr = {(f , m )}k from LP solution r r 0 ˆ 0 dom variables Yr = Yr/b, such that Yr ∈ [0, 1] holds. 7 // perform randomized rounding do 0 P 0 ˆ ˆ k k Let B = r∈R Yr = B/b. As E(B) = BLP ≥ b 8 foreach r ∈ R select mr with probability fr 0  solution is not (α, β, γ)-approximate and  holds (cf. Lemma 12), we have E(B ) ≥ 1. Choos- 9 while 0 maximal rounding tries are not exceeded ing ε = 2/3 and applying Theorem 13 on B we ob- 0 0
0 tain P B ≤ (1/3) · E(B ) ≤ exp(−2 · E(B )/9). Plugging 0 in the minimal value of E(B ), i.e. 1, into the equation we the expected allocation E(Ax,y) is upper bounded by the 0 0
obtain: P B ≤ (1/3)·E(B ) ≤ exp(−2/9) and by linearity resource’s capacity dS(x, y) to obtain Equation 19. B ≤ (1/3) · (B)
≤ exp(−2/9). P E Given Lemma 16, we obtain the following corollary. Denoting the profit of an optimal solution by BIP and observing that BIP ≤ BLP holds as the linear relaxation yields Corollary 17. Let 0 < ε ≤ 1 be chosen minimally, such that an upper bound, we have BIP/3 ≤ E(B)/3. Accordingly, we dmax(r, x, y)/dS(x, y) ≤ ε holds for all resources (x, y) ∈ RS
conclude that, P B ≤ (1/3) · BIP ≤ exp(−2/9) holds. and all requests r ∈ R. Let ∆(X) = max(x,y)∈X ∆(x, y), q B. Probabilistic Guarantee for Resource Augmentations V β =(1 + ε · 2 · ∆(RS ) · log(|VS| · |T |)) , and In the following, we analyze the probability that a rounded p solution exceeds substrate capacities by a certain factor. γ =(1 + ε · 2 · ∆(ES) · log(|ES|)) . V We first note that dmax(r, x, y) ≤ dS(x, y) holds for The following holds for all node resources (τ, u) ∈ RS and all resources (x, y) ∈ RS and all requests r ∈ R. We edge resources (u, v) ∈ ES, respectively: model the allocations on resource (x, y) ∈ RS by re- −4 P(Aτ,u ≥ β · dS(τ, u)) ≤(|VS| · |T |) (20) quest r ∈ R as random variable Ar,x,y ∈ [0,Amax(r, x, y)]. k k −4 By definition, we have P(Ar,x,y = A(mr , x, y)) = fr P(Au,v ≥ γ · dS(u, v)) ≤|ES| (21) and (A = 0) = 1 − P f k. Furthermore, we denote by P r,x,y k r Proof. First, note that ε is chosen over all resources and A = P A the random variable capturing the x,y r∈R r,x,y requests and that ∆(RV ) ≥ ∆(τ, u) and ∆(E ) ≥ ∆(u, v) overall allocations on resource (x, y) ∈ R . As (A ) = S S S E x,y hold for (τ, u) ∈ RV and (u, v) ∈ E , respectively. Equa- P P f k · A(mk, x, y) holds by Theorem 11, we obtain S S r∈R k r r tions 20 and 21 are then obtained from Lemma 16 by setting E(Ax,y) ≤ dS(x, y) for all resources (x, y) ∈ RS. λ = |VS| · |T | for nodes and λ = |ES| for edges. We employ Hoeffding’s inequality to upper bound Ax,y. Pn C. Approximation Result Theorem 15 (Hoeffding’s inequality [20]). Let X = i=1 Xi, Xi ∈ [ai, bi], be a sum of n independent random variables. Given the probabilistic bounds established above, the main The following holds for any t ≥ 0: approximation result is obtained via a union bound. 