Efficient Techniques and Tools for Intra-Domain Traffic ... - CiteSeerX

Efficient Techniques and Tools for Intra-Domain Traffic Engineering Bernard Fortz Department of Computer Science, Universit´e Libre de Bruxelles, Brussels, Belgium and CORE, Universit´e catholique de Louvain, Louvain-la-Neuve, Belgium, [email protected]

¨ Hakan Umit IAG-Louvain School of Management and CORE, Universit´e catholique de Louvain, Louvain-la-Neuve, Belgium, [email protected]

In an Autonomous System (AS), links that connect routers have certain metric (administrative weight or cost of transferring an IP packet) that can be modified by network administrators, usually for the purpose of minimizing traffic congestion or minimizing the cost of overall routing. As traffic is routed along the shortest paths in the network, finding an optimum combination of these integer weights is proved to be NP-Hard (Fortz and Thorup 2000) due to hard protocol constraints, e.g. a flow is evenly distributed along the shortest paths between its origin and destination nodes, also referred to as equal cost multi-path (ECMP) rule. We study the link weight optimization problem in intra-domain networks. We present an open source Internet Traffic Engineering toolbox (TOTEM) and its link weight optimizer (IGP-WO) that we have developed and implemented. Furthermore, we present two heuristic approaches to generate fast and efficient link metric for intra-domain routing. We also present some promising experimental results that were done using various sizes of networks and traffic matrices. Key words: IGP; OSPF; Internet Traffic Engineering; shortest path routing; column generation; open source

1.

Introduction

In today’s internet service providing domain, supplying an appropriate level of service has become a big challenge due to growing customer needs and Quality of Service (QoS) terms in Service Level Agreements (SLAs). The usual method to provide a good level of service is to overprovision the network compared to the real needs. However, overprovisioning does not necessarily ensure the required QoS. Application of Traffic Engineering (TE) techniques 1

proves to be a popular alternative. Nevertheless the problems that are encountered in this field are combinatorial and of large size, which implies to make use of heuristic and suboptimal solutions. Interior gateway protocols (IGP) such as Open Shortest Path First (OSPF), Intermediate System-Intermediate System (IS-IS) and Routing Information Protocol (RIP) are among the most popular routing protocols used by Internet Service Providers. Traffic routing is based on shortest paths that are calculated and regularly maintained by the routers using link metric, i.e. a positive integer value between [1, 216 ). These link metric or weights can be modified by network administrators for administrative purposes (Moy 1998). Intra-domain routing operates under various protocol rules, the most important being the “traffic splitting” rule. According to the OSPF standards in RFC 2328, any traffic that arrive a router is split equally over all the outgoing links of that router that belongs to a shortest path to the destination (Moy 1998). Due to this protocol rule, also referred to as equal cost multi-path (ECMP), optimizing IGP weights is shown to be NP-Hard (Fortz and Thorup 2000). In Figure 1, a small shortest path network is depicted, where 2 units of data is routed according to the ECMP rule.

2

1/2 88 ◦ NNN NN ppp p p 1/2NN 1p NNN ppp p p p && // • N NNN p88 ◦ p p NN ppp 1NNN pp1 NNN p p && pp

//

◦ NNN

NN

1/2NN

3/2

NNN //&& •

2

//

◦

Figure 1: Equal cost multi-path in intra-domain routing To the best of our knowledge first studies on OSPF link weight optimization were done by Fortz and Thorup (Fortz and Thorup 2000), where the authors develop a tabu search heuristic in order to optimize the weight setting of a proposed AT&T WorldNet backbone. They show near optimal results where their heuristic can support up to 110% increase in demand compared to a standard heuristic, i.e. unit weights inveresely proportional to the link capacities, as suggested by Cisco (1997). Further studies on the IGP weight setting problem continue drawing an attention on the optimization by using various heuristic approaches. A recent example is a Hybrid Genetic 2

