I Introduction
Segment routing (SR) combines the advantages of source routing supported by centralized softwaredefined networking (SDN) paradigm and hopbyhop routing applied in distributed IP network infrastructure [1, 2, 12]. The key idea behind SR is to break a routing path into multiple segments using a sequence of SRnode (a.k.a. intermediate node) in order to control the routing path more flexibly and hence improve network utilization.
In parallel with the SR, middleboxes have become ubiquitous in SDN as well as data center networks (DCN). Service function chaining (SFC) is a set of operations to steer traffic through an ordered list of physical or virtual middleboxes which provide network functions such as VPN, NAT, DPI, and firewall. From another perspective, SR can be viewed as the supporting technology of a large variety of novel network technologies at the network layer, such as SFC [4], network function virtualization (NFV) [11], 5G [13], SDWAN [2]. They share a common technical ground on the modeling and algorithm design from the multicommodity flow (MCF) theory, while having disparate orientations.
Existing literature claim that 2SR (SR using at most 2 segments) can achieve nearoptimal performance [8, 10, 22, 23]. Thus, the routing paths between the source and target nodes usually take on a short and wide shape, i.e., there are usually a large number of 2hop paths; here each hop means a shortest path routing. This is quite different from the case in SFC, where the routing paths are often long and narrow. Due to the computation inefficiency, it is nearly impossible to evaluate whether various types of networks will benefit from the SR with multiple segments using conventional approaches [22]. To this end, in this paper, we try to address the following challenges in SR networks.

How to establish a flexible and extensible model to optimize network throughput by leveraging SR with multiple segments?

How to design efficient algorithms to solve the model in both offline and online scenarios?
To tackle the above challenges, we aim to fully explore the potential of SR from an algorithmic perspective. Similar to the methodology in [6], we will not consider practical hardware (e.g. routers) or software (e.g. protocols) limits on the segment number. Specifically, the main contributions of this paper are the following:

We propose a flexible SR model as well as its formulation where segment number, SRnode number, intrasegment routing policy are all parameterized. The model leads to a highly extensible framework to design and evaluate algorithms that can be adapted to various network topologies and traffic matrices.

For the offline setting, we develop an fully polynomial time approximation scheme (FPTAS) which can finds a approximation solution for any specified in time that is a polynomial function of the problem size. The proposed FPTAS is the first algorithm that can compute arbitrarily accurate solution and only on this basis can we further evaluate whether using multiple segments is inevitable on various types of networks.

For the online setting, we develop an online primaldual algorithm that proves competitive and violates link capacities by a factor of , where is the node number.

We prove performance bounds for the proposed algorithms. We conduct simulations on realistic topologies to validate SR related parameters and algorithmic parameters in both offline and online settings.
The rest of this paper is organized as follows. We review related work in Section II. We introduce the system model and preliminaries in Section III. We formulate the offline and online network throughput maximization problems for SR and develop approximation and online algorithms in Sections IV and V, respectively. The mincost SRpath computation module is presented in Section VI. We present simulation results in Section VII. Finally, we discuss important extentions in Section VIII and conclude in Section IX. All proofs are presented in the appendix. Main notation is summarized in Table I.
Ii Related Work
Iia Segment Routing Using Multiple Segments
Bhatia et al. [8] for the first time formulate a generic SRTE problem to minimize maximum link utilization, where all intermediate nodes are used to construct optimal segment routing paths and they also propose a 2SR online algorithm.
To jointly optimize the efficiency of intermediate nodes selection and the subsequent flow assignment, Settawatcharawanit et al. [10] formulate a biobjective mixedinteger nonlinear program (BOMINLP) to investigate the tradeoff between link utilization and computation time. They conclude that the maximum link utilization performance of 2SR is indentical to SR. Thus, they focus on limiting the candidate paths lengths as well as reducing the computation overheads by a stretch bounding method.
Pereira et al. [3]
propose a single adjacency label path segment routing (SALPSR) model that forwards traffic flows using at the most three segments. They also propose an evolutionary computation approach to improve traffic distribution. As applications, they also extend the model to handle semioblivious traffic matrices and address link failures.
Jadin et al. [6] formulate the SRTE problem into an ILP, and propose a CG4SR approach that combines the column generation and dynamic program techniques. The approach can only obtain near optimal solutions with gap guarantees by realistic experiments. They also compute a stronger lowerbound than traditional MCF approach through experiments.
Li et al. [5] find that SR without support of adjacency segments cannot reach the optimum. To fully support adjacency segments in SR, they propose an extended LP formulation for 2SR and an MILP formulation for SR. Due to the computation complexity, the MILP is further simplified to prevent excessive flow splitting or using long paths.
SR combines the advantages of centralized intersegment source routing and distributed intrasegment hopbyhop routing. Unlike the above works, the proposed SR model in this paper provides the maximum freedom to deploy SRnodes and predict the performance. Our algorithm proves a approximation solution, that is, it can be arbitrarily close to the optimum. In our simulation, the proposed algorithms can even support rapid computation for the number of segments as large as .
IiB Service Function Chain
Starting from the classical MCF model, Cao et al. [15] consider the policyaware routing problem in both offline and online settings, where a traffic demand must traverse a predetermined ordered list of middleboxes. However, the only resource constraint is put on link capacities; the middleboxes do not consume any resources.
Further, Charikar et al. [16] propose a new kind of MCF problem, where the traffic flows consume bandwidth on the links as well as processing resources on the nodes. They also formulate the problem via an LP model and develop an efficient combinatorial algorithm to solve the model approximatedly with arbitrary accuracy.
Recently, more realistic SFC models for unicast [17] and multicast [18], which incorporate link bandwidth, residual energy in mobile devices and cloudlet computing capacity constraints in the context of NFVenabled MEC networks are proposed. Notice that these models assume that the network elements (APs or cloudlets) in the backbone are connected via wireline links.
Although SR and SFC belong to different areas of research, they are very close in terms of the MCFbased models and algorithms. In SR, all the available resources, including network links and SRnodes, should be jointly managed to optimize the TE objectives at the network layer [21, 22]. While in SFC, the middleboxes impose extra computation and storage constraints on a wider range of objectives from network layer to application layer. Therefore, the two areas can borrow ideas and merit from each other.
Notation  Description 