2 X 2 Theorem 18. Assume |V | ≥ 3. Let β and γ be defined as P(X − E(X) ≥ t) ≤ exp(−2t /( (bi − ai) )) S i in Corollary 17. Algorithm 2 returns (α, β, γ)-approximate Lemma 16. Consider a resource (x, y) ∈ RS and 0 < ε ≤ 1, solutions for the VNEP (restricted on cactus request graphs) such that dmax(r, x, y)/dS(x, y) ≤ ε holds for r ∈ R. Let of at least an α = 1/3 fraction of the optimal profit, and ∆(x, y) = P (A (r, x, y)/d (r, x, y))2. allocations on nodes and edges within factors of β and γ of r∈R:dmax(r,x,y)>0 max max −4 the original capacities, respectively, with high probability. P(Ax,y ≥ δ(λ) · dS(x, y)) ≤ λ (19) Proof. We employ the following union bound argument. Em- holds for δ λ ε · p · x, y · λ and any λ > . ( ) = 1 + 2 ∆( ) log( ) 0 ploying Corollary 17 and as there are at most |VS| · |T | node 2 Proof. We apply Hoeffding with t = (1 − δ(λ)) · dS(x, y): resources and at most |VS| edges, the joint probability that   any resource exceeds their respective capacity by factors of β 3 2 P Ax,y − E(Ax,y) ≥ (1 − δ(λ)) · dS (x, y) or γ is upper bounded by (|VS| · |T |) + |VS| ≤ 1/27 + 1/9 for |VS| ≥ 3. By Theorem 14 the probability of not finding a  2 2  −4 · ε · log(λ) · ∆(x, y) · dS (x, y) solution achieving an α = 1/3 fraction of the optimal objective ≤ exp P 2 (Amax(r, x, y)) r∈R is upper bounded by exp(−2/9). Hence, the probability to not  −4 · ε2 · log(λ) · ∆(x, y) · d2 (x, y)  find a (α, β, γ)-approximate solution within a single round ≤ exp S P 2 is upper bounded by exp(−2/9) + 1/9 + 1/27 ≤ 19/20. (Amax(r, x, y)) r∈R:dmax(r,x,y)>0 Hence, as the probability to return a suitable solution within a 2 2  −4 · ε · log(λ) · ∆(x, y) · dS (x, y)  single round is at least 1/20, the probability that the algorithm ≤ exp P 2 (ε · dS (x, y) · Amax(r, x, y)/dmax(r, x, y)) returns an approximate solution within N ∈ N rounding tries r∈R:dmax(r,x,y)>0 is lower bounded by 1 − (19/20)N . Thus, Algorithm 2 yields  −4 · log(λ) · ∆(x, y)  −4 approximate solutions for the VNEP with high probability. ≤ exp P 2 = λ (Amax(r, x, y)/dmax(r, x, y)) r∈R:dmax(r,x,y)>0 D. Discussion & Proposed Heuristics