Algorithm (HGA) approach by Buriol et al. (2005) that is based on a Genetic Algorithm (GA) heuristic and a local search procedure proposed by Ericsson et al. (2002). Experiments show that HGA performs better and faster than GA, where the results of HGA are equally competitive with that of Fortz and Thorup (2000). More different approaches were studied by Pi´oro et al. (2002) and Wang et al. (2001), where the authors use the idea of optimal dual link weights resulting from linear relaxation of a bifurcated flow allocation problem in the context of single shortest-path routing. There are also some studies where additional objectives are taken into account, e.g. optimizing weights with few possible changes and also robust optimization approaches (Fortz and Thorup 2002). In this study, the authors introduce multiple demand matrices in order to obtain one generic weight setting that suits changing network conditions. In networks, it is sometimes required that the paths between demand end nodes are unique. In the so-called Inverse Shortest Paths (ISP) problem, the lengths of the links must be chosen in such a way that all end-to-end traffic demands must be connected by unique shortest paths. Recently, Bley has shown that minimizing the longest arc or the longest path length is APX-Hard to approximate (Bley 2005). Previously Ben-Ameur and Gourdin (2003) formulated this problem as a linear program and developed heuristics to compute small integer weights with guaranteed worst-case situations. This paper is structured as follows. Section 2 gives the definiton of the IGP weight optimization problem and presents the assumed piecewise linear objective function. Section 3 presents the architecture and features of the open source Traffic Engineering tool (TOTEM). We also present the TE algorithms that are embedded within the toolbox as well as the weight optimizer, IGP-WO, which we have developed and implemented. In Section 4, we propose two heuristic methods to generate IGP weights using a column generation method. In section 5, we present various numerical experiments along with comparisons of different methods. We conclude the study with section 6.

2.

Problem Definition

Given an internetwork (routers, hosts and links that connect them) and a traffic matrix (positive entries are defined to be a “demand” between any pair of nodes), the intra-domain routing metric optimization problem is to find the optimum set of link weights for IGP routing such that the cost of the routing is minimized. The objective in IGP routing is 3

usually to avoid link congestion, hence over-utilized links must be avoided. Let G = (N, A) be a directed graph, where N denotes the set of nodes and A represents the set of capacitated arcs connecting these nodes. The capacity of an arc (i, j) ∈ A is denoted by cij . We have a traffic matrix F of size (|N | × |N |), where each positive entry of F (o, d) is called a demand between origin o ∈ N and destination d ∈ N . We have a decision variable fijd which represents the flow on arc (i, j) destined for node d. P Let Φ = (i,j)∈A φ(cij , lij ) be the objective function with derivative below, where lij , the total flow on arc (i, j) ∈ A, is a decision  1     3    0 10 φ (cij , lij ) = 70     500    5000

variable representing the load on arc (i, j) ∈ A: for 0≤ for 1/3 ≤ for 2/3 ≤ for 9/10 ≤ for 1≤ for 11/10 ≤

lij /cij lij /cij lij /cij lij /cij lij /cij lij /cij

< 1/3, < 2/3, < 9/10, < 1, < 11/10, < ∞.

(1)

Note that φ(cij , 0) = 0. It is clear from (1) that it forms an increasing piece-wise linear convex function (see Figure 2). In practical terms this objective means that over-utilized links are heavily penalized, anyhow they are still allowed to transmit traffic.

Figure 2: Arc cost with respect to Φ(1, lij )

3.

A Traffic Engineering Toolbox

TOTEM (TOolbox for Traffic Engineering Methods) is an open source software for Internet Traffic Engineering (TE) purposes. It unites a set of algorithms and tools that allow netwok 4

administrators to optimize their networks. These methods cover intra-domain, inter-domain TE, IP based and Multiprotocol Label Switching (MPLS)-based TE, which can be used for better routing of traffic for providing Quality of Service (QoS), load balancing, protection and restoration in case of failure, etc. The toolbox can both be used as an on-line tool as well as off-line optimization and simulation tool (Leduc et al. 2006).

3.1.

Technical Features and Architecture

TOTEM embodies a wide range of TE algorithms. The kernel of the toolbox is the repository of TE methods grouped into several categories (see Figure 3): • Internet Protocol (IP): algorithms using only IP information, e.g. IGP weight optimization • Multiprotocol Label Switching (MPLS): algorithms using MPLS TE functionalities, e.g. Label Switched Path (LSP) primary or backup computation algorithms • Border Gateway Protocol (BGP): inter-domain algorithms, e.g. traffic redistribution • Generic: classical optimization and search algorithms useful for other parts of the toolbox, e.g. tabu search framework Besides this kernel, the topology manager contains all the topological data (i.e. node, link, IGP, BGP and MPLS information). This module is the reference access point to the topology representation in the toolbox, which is realized by the XML language (see Figure 4). The configuration manager configures the global toolbox parameters and the different algortihms. Finally the web service interface module provides the standard interface for interoperability with existing external tools. We have developed the toolbox in Java since it allows rapid and structured development. Moreover, the Java Native Interface (JNI) library allows us to integrate C and C++ algorithms in the toolbox. The toolbox has been designed to facilitate the integration of new algorithms by providing different generic services. It provides topology information (nodes, links, LSPs, ...) to the algorithm to be integrated. It also provides a scenario execution service. This parses an XML file describing scenario and then calls the appropriate algorithm to execute the scenario. As of version 2.0, the toolbox has been improved with the addition of Graphical User 5

Figure 3: TOTEM architecture

Figure 4: Example of the XML DOMAIN element

6

Interface (GUI) that uses the Java Universal Network/Graph Framework (JUNG) library (see Figure 5).