SR network , where is the node set and the link set.  
Auxiliary graph constructed for request .  
Node number and link number of .  
Capacity of link .  
, ,  Request , its source node and target node. 
Size of request .  
Set of all possible SRnode lists for request .  
Available SRnodes for request .  
Maximum number of segments for request .  
SRpath via SRnode list due to request .  
Mapping coefficient from to link , i.e. the amount of flow routed on link through SRnode list due to a unit request .  
Fraction of request routed through SRnode list .  
Flow amount of request routed through SRnode list .  
Dual variable associated with each link .  
Dual variable associated with each request .  
Multiplier that request size can be supported for .  
Tunable parameter of FPTAS.  
Tunable parameter of online algorithm.  
SR  Segment routing. 
LP  Linear program. 
MCF  Multicommodity flow. 
ECMP  Equalcost multipath. 
MF  Middlebox fabric. 
Iii System Model
We model an SR network with a directed graph , where represents the node set and the link (edge) set. The number of nodes and links are denoted by and , respectively. The network is not necessarily assumed symmetric, i.e., some links may not be bidirectional.
We introduce the following SR parameters in the SR framework, which can be illustrated by Fig. 3 in Section VI.
Given a request , assume the set of available SRnodes is and the (maximum allowable) segment number is , then .
By definition, SR may even degenerate to 1SR, i.e. the shortest path routing, or generalize to SR, i.e. the MCF routing, which can use all simple paths available to achieve the highest performance in theory while suffering from the largest cost.
Define as the set of all possible SRnode lists, i.e.:
The framework is flexible and extensible to support innovations in SR. For instance, under this framework, both node segment and adjacency segment in SR can be supported. In this paper, we restrict our attention to intersegment routing, assuming that the intrasegment routing multpaths are predetermined using some linkstate routing protocols.
Iiia SRFunction for IntraSegment Routing
Let represent the flow on link when unit flow is routed from to according to some linkstate routing policy. The routing policy may be the shortest path algorithm (ECMP permitted), DEFT [24], PEFT [25], and etc. Note that is uniquely determined by the IGP link weights, which have no relation with the link length system in the dual problem. The linkstate based routing policy based on the IGP link weights. Therefore, we treat as input parameters of the solution algorithms. The definition and computation of consider the hierarchical structure of the Internet. Unlike previous works, we need not to compute over the entire node set . This provides more operational flexibility of the SRnodes deployment.
Given an SRnode list for request and the notations and . Define the SRfunction as
Therefore, calculates the flow on link if a unit flow is routed from to through SRnode list . It is predetermined by the network topology, link weights and intrasegment routing policy including but not limited to the shortest path policy. Note that if the equalcost multipath (ECMP) routing is employed, then can be fractional and that if there is a link traversed more than once, can be larger than one. Thus, the path, referred to as SRpath, is a generic path rather than a simple path. In the following sections we will see that the SRpath constitutes the metastructure of the proposed algorithms. The examples in the next section illustrate how traffic will be split across the SRpath.
It is not hard to see that adjacency segments [6, 3] can also be supported in the SR mode as well as the proposed algorithms in the following sections. For instance, suppose link is an adjacency segment, we only need to make a simple assignment . For clarity and without loss of generality, we focus on the node segments in this paper.
IiiB Illustrative Examples
Example 1: The closetooptimal performance of the 2SR setting, when being applied to real networks, has been claimed in many literature, e.g. [8, 10, 22]. In [22], an unrealistic topology is constructed to validate this point. Here we give another counter example to illustrate the inefficiency of 2SR, see Fig. 1. The topology we used here, however, can be seen as a highly abstract multidomain network structure. Suppose all links have identical capacities 100 and weights 