The second inequality holds, as Amax(r, x, y) > 0 Theorem 18 yields the first approximation algorithm for the implies dmax(r, x, y) > 0. For the third inequality, profit variant of the VNEP. However, the direct application Amax(r, x, y) ≤ ε·dS(x, y) · Amax(r, x, y)/dmax(r, x, y) is used, of Algorithm 2 to compute (α, β, γ)-approximate solutions which follows from the assumption dmax(r, x, y) ≤ ε·dS(x, y) is made difficult by the cumbersome definition of the terms 2 2 V and dmax(r, x, y) > 0. In the next step, ε ·dS(x, y) is reduced ∆(RS ) and ∆(ES). Specifically, computing β and γ exactly from the fraction. As the denominator equals ∆(x, y) by requires enumerating all valid mappings, which is not feasible. definition, the final equality follows. Lastly, we utilize that Hence, to directly apply Algorithm 2, the respective values Request Graph Characteristics 1.0 0.9 0.8 0.7 0.6 0.5

ECDF 0.4 0.3 number of nodes: |Vr|

0.2 number of edges: |Er| 0.1 number of cycles: |Er| |Vr| + 1 0.0 0 3 6 9 12 15 18 21

Fig. 3. Visualization of the GEANT´ network chosen as substrate network. Fig. 4. Characteristics of the generated cactus requests graphs, namely The GEANT´ network connects several research, education and innovation the number of nodes, edges and number of cycles. The depicted empirical institutes across Europe. cumulative distribution function (ECDF) is based upon 100k sampled cactus requests graphs. have to be estimated. The following upper bounds can be resource capacity, then this mapping is simply discarded easily established: and request r is not embedded. To increase the diversity of V found solutions, the order in which requests are processed ∆(RS ) =|R| · max |Vr| (22) r∈R is permuted before each rounding iteration. A fixed number ∆(ES) ≤|R| · max |Er| . (23) r∈R of solutions is rounded and the one maximizing the profit is returned. Both inequalities follow from the observation that V. EXPLORATIVE COMPUTATIONAL STUDY Amax(r, x, y)/dmax(r, x, y) is upper bounded by the number of virtual nodes and edges, respectively, as Amax(r, τ, u) ≤ We now complement our formal approximation result in dmax(r, τ, u) · |Vr| and Amax(r, u, v) ≤ dmax(r, u, v) · |Er| the standard multi-criteria model with resource augmentation V holds for node resources (τ, u) ∈ RS and edge resource with an extensive computational study. Specifically, we study (u, v) ∈ ES, respectively. Now, plugging these upper bounds the performance of vanilla rounding and heuristical rounding into the definition of β and γ yields large resource augmenta- (i.e., without resource augmentations) as introduced above. p tions of β ∈ O(ε · |R| · maxr∈R |Vr| · log(|VS| · |T |)) and As we are not aware of any systematic evaluation of p γ ∈ O(ε · |R| · maxr∈R |Er| · log(|ES|)), respectively. the profit maximization in the offline settings, we present a To overcome estimating β and γ by the above (or simi- synthetic but extensive computational study. Specifically, we lar) bounds, we propose the following to apply randomized have generated 1,500 offline VNEP instances with varying rounding in practice. Concretely, we consider two types of request numbers and varying demand-to-capacity ratios. For adaptations to our Algorithm 2. The first adaptation is to all instances, baseline solutions were computed by solving the discard the consideration of respective approximation factors (Mixed-)Integer Programming Formulation 1. α, β, γ and simply returning the best solution found within We have implemented all presented algorithms in Python 2.7 a fixed number of rounding iterations. We refer to this type employing Gurobi 7.5.1 to solve the Mixed-Integer Programs of rounding heuristic as vanilla rounding, as the rounding and Linear Programs. Our source code is freely available procedure itself is not changed. The other type of adaptation at [21]. All experiments were executed on a server equipped is the avoidance of resource augmentations by considering with Intel Xeon E5-4627v3 CPUs running at 2.6 GHz and heuristical rounding. reported runtimes are wall-clock times. Concretely, we consider two vanilla rounding heuristics, A. Instance Generation namely RR and RR , as well as one heuristi- MinLoad MaxProfit a) Substrate Graph: We use the GEANT´ topology1 as cal rounding variant RR , as described below: Heuristic substrate network. It consists of 40 nodes and 122 edges (see RR A fixed number of solutions is rounded at random MinLoad Figure 3). We consider a single node type and set node and as in Line 8 of Algorithm 2. The solution minimizing edge capacities uniformly to 100. resource augmentations (and among those the one of the b) Request Topology Generation: Cactus graph requests highest profit) is selected. are generated by (i) sampling a random binary tree of maxi- A fixed number of solutions is rounded at ran- RRMaxProfit mum depth 3, (ii) adding additional edges randomly as long dom as in Line 8 of Algorithm 2. The solution maximizing as they do not refute the cactus property as long as such edges the profit (and among those the one of the least resource exist, and (iii) orienting edges arbitrarily. augmentations) is selected. Concretely, the sampling process of binary trees works RRHeuristic Line 8 of Algorithm 2 is adapted in such a as follows: starting with an initial root node, the num- way that selected mappings are only accepted, when its ber of children is drawn using the discrete distribu- incorporation into the solution does not exceed resource k tion P(#children = 0) = 0.15, P(#children = 1) = 0.5, and capacities. In other words, if a mapping mr is selected for request r ∈ R whose addition would exceed any 1Obtained from http://www.topology-zoo.org/ (version March 2012) . MIPMCF: #Feasible Requests MIPMCF: #Embedded / #Feasible [%] MIPMCF: Objective Gap [%] MIPMCF: Objective Gap [%] 100 100 20 20 4.0 4.0 4.0 4.0 80 80 16 16 2.0 2.0 2.0 2.0 60 60 12 12 1.0 1.0 1.0 1.0 40 40 8 8 0.5 0.5 0.5 0.5 20 20 4 4 Edge Resource Factor Edge Resource Factor Edge Resource Factor 0.25 0.25 0.25 Edge Resource Factor 0.25 0 0 0 0 40 60 80 100 40 60 80 100 40 60 80 100 0.2 0.4 0.6 0.8 1.0 Number of Requests Number of Requests Number of Requests Node Resource Factor