3.2.

TE Algorithms of the Toolbox

The TE methods that exist in TOTEM can be classified into three categories: (1) intradomain IP-based, (2) inter-domain IP-based and (3) MPLS-based. The list of the algorithms and tools in the toolbox is as follows (Balon et al. 2007): 3.2.1.

Shortest Path First Algorithm

The toolbox contains a flexible implementation of the SPF (Shortest Path First) algorithm and its constrained extension CSPF (Constrained Shortest Path First). The implementation is very efficient and uses a priority queue to store the list of temporary visited nodes. For computing the path of a LSP (Label Switched Path) with a given reservation, the CSPF skips links that do not meet the bandwidth requirement. 3.2.2.

DAMOTE

DAMOTE (Decentralized Agent for MPLS Online Traffic Engineering) is a generic tool with the following functionalities: • QoS-based routing Diffserv LSPs under constraints, i.e. compute primary paths at ingress nodes in a way similar to the classical CSPF. • Local and global back-up LSP routing for fast restoration 3.2.3.

SAMCRA

This algorithm can be used to compute LSPs between two nodes in the network. SAMCRA is an exact multi-constrained shortest path algorithm that was originally proposed by Van Mieghem et al. (2001) and later extended by Van Mieghem and Kuipers (2004). The implementation of SAMCRA included in the toolbox is the one described in the latter. 3.2.4.

optDivideTM

This algorithm consists of dividing the traffic matrix into N sub-matrices, called strata, and route each of these independently. The method can also be used to compute a very precise

7

Figure 5: Graphical User Interface of TOTEM

8

approximation of the optimal value of a given objective function to compare TE algorithms (Balon and Leduc 2006). 3.2.5.

C-PGP

C-BGP is a BGP simulator. It aims at the interdomain routes selected by BGP routers in a domain. The route computation is an accurate model of the BGP decision process as well as several sources of input data. The model of the decision process takes into account every decision rule present in genuine BGP decision process as well as the iBGP hierarchy (route-reflectors) (Quoitin 2007). 3.2.6.

SAMTE

SAMTE (Scalable Approach for MPLS Traffic Engineering) is a hybrid IP/MPLS optimization method. The idea of SAMTE is to combine both the simplicity and robustness of IGP routing and the flexibility of MPLS. This approach lies between the pure IP metric based optimization and the full mesh of LSPs. SAMTE uses the simulated annealing meta-heuristic to find a small number of LSPs to establish in the network. The combination of the set of LSPs computed by SAMTE and the IGP routing for the remaining flows optimize a given operational objective. 3.2.7.

IGP-WO

IGP-WO is the intra-domain routing metric optimizer of TOTEM. The basic model in the weight optimization problem assumes a given topology and a traffic matrix. The objective is to maintain the utilization of links within given link capacities. The tool implements the heuristic algorithm introduced by Fortz and Thorup (2000). The piecewise linear cost function presented in section 2 is used as the objective function. The search procedure includes a heuristic algorithm based on tabu search (Glover and Laguna 1997). A solution is represented by an integer weight vector, (wij ). Two functions are defined to build the neighborhood of a solution: 1. Single weight change: The weight of a single link is changed at each iteration 2. Evenly balancing flows: Given a destination node d, a node u is selected randomly among the ones that are on any shortest path towards d. The weights of the arcs outgoing from u is adjusted in such a way that the traffic from u to d is split as evenly 9

as possible among multiple arcs. The weight change is retricted to the arcs that have less load than the treshold. The changes leading to infeasible weight values are also avoided. The tabu search algorithm is structured with the following specifications. Tabu lists are used to avoid cycling during the whole run. Special hash functions are used to facilitate the tabu aspect of the heuristic, as well as to improve the running time. Diversification is carried out when a working solution is not improved for 300 iterations. During the diversification, each link weight is changed with probability rate 10% by adding a randomly chosen integer between [−2, +2]. If the resulting weight is infeasible (less than 1 or larger than 216 − 1), it is forced to the corresponding bound value. At each iteration, a proportion of the neighborhood is evaluated due to the large size of the problems. The initial rate by which the neighborhood is sampled is determined by the users. During the algorithm run, the value of the sampling rate is updated. If the current solution is improved, the sampling rate is divided by three, if not it is multiplied by two. The upper and lower bounds of the sampling rate are determined by the users, too. Cost evaluation is a bottleneck throughout the local search as the algorithm necessitates calculation of shortest path graphs using Dijkstra’s algorithm (1959). Therefore a dynamic shortest path algorithm (Ramalingam and Reps 1996) is embedded in the cost evaluation procedure. This addition saves up to 70% of the total CPU time. As observed in Fortz and Thorup’s work (2000), OSPF performs well with optimized weights in realistic network topologies. The results have shown that maximum link utilization rate in OSPF networks with optimized weights is generally close to the one in the ideal case, where the traffic is split freely.

4.

Heuristics for IP Metric

The multicommodity network flow problem (MCNFP) is assumed to be a good lower bound for the IGP weight setting problem (Fortz and Thorup 2000) and we opt for the same approach in this study.

10

4.1.

A Lower Bound for IGP Routing

Given the cost function Φ, the linear programming formulation of the MCNFP is as follows: (GRPa ) X min Φ = φij (i,j)∈A

subject to X

fijd

−

i∈N

X

fjid

=



−

P

o∈N F (o, d) if i = d F (i, d) if i 6= d

i, d ∈ N

(2)

(i, j) ∈ A

(3)

φij ≥ αz lij − βz cij

(i, j) ∈ A, z ∈ Z

(4)

fijd ≥ 0

(i, j) ∈ A, d ∈ N

(5)

i∈N

lij =

X

fijd

d∈N

Within the above egress-centric flow formulation, constraints (2) are the flow conservation constraints. Equations (3) define the load on each arc and inequalities (4) define the cost on each arc, where Z is the set of break points of the piece-wise linear function and αz and βz are the coefficients of the corresponding segment. As an alternative to defining the flows on arcs, we now define flows on directed paths for each commodity F (o, d) : o, d ∈ N . Let P (o, d) be the set of directed paths between origin o ∈ N and destination d ∈ N . Then fp represents the flow on path p ∈ P (o, d). For the path-based formulation of the MCNFP we use the same piece-wise linear objective function: (GRPp ) X

min Φ =

φij

(i,j)∈A

subject to (i, j) ∈ A, z ∈ Z

z ) (νij

(6)

(i, j) ∈ A

(ωij )

(7)

fp = F (o, d)

o, d ∈ N

(σod )

(8)

fp ≥ 0

o, d ∈ N, p ∈ P (o, d)

φij ≥ αz lij − βz cij X lij = fp p∈P (o,d):(i,j)∈A

X

p∈P (o,d)

11

(9)

Constraints (6) are basically the same with constraints (4) in the previous formulation. Following equations (7) define the load on each arc, whereas constraints (8) make sure that each demand is satisfied.

4.2.

A Column Generation Approach

Let us reconsider GRPp , the path-based formulation of the MCNFP. We will solve this model using column generation due to exponential number of paths. In this formulation we have three basic set of constrains. So, the dual linear program of this model has the dual variables that are shown in the formulation above. With path flow complementary slack conditions, the dual variable ωij corresponding to the flow constraints is the optimum arc price for (i, j) ∈ A and the dual variable σod corresponding to the demand constraints is the optimum shortest path cost for the commodity o, d ∈ N , for the MCNFP (Ahuja et al. 1993). Hence the dual variables are used to check optimality with the following pricing: min

p∈P (o,d)

X

ωij ≤ σod

(10)

(i,j)∈p

It is clear from the above equation that the pricing problem is indeed a shortest path problem. Let S be an arbitrary subset of P , then the column generation procedure is the following: P rocedure CG I Initialize S(o, d) ⊂ P (o, d). II Solve the LP obtaining ωij for (i, j) ∈ A and σod for (o, d) ∈ N × N . III If (minp∈P (o,d)

P

0

(i,j)∈P

0

ωij ≤ σod ), then add S (o, d) to GRPp , where S (o, d)

contains the shortest paths from o to d, and go to Step II. Else an optimal solution is obtained. Stop.

4.3.

Heuristics

We propose two ideas to be used as IP metric for intra-domain routing, which can be obtained as follows: 1. The dual vector of (7), denoted by Ωp , that is obtained as a byproduct at step II of Procedure CG. 12

2. The dual vector of (3), denoted by Ωa , of GRPa . In fact this idea stems from the fact that the MCNFP or the General Routing Problem (GRP) is assumed to be a good lower bound for IGP routing. Thus, we would like to see whether we could use it as a heuristic for IGP routing. This is a heuristic because in the General Routing Problem the flow is distributed freely disregarding the ECMP requirement of IGP routing. Further optimization strategies are also possible by combining existing tools with new ones: the above heuristic weights, i.e. Ω, can be used as a good starting point in TOTEM (IGP-WO) as it is normally initialized by random or unit weights.

5.