Fig. 5. Overview of the feasibility of generated requests and the baseline’s Fig. 6. Overview of the objective gap achieved by the baseline algorithm acceptance ratio. Each cell averages the result of 75 instances. MIPMCF after upto 3 hours of computation time. Note the different x-axes. Left: The feasibility of requests is obtained from (cost-optimally) embedding The left plot averages the results of 75 and the right one the results of 60 the requests to compute the profit a priori. Note absolute numbers are depicted. instances per cell. The number of requests, i.e. the problem size, has a less Right: The acceptance ratio of the baseline MIPMCF with respect to the distinct impact on the objective gap than the resource factors alone. requests that were found to be feasible (infeasible requests are not considered).

LPnovel: Total Runtime [min] MIPMCF: Runtime [min] . 180 10 tion, a resource factor NRF = 0 6 implies that the node load 4.0 4.0 – averaged over all substrate nodes – equals exactly 60%. As 144 8 2.0 2.0 virtual edges can be mapped on arbitrarily long paths (even 108 6 of length 0), the edge resource factor should be understood as 1.0 1.0 72 4 follows: the ERF equals ‘the number of substrate edges that 0.5 0.5 . 36 2 each virtual edge may use’. In particular, a factor ERF = 0 5 Edge Resource Factor Edge Resource Factor 0.25 0.25 implies that if each virtual edge spans exactly 0.5 substrate 0 0 0.2 0.4 0.6 0.8 1.0 40 60 80 100 edges (and all requests are embedded), then the (averaged) Number of Requests Node Resource Factor edge resource utilization equals exactly 100%. Hence, while

Fig. 7. Overview of the runtimes of the baseline algorithm MIPMCF and increasing the NRF renders node resources more scarce, our novel LP formulation LPnovel. Note the different x-axes. On the left each increasing the ERF reduces edge resource scarcity. cell averages 60 results while on the right 75 results are averaged per cell. Left: The runtime of the MCF Formulation 1 being solved by Gurobi 7.5.1. e) Profit Computation: To correlate the profit of a request Copmutations are terminated after 180 minutes. with its size, its resource demands, and its mapping restric- Right: The runtime of the novel LP Formulation 2 – solved using the Barrier tions, we compute for each request its minimal embedding algorithm of Gurobi 7.5.1. – including the constructing time of the LP. costs as follows. The cost c(u, v) of using an edge (u, v) ∈ VS equals the geographical distance of its endpoints. The cost of P nodes is set uniformly to c(·, u) = c(u, v)/|VS| P(#children = 2) = 0.35. For each (newly) generated node (u,v)∈ES (of depth less than 3) further children are generated according for all u ∈ VS. Hence, the total node cost equals the to the same distribution. We discard graphs having less than total edge cost. Defining the cost of a mapping mr to be P A(m , x, y) · c(x, y), we compute the minimum 3 nodes. According to the above generation procedure, the (x,y)∈RS r expected number of nodes and edges is 6.54 and 7.28, respec- cost embedding for each request r ∈ R using an adaption tively. On average, 61% of the edges lie on a cycle. Figure 4 of Mixed-Integer Program 1 and set br accordingly. offers a more in-depth view on the request characteristics. f) Parameter Space: We consider the following param- c) Mapping Restrictions: To force the virtual networks eters |R| ∈ {40, 60, 80, 100}, NRF ∈ {0.2, 0.4, 0.6, 0.8, 1.0}, to span across the whole substrate network, we restrict the ERF ∈ {0.25, 0.5, 1.0, 2.0, 4.0} and generate 15 instances per mapping of virtual nodes to one quarter of the substrate parameter combination, yielding 1, 500 instances overall. nodes. Hence, the mapping of virtual nodes is restricted to ten B. Computational Results substrate nodes. The mapping of virtual edges is not restricted. We first present our baseline results computed by solving d) Demand Generation: We control the demand-to- the MCF Integer Programming Formulation 1 and then study capacity ratio of node and edge resource using a node resource the performance of vanilla rounding and heuristical rounding. factor NRF and an edge resource factor ERF. The request’s a) Baseline MIP : To obtain a near-optimal baseline demands are drawn from an exponential distribution and MCF solution for each of the 1,500 instances, we employ Gurobi afterwards normalized, such that the following holds: 7.5.1 to solve the Mixed-Integer Programming Formulation 1 X X X (using a single thread). We terminate the computation after dr(i) =NRF · dS (u) (24) 3 hours or when the objective gap falls below 1%, i.e. r∈R i∈Vr u∈VS X X X when the constructed solution is provably less than 1% off ERF · dr(i, j) = dS (u, v) (25) r∈R (i,j)∈Er (u,v)∈ES the optimum. On average the runtime per instance is 129.8 minutes (cf. Figure 7, left). The resource factors can be best understood under the Figure 5 gives an initial overview of the number of requests assumption that all requests are embedded. Under this assump- for which feasible embeddings exist and the acceptance ratio MIPMCF: Avg. Node Load [%] MIPMCF: Max. Node Load [%] MIPMCF: Avg. Edge Load [%] MIPMCF: Max. Edge Load [%] 60 100 30 100 1.0 1.0 4.0 4.0 50 80 24 80 0.8 40 0.8 2.0 2.0 60 18 60 0.6 30 0.6 1.0 1.0 40 12 40 0.4 20 0.4 0.5 0.5 10 20 6 20 Edge Resource Factor Edge Resource Factor Node Resource Factor 0.2 Node Resource Factor 0.2 0.25 0.25 0 0 0 0 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 Number of Requests Number of Requests Number of Requests Number of Requests