Numerical Experiments and Discussion

We have done numerous experiments using 2-level hierarchical graphs, which were also used by Buriol et al. (2005), Ericsson et al. (2002), Fortz and Thorup (2000) and Fortz and Thorup (2002). Hierarchical graphs are considered to be the most realistic representation of internetworks. The arcs are clustered into two groups according to their capacities, i.e. local access and long distance arcs. We have a complete traffic matrix to define demands, i.e. from each node to every other node there exists a positive demand, which is derived from multiplication of three random variables between (0, 1]. We use XPress-MP (optimizer version 14.27) to solve the LPs on a Pentium PC with 512 Mb of memory running at 1.40 Ghz. In order to generate columns, the Dijkstra Algorithm is used with an implementation in C. In Procedure CG, a run is initialized with |N |(|N | − 1) shortest paths that are calculated using unit weights. During this procedure, a single path is added for each commodity (o, d) ∈ N , provided that the optimality condition in (10) is satisfied. At each iteration, the dual vector Ωp is retrieved to be used as heuristic weights in IGP routing. We use XPress-MP to obtain Ωa by solving GRPa .

5.1.

Performance Evaluation

In order to evaluate the real performances of Ωp and Ωa , we calculate the actual IGP routing cost as follows (Fortz and Thorup 2000):

13

P rocedure IGP cost I Compute a shortest path (SP) graph for each node n ∈ N using Ω n o n n n II Compute the set arcs in SP graph of n ∈ N : A = (i, j) ∈ A : di − dj = ωij , where dn is the shortest path distance of a node to n

n III For each (i, j) ∈ An , compute the partial load lij =

1 |δi+ |

where δi+ is the outdegree of node i ∈ An .



F (i, n) +

P

(k,i)∈An

 n lki ,

Note that this computation is initialized from the farthest node to the root node. IV Aggregate the load on arc lij =

P

n n∈N lij

V Compute the total routing cost Φ =

P

(i,j)∈A

φ(cij , lij ).

We have two criteria to evaluate the performance of our heuristics: maximum utilization and normalized cost. Maximum utilization refers to the maximum loaded link from the resulting weight set. Normalized cost is a transformation of the cost function value into a more universal one by using a special scaling factor. This scaling factor is calculated by: 1. Computing the total cost, Φ1 , of routing the commodities over the network using unit weights, 2. Multiplying Φ1 by a cost factor, 32/3. This value comes from the piece-wise linear cost function: basically, if a commodity follows a shortest path, and if all arcs are exactly full, the cost factor is 32/3. For more details, the user is referred to the article by Fortz and Thorup (2002). The significance of the normalized cost is: if it is more than 1, the routing will result in congestion. We present the numerical results in Tables [1-4]. Maximum link utilization and normalized cost results are given for each approach and for a range of demands that increase gradually. The significance of the abbreviations are the following: • Opt: Optimum solution of the MCNFP • IGP-WO: Result obtained using the IGP weight optimizer of TOTEM v2.4

14

• CG: Best result obtained by using heuristic weights of the column generation approach: At each iteration of Procedure CG, the corresponding dual vector Ωp is evaluated in IGP routing and the best Ωp is chosen among these results • DV: Result obtained by using the heuristic weights of Ωa . Table 1: 2-Level Hierarchical Graph with 50 nodes and 148 arcs Maximum Utilization Normalized Cost # Opt IGP-WO DV CG Opt IGP-WO DV CG Sum of Demands 1 0.18 0.18 0.18 0.18 0.09 0.09 0.09 0.09 410.64 2 0.33 0.33 0.37 0.33 0.09 0.09 0.09 0.09 821.28 3 0.33 0.33 0.43 0.43 0.09 0.09 0.09 0.09 1231.92 4 0.59 0.66 0.47 0.47 0.09 0.09 0.10 0.10 1642.56 5 0.66 0.68 0.59 0.59 0.10 0.10 0.11 0.11 2053.20 6 0.66 0.70 0.73 0.73 0.10 0.11 0.12 0.12 2463.84 7 0.66 0.86 0.93 0.92 0.12 0.12 0.14 0.14 2874.49 8 0.90 0.90 0.97 0.93 0.13 0.14 0.16 0.16 3285.13 9 0.90 1.01 1.02 1.02 0.14 0.17 0.20 0.20 3695.77 10 1.00 1.01 1.14 1.14 0.17 0.20 0.88 0.84 4106.41

Table 2: 2-Level Hierarchical Graph with 50 nodes and 212 arcs Maximum Utilization Normalized Cost # Opt IGP-WO DV CG Opt IGP-WO DV CG Sum of Demands 1 0.19 0.19 0.19 0.19 0.09 0.09 0.09 0.09 280.22 2 0.33 0.33 0.27 0.27 0.09 0.09 0.09 0.09 560.44 3 0.33 0.33 0.38 0.38 0.09 0.09 0.09 0.09 840.67 4 0.39 0.66 0.45 0.45 0.09 0.09 0.10 0.10 1120.90 5 0.66 0.61 0.54 0.54 0.09 0.10 0.10 0.10 1401.12 6 0.66 0.66 0.65 0.65 0.10 0.10 0.11 0.11 1681.35 7 0.66 0.67 0.71 0.71 0.10 0.11 0.12 0.12 1961.57 8 0.66 0.87 0.89 0.89 0.11 0.12 0.13 0.13 2241.80 9 0.90 0.89 0.97 0.97 0.12 0.14 0.16 0.16 2522.02 10 0.90 0.96 1.06 1.06 0.13 0.16 0.24 0.24 2802.24 Considering the overall results in Table 1 and Table 2, we observe that both of our heuristic approaches yield in highly good results in general, with superior positions that are printed in bold. In almost all runs except #10 in Table 1, the maximum utilization and normalized cost results of DV and CG are equally competitive with that of IGP-WO and Opt. The advantage of our heuristic is that we are able to generate weights quite fast. Using GRPa , the optimum solution (hence the heuristic weights) is attained in just a few seconds 15

Table 3: 2-Level Hierarchical Graph with 100 nodes and 280 arcs Maximum Utilization Normalized Cost # Opt IGP-WO DV CG Opt IGP-WO DV CG Sum of Demands 1 0.17 0.17 0.17 0.17 0.09 0.09 0.09 0.09 383.76 2 0.33 0.33 0.35 0.25 0.09 0.09 0.09 0.09 767.53 3 0.35 0.35 0.38 0.35 0.09 0.09 0.09 0.09 1151.30 4 0.46 0.46 0.46 0.46 0.09 0.09 0.09 0.09 1535.07 5 0.60 0.60 0.64 0.62 0.09 0.09 0.10 0.10 1918.84 6 0.67 0.67 0.70 0.71 0.10 0.10 0.11 0.11 2302.61 7 0.78 0.78 0.82 0.81 0.11 0.11 0.12 0.11 2686.37 8 0.89 0.89 0.96 0.96 0.12 0.12 0.13 0.13 3070.14 9 1.00 1.00 1.00 1.00 0.14 0.14 0.15 0.15 3543.91 10 1.11 1.11 1.34 1.11 0.23 0.32 1.34 0.31 3837.68

# 1 2 3 4 5 6 7 8 9 10

Table 4: 2-Level Hierarchical Graph with 100 nodes and 360 arcs Maximum Utilization Normalized Cost Opt IGP-WO DV CG Opt IGP-WO DV CG Sum of Demands 0.22 0.21 0.22 0.22 0.09 0.09 0.09 0.09 1033.88 0.33 0.33 0.45 0.35 0.09 0.09 0.09 0.09 2067.76 0.33 0.41 0.39 0.39 0.09 0.09 0.09 0.09 3101.64 0.48 0.55 0.63 0.60 0.10 0.10 0.10 0.10 4135.51 0.67 0.67 0.79 0.79 0.10 0.10 0.10 0.10 5169.39 0.67 0.68 0.71 0.71 0.10 0.11 0.10 0.10 6203.27 0.76 0.76 0.79 0.86 0.11 0.11 0.11 0.11 7237.15 0.90 0.90 1.02 0.97 0.11 0.13 0.14 0.12 8271.03 0.93 0.96 1.38 1.14 0.12 0.16 0.81 0.28 9304.91 1.04 1.04 1.30 1.20 0.17 0.22 0.58 0.20 10338.80

16

for all the instances of the 50-node networks. In order to obtain the same quality of result for the same data set using IGP-WO, a running time around 45 minutes is needed depending on the tabu search parameters. The CPU times of the column generation approach are also quite reasonable: they vary from 20 to 113 seconds for both instances of the 50-node graphs. Our results show that the column generation procedure stops, i.e. optimal solution is attained, after maximum 7 iterations for the 50-node and 9 iterations for the 100-node networks. In these iterations column generation procedure itself takes approximately 3% of the total running time. Table 3 and Table 4 present the results for the 100-node networks. These results are parallel to the results shown in Table 1 and 2, i.e. both of our heuristic approach results are either equaly competitive or superior with that of IGP-WO. It takes us maximum 106 seconds to compute the optimum solution of GRPa for these large networks. The column generation approach requires more CPU time for these instances: a single run takes from 8 to 47 minutes for the examples #1 to #10. We now would like to go further and combine our heuristic approaches with IGP-WO. From the observations in Tables [1-4], we see that there is still some margin to converge the normalized cost results of Opt, e.g. for the instances #7 - #10 in Table 1 and the instances #8 - #10 in Table 4. To capture this gap we would like to perform further experiments with IGP-WO using Ωa and Ωp . By default IGP-WO initializes tabu search with random weights. Instead of this approach, we start the neighborhood search with Ω. To maintain consistency, we have made the experiments in TOTEM v2.4 with the following fixed parameters for all the instances: • Iteration number: 100 • Neighborhood sampling rate: 20% • Weight range: [1, 20] The columns that are shown in Tables [5-8] represent the following. Bold entries show superiority of our results. • IGP-WO: Tabu search results initialized with random weights • IGP-WO + CG: Tabu search results initialized with Ωp • IGP-WO + DV: Tabu search results initialized with Ωa 17