Fig. 8. Overview of the resource loads of the solutions computed by the baseline algorithm MIPMCF. Each cell averages the results of 75 solutions. Depicted are the averaged node/edge loads and the averaged maximum node/edge loads as a function of the node/edge resource factor and the number of requests. The respective resource factors have a distinct impact on the loads. Node resources are in general more scarce than edge resources.

Vanilla Rounding RRMinLoad Vanilla Rounding RRMaxProfit Heuristic Rounding Performance Heuristic Rounding Performance ]

] 500 225 ERF Profit(RRHeuristic)/Profit(MIPMCF) [%] Profit(RRHeuristic)/Profit(MIPMCF) [%] %

% 100 100 [ [

0.25 ) )

t 4.0 4.0 i d 200 f

a 0.5 o 400 r o P L 90 90 x n 1.0 i 2.0 2.0 175 a M 2.0 M ERF R R 300

R 0.25 R 1.0 80 ( 4.0 1.0 80

150 ( d

d 0.5 a a o

125 o 200 1.0 0.5 0.5 L L 70 70 x

x 2.0 a 100 a 0.25 0.25 Edge Resource Factor M 4.0 Edge Resource Factor M 100 60 60 60 80 100 120 140 100 125 150 175 200 40 60 80 100 0.2 0.4 0.6 0.8 1.0 Profit(RRMinLoad)/Profit(MIPMCF) [%] Profit(RRMaxProfit)/Profit(MIPMCF) [%] Number of Requests Node Resource Factor

Fig. 9. Depicted are the solutions obtained via the two vanilla rounding Fig. 10. Overview of the averaged relative performance achieved by the schemes RRMinLoad (left) and RRMaxProfit (right). Each point corresponds heuristic rounding algorithm as a function of the edge resource factor and to a single instance and is colored according to the instance’s edge resource the number of requests (left) and the node resource factor (right). Each cell factor. The left and right plots shows results for 1,493 and 1,499 of the 1,500 averages the result of 75 (left) and 60 (right) results. The performance of the requests, respectively; the other results lie outside the depicted area. heuristic depends both on the number of requests and the resource availability. of the best solution as a function of the number of requests resource load is always close to 100%. For the edge resource and the edge resource factor. The number of feasible requests loads, the picture is a different one. In particular, the averaged is determined during the a priori profit computation and may edge load lies significantly below the averaged node load. (on average) lie below 50% when edge resources are very However, the impact of the edge resource factor is again scarce (ERF = 0.25) but otherwise consistently lies above apparent. Interestingly, for edge resource factors of 2.0 and 4.0, 75%. Similarly, the acceptance ratio of the baseline solution i.e. when edge resources are abundant, the maximal edge load highly depends on the edeg resource factor, ranging from close only lies around 40%, indicating that in these cases mostly the to 53% to roughly 98% (on average). node resources are constraining the solution space. Figure 6 depicts quality guarantees for the baseline solutions The above can also be observed by the maximum load obtained during the solution process of IP Formulation 1. ECDFs depicted in Figure 11 (left). In particular, the maximum Concretely, the formulation is solved using Gurobi’s branch- edge load of the baseline MIP lies consistently below the and-bound implementation, consistently yielding upper bounds maximum node load: While 40% of the solutions exhibit a on the attainable profit (by considering the LP relaxation). maximum edge load of 60% or less, only roughly 10% of the Accordingly, the objective gap depicted in Figure 6 gives solutions exhibit a maximum node load of 90% or less. guarantees on how far (at most) the found solutions are b) Solving LP Formulation 2: To apply the rounding off optimality (on average). While increasing the number of algorithms presented in Section IV-D, we solve our novel LP requests does not increase the objective gap per se, both Formulation 2 by employing again Gurobi 7.5.1, specifically the node and edge resource factors have a distinct impact. its Barrier algorithm with crossover. Figure 7 (right) depicts Particularly, for the maximal node resource factor of 1.0 and a the averaged runtime to solve the LP, including the time to medium edge resource factor of 1.0, the averaged objective gap construct the (potentially very large) LP. The latter is not lies slightly above 19%. The mean objective value is less than negligible as the formulation contains up to 1,000k variables 7% and the maximum observed gap across all 1,500 instances for some instances. The runtime increases from around 2 was roughly 59%. minutes for |R| = 40 to around 7 minutes for |R| = 100. The Figure 8 validates the impact the node and edge resource maximally observed runtime in our experiments amounted to factors have on the respective resource loads. Depicted are the roughly 18 minutes. averaged and maximal node and edge loads as a function of the c) Vanilla Rounding: With respect to the results of our respective resource factors and the number of requests. As can rounding heuristics, we first discuss the results of our vanilla be clearly seen, increasing the node resource factor increases rounding heuristics RRMinLoad and RRMaxProfit. Concretely, the averaged node resource loads, while the maximum node we report on the best solution found within 1,000 rounding ECDF of Resource Loads ECDF of Relative Achieved Profit LPnovel: Formulation Strength 1.0 Algorithm 1.0 1.0

RRMinLoad 0.8 0.8 RRMaxProfit 0.8

RRHeuristic 0.6 0.6 0.6 MIPMCF #Requests ECDF ECDF 0.4 0.4 ECDF Algorithm 0.4 40

Resource RRMinLoad Bound(MIPMCF) 60 0.2 0.2 0.2 node RRMaxProfit initial 80 RR 0.0 edge 0.0 Heuristic 0.0 final 100 10 50 100 200 500 50 100 150 200 0.75 1.00 1.50 2.00 2.50 3.00 3.50 Maximum Resource Load [%] Profit(RRAlg)/Profit(MIPMCF) [%] Bound(MIPMCF) / Bound(LPnovel)