Table 5: Tabu Search Results for 2-Level Hierarchical Graph with 50 nodes and 148 arcs Maximum Utilization Normalized Cost IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO # + CG + DV + CG + DV 7 0.76 0.70 0.87 0.12 0.12 0.12 8 0.89 0.88 0.87 0.13 0.13 0.14 9 0.89 0.89 0.96 0.15 0.15 0.17 10 0.99 0.99 1.02 0.19 0.19 0.22

Table 6: Tabu Search Results for 2-Level Hierarchical Graph with 50 nodes and 212 arcs Maximum Utilization Normalized Cost IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO # + CG + DV + CG + DV 7 0.71 0.71 0.89 0.11 0.11 0.12 8 0.78 0.78 0.89 0.12 0.12 0.13 9 0.89 0.89 1.00 0.14 0.14 0.19 10 0.91 0.91 1.05 0.16 0.16 0.32

Table 7: Tabu Search Results for 2-Level Hierarchical Graph with 100 nodes and 280 arcs Maximum Utilization Normalized Cost IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO # + CG + DV + CG + DV 7 0.78 0.78 0.78 0.11 0.11 0.11 8 0.89 0.89 0.89 0.12 0.12 0.13 9 1.00 1.00 1.00 0.14 0.14 0.15 10 1.11 1.11 1.11 0.30 0.30 0.33

Table 8: Tabu Search Results for 2-Level Hierarchical Graph with 100 nodes and 360 arcs Maximum Utilization Normalized Cost IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO IGP-WO # + CG + DV + CG + DV 8 0.85 0.88 1.00 0.12 0.12 0.17 9 0.99 0.97 1.10 0.14 0.15 0.27 10 1.06 1.03 1.09 0.20 0.21 0.39

18

As observed from Tables [5-8], when IGP-WO is initialized with Ωa or Ωp , normalized cost and maximum utilization values converge to the lower bound better and faster than IGP-WO alone, even for a relatively small number of iteration. Note that the results of IGP-WO in Tables [1-4] are obtained with 5000 iterations, whereas the experiments shown in tables [5-8] are the results of only 100 iterations. This is a substantial gain in terms of solutions attained as well as the running time needed. In addition to these improvements we are still interested to know how far we can go with the hybrid approach. We know that in order to obtain “good” solutions with this stochastic search, an experiment must be made with at least 3000 iterations (Fortz and Thorup 2000). Thus, we perform one more experiment with 3000 iterations for the example that has the highest gap, i.e. #10 of the 100-node network 360-arc network presented in Table 8. We initialize this example with Ωp . As a consequence of this experience, we still get highly improved results: 0.19 for normalized cost and 1.03 for maximum link utilization.

6.

Conclusion

In this study, we bring some promising solutions to the IGP weight optimization problem. We introduce an open source Traffic Engineering Toolbox, i.e. TOTEM, that embodies a set of TE tools including an intra-domain IP metric optimizer, called IGP-WO. IGP-WO can be used as an offline link metric optimization tool by intra-domain network administrators. Additionally, we present two heuristic methods for the same problem. We use the Minimum Cost Network Flow Problem as a lower bound and a base for our heuristics. We formulate the MCNFP as arc-based and path-based models and use the corresponding dual variables as heuristic weights for IGP routing. We demonstrate that both 50-node and 100-node 2-level hierarchical networks can support up to ten times more traffic when heuristic weights are used. We perform further experiments by combining our heuristics with the IGP weight optimizer of TOTEM, i.e. IGP-WO. In this hybrid approach we intialize IGP-WO with our heuristic weights. We show that we are able to get even better results with a substantial gain in CPU time, i.e. 50 times less number of iterations. Improvement of these results and obtaining a robust solution that suits a wider span of topologies are considered as future work of this study. Further methods can be sought on how to minimize the number of weight changes of the initial set. Moreover, router and link 19

failure scenarios can be incorporated in order to represent more realistic situations in internetworks. Demand uncertainty is a further stochastic extension that could be considered for application within TOTEM (IGP-WO). This topic has been recently studied by Altın et al. (2007), where the authors use two uncertainty models, i.e. Hose model and Bertsimas-Sim model, to define traffic uncertainty. We also plan to improve the path-based formulation by adding cuts valid for the equal-cost multipath constraint of IGP routing. This approach would hopefully generate better lower bounds.