Fig. 11. Comparison of the different algorithms in terms of (maximal) Fig. 12. Comparison of the bounds on the profit computed by the novel resource usage and the relative achieved profit for all 1,500 results. LP Formulation 2 and the classic MCF Formulation 1. As objective bounds Left: ECDF of maximal node and edge resource loads for all algorithms. For are continuously improved during the solution process of the Mixed-Integer the randomized rounding algorithms, the maximum edge load is in general Program MIPMCF, we report on the initial (weakest) bound and the final higher than the maximum edge load. Note the logarithmic x-axis. (best) bound. The initial bound is essentially the objective value of the LP Right: ECDF of the profit achieved by the randomized rounding algorithms relaxation of Formulation 1, but might be improved by the solver Gurobi based compared to the profit achieved by the baseline solution. on the introduction of cutting planes valid only for the integer variant. iterations. Figure 10 depicts the respective results as a scatter For any combination of resource factors, the virtual resource plot while Figure 11 depicts empirical cumulative distribution demands are computed according to Equations 20 and 21 functions (ECDF) for both the achieved profit and the maximal independently of how many requests are considered. Hence, resource loads. As can be seen, for RRMinLoad the algorithm when increasing the number of requests, the resource demands achieves a profit between 50% and 140% compared to the of each single request become smaller. Hence, the maximal best solution constructed by the MIP, while exceeding resource demand-to-capacity ratio dmax(r, x, y)/dS(x, y) decreases for capacities mostly by 25% to 125% of the resource’s capacity. all substrate resources (x, y) ∈ RS when increasing the For RRMaxProfit, the achieved profit always exceeds the number of requests. Accordingly, the value of ε (cf. Lemma 16 baseline’s profit. This is to be expected based on our analysis and Corollary 17) can be chosen smaller and the expected re- in Section IV-A, as the expected benefit is at least as large as source augmentations decrease. Thus, the heuristical rounding the optimal Mixed-Integer Programming formulation’s value. algorithm is able to more easily find solutions of high profit The resource loads mostly lie below 400% with the maximum not augmenting resources. coming close to 500%. Overall, the average relative performance with respect to For both selection criteria, the edge resource factor has a the baseline solutions is 73.8%, with the minimal one being distinct impact on the (overall) maximum load. This can be 22.3% and only around 5% of constructed solutions achieving explained as follows. As each request edge may use any of the less than 50% of the baseline’s profit. substrate edges (compared to the restricted node mappings), the chances of ‘collisions’, i.e. multiple request edges being C. Comparisong of Formulation Strengths mapped on the same substrate edge, is higher. Additionally, Lastly, we empirically study the strength of our novel the fact that the number of virtual edges is (always) at LP Formulation 2 and compare it with the classic MCF least as large as the number of edges (cf. Figure 4) and Formulation 1. Concretely, as proven in Theorems 8 and 9, that a single request edge may use multiple substrate edges, the classic MCF formulation has an unbounded or very large complicates the rounding of edge mappings. This observation integrality gap in general, i.e. the bound on the profit returned is further substantiated by the ECDF presented in Figure 12: by the solution can be arbitrarily far off the optimal attainable the maximal edge load are consistenly larger than the maximal profit. Our novel LP formulation is provably stronger than the node resource load for all randomized rounding algorithms. old formulation, as it only allows for (fractional) solutions, d) Heuristical Rounding RRHeuristic: The results of the which can be decomposed into valid mappings. Thus, it will heuristical rounding, which does not exceed resource capac- always yield (equal or) better bounds on the attainable profit. ities, are presented in Figure 10 (heatmaps) and Figure 12 Figure 12 presents the experimental comparison of both (ECDFs). Again, 1,000 rounding iterations were considered. formulations. In particular, for each of the 1,5000 instances While for low edge resource factors, i.e. scarce edge resources, we compare the objective of our novel Linear Programming the solutions achieve around 65% of the profit of the MIP Formulation 2 to two bounds computed during the solution baseline, for larger edge resource factors, the relative perfor- process of the baseline MIPMCF: the initial bound, i.e. the mance exceeds 80%. Both resource factors have a significant objective of the Linear Programming Formulation 1, and the impact and the heuristical rounding procedure works best final (best) bound computed during the solution process of the if neither of the resources is scarce, achieving on average (Mixed-)Integer Program. around 97.5% of the baseline’s profit for NRF = 0.2 and As can be seen, the initial LP bounds of the classic ERF = 4.0. Also, and in contrast to the performance of the formulation at times exceeds our formulation’s objective by baseline MIPMCF, the performance improves when increasing more than 300%, i.e. the classic formulation ‘overestimates’ the number of requests. This can be explained as follows. the maximal attainable profit by at least a factor of 3. For roughly 60% of the instances, the novel LP improves the [2] S. Mehraghdam, M. Keller, and H. Karl, “Specifying and placing chains bounds by a factor of 1.5. Clearly, the more requests are of virtual network functions,” in Proc. 3rd IEEE CloudNet, October 2014. considered, the less accurate the classic MCF formulation is. [3] H. Ballani, P. Costa, T. Karagiannis, and A. Rowstron, “Towards Considering the final bound computed during the execution of predictable datacenter networks,” in ACM SIGCOMM Computer Com- the (Mixed-)Integer Program, we see that these bounds always munication Review, vol. 41, no. 4. ACM, 2011, pp. 242–253. [4] J. Napper, W. Haeffner, M. Stiemerling, D. R. Lopez, and improve upon upon the novel LP’s bound. However, for 80% J. Uttaro, “Service Function Chaining Use Cases in Mobile of the instances, our LP’s bound is only improved by roughly Networks,” Internet-Draft, Apr. 2016. [Online]. Available: https: 15% with the maximal improvement being less than 33%. //tools.ietf.org/html/draft-ietf-sfc-use-case-mobility-06 [5] A. Fischer, J. Botero, M. Till Beck, H. de Meer, and X. Hesselbach, Concluding, we note that our novel LP formulation is much “Virtual network embedding: A survey,” Comm. Surveys Tutorials, IEEE, stronger than the classic formulation: it consistently yields vol. 15, no. 4, 2013. significantly better bounds in practice compared to the classic [6] M. Rost and S. Schmid, “Virtual Network Embedding Approximations: Leveraging Randomized Rounding,” in Proc. IFIP Networking, 2018. LP formulation and comes close to the bounds obtained by [7] ——, “Charting the Complexity Landscape of Virtual Network Embed- solving the (Mixed-)Integer Program for upto 3 hours. dings,” in Proc. IFIP Networking, 2018. [8] N. Chowdhury, M. Rahman, and R. Boutaba, “Virtual network em- VI.CONCLUSION bedding with coordinated node and link mapping,” in Proc. IEEE INFOCOM, 2009. This paper has initiated the study of approximation algo- [9] R. Hartert et al., “A declarative and expressive approach to control rithms for the Virtual Network Embedding Problem supporting forwarding paths in carrier-grade networks,” in SIGCOMM, 2015. [10] A. Jarray and A. Karmouch, “Decomposition approaches for virtual arbitrary substrate graphs and supporting cyclic request graphs, network embedding with one-shot node and link mapping,” IEEE/ACM specifically cactus request graphs. To obtain the approximation Transactions on Networking, vol. 23, no. 3, pp. 1012–1025, 2015. result, we have derived a novel strong LP formulation. Our [11] M. Rost, S. Schmid, and A. Feldmann, “It’s About Time: On Optimal Virtual Network Embeddings under Temporal Flexibilities,” in Proc. computational evaluation shows the practical significance of IEEE IPDPS, 2014, pp. 17–26. our work: obtained solutions achieve (on average) around 74% [12] M. Rost, C. Fuerst, and S. Schmid, “Beyond the stars: Revisiting virtual of the baseline’s profit while not augmenting capacities. cluster embeddings,” in Proc. ACM SIGCOMM Computer Communica- tion Review (CCR), 2015. We note that the developed approximation framework – [13] G. Even, M. Medina, and B. Patt-Shamir, “Online path computation including the proposed rounding heuristics – is independent of and function placement in sdns,” in Proc. International Symposium on how LP solutions are computed and decomposed. In particular, Stabilization, Safety, and Security of Distributed Systems (SSS), 2016. [14] G. Even, M. Rost, and S. Schmid, “An approximation algorithm for path while the LP formulation presented in this paper is only computation and function placement in sdns,” in Proc. SIROCCO, 2016. applicable for cactus request graphs, our formulation can be [15] T. Lukovszki and S. Schmid, “Online admission control and embedding generalized to arbitrary request graphs [18]. of service chains,” in Proc. 22nd SIROCCO, 2015. [16] N. Bansal, K.-W. Lee, V. Nagarajan, and M. Zafer, “Minimum conges- ACKNOWLEDGEMENTS tion mapping in a cloud,” in Proc. ACM PODC, 2011. [17] M. Rost and S. Schmid, “Service chain and virtual network em- This work was partially supported by Aalborg University’s beddings: Approximations using randomized rounding,” Tech. Rep. PreLytics project as well as by the German BMBF Software arXiv:1604.02180 [cs.NI], April 2016. [18] ——, “(FPT-)Approximation Algorithms for the Virtual Network Em- Campus grant 01IS1205. bedding Problem,” Tech. Rep. arXiv:1803.04452 [cs.NI], March 2018. We thank Elias Dohne,¨ Alexander Elvers, and Tom Koch [19] P. Raghavan and C. D. Thompson, “Provably good routing in graphs: for their significant contribution to our implementation [21]. Regular arrays,” in Proc. 17th ACM STOC, 1985, pp. 79–87. [20] D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. REFERENCES [21]E.D ohne,¨ A. Elvers, T. Koch, and M. Rost, “Source code for [1] J. C. Mogul and L. Popa, “What we talk about when we talk about cloud the evaluation presented in this work,” https://github.com/vnep-approx/ network performance,” ACM SIGCOMM CCR, vol. 42, no. 5, 2012. evaluation-ifip-networking-2018.