Acknowledgments This work has been partially supported by the Walloon Region (DGTRE) in the framework of the TOTEM project and the Communaut´e fran¸caise de Belgique - Actions de Recherche Concert´ees (ARC).

References Ahuja, R.K., T.L. Magnanti, J.B. Orlin, 1993. Network Flows: Theory, Algorithms and Applications, Prentice-Hall. Englewood Cliffs, NJ. Altın, A., P. Belotti, M. C ¸ . Pınar, 2007. OSPF Routing with Optimal Oblivious Performance Ratio under Polyhedral Demand Uncertainty. Proceedings of International Network Optimization Conference (INOC 2007), Spa, Belgium. Balon S., S. Cerav-Erbas, O. Delcourt, J. Lepropre, G. Monfort, B. Quoitin, F. Skiv´ee, H. ¨ Umit, 2007. TOTEM 2.4 - User Guide. http://totem.run.montefiore.ulg.ac.be/documentation.html/. Balon, S., G. Leduc, 2006. Dividing the Traffic Matrix to Approach Optimal Traffic Engineering. Proceedings of 14th IEEE International Conference on Networks September 2006, IEEE Xplore, Singapore. Ben-Ameur, W., E. Gourdin, 2003. Internet Routing and Related Topology Issues. SIAM Journal on Discrete Mathematics 17 18-49. Bley, A., 2005. Finding Small Administrative Lengths for Shortest Path Routing. Proceedings of International Network Optimization Conference (INOC 2005) 121-128, Lisbon, Portugal.

20

Buriol, L.S., M.G.C. Resende, C.C. Ribeiro, M. Thorup, 2005. A Hybrid Genetic Algorithm for the Weight Setting Problem in OSPF/IS-IS Routing. Networks 46(1) 36-56. Cisco, 1997. Configuring OSPF, http://www.cisco.com. Dijkstra, E.W., 1959. A Note on Two Problems in Connexion with Graphs. Numerische Mathematik 1 269-271. Ericsson, M., M.G.C. Resende, P. M. Pardalos, 2002. A Genetic Algorithm for the Weight Setting Problem in OSPF Routing. J. Combinatorial Optimization 6 299-333. Fortz, B., M. Thorup, 2000. Internet Traffic Engineering by Optimizing OSPF Weights. Proc. 19th IEEE Conf. on Computer Communications (INFOCOM) 519-528. Fortz, B., M. Thorup, 2002. Optimizing OSPF/IS-IS Weights in a Changing World. IEEE Journal on Selected Areas in Communications 20(4) 756-767. Glover, F., M. Laguna, 1997. Tabu Search. Kluwer Academic Publisher. Java Native Interface (JNI), 1997. http://java.sun.com/j2se/1.4.2/docs/guide/jni/. Java Universal Network/Graph Framework (JUNG). http://jung.sourceforge.net/. Leduc, G., H. Abrahamsson, S. Balon, S. Bessler, M. D’Arienzo, O. Delcourt, J. DomingoPascual, S. Cerav-Erbas, I. Gojmerac, X. Masip, A. Pescap`e, B. Quoitin, S.P. Romano, ¨ E. Salvadori, F. Skiv´ee, H.T. Tran, S. Uhlig, H. Umit, 2006. An Open Source Traffic Engineering Toolbox. Computer Communications 29(5) 593-610. Moy, J., 1998. OSPF Version 2 RFC 2328. Internet RFC/STD/FYI/BCP Archives. http://www.ietf.org/rfc/rfc2328.txt. Pi´oro, M., A. Szentesi, J. Harmatos, A. J¨ uttner, P. Gajowniczek, S. Kozdrowski, 2002. On Open Shortest Path First Related Network Optimization Problems. Performance Evaluation 48 201-223. Quoitin, B., 2007. C-BGP, An Efficient BGP Simulator, http://cbgp.info.ucl.ac.be/. Ramalingam, G., T. Reps, 1996. An Incremental Algorithm for a Generalization of the Shortest-Path Problem. Journal of Algorithms 21(2) 267-305. TOTEM Toolbox v2.4, 2007. http://totem.run.montefiore.ulg.ac.be/. Van Mieghem, P., H. De Neve, F.A. Kuipers, 2001. Hop-by-hop Quality of Service Routing. Computer Networks 37(3-4) 407-43. Van Mieghem, P., F.A. Kuipers, 2004. Concepts of Exact Quality of Service Algorithms. 21

IEEE/ACM Transaction on Networking. XML Schema, 2004. http://www.w3.org/XML/Schema.html. XPress-MP, 2007. http://www.dashoptimization.com. Wang, Y., Z. Wang, L. Zhang, 2001. Internet Traffic Engineering without Full Mesh Overlaying. Proc. of INFOCOM April 2001.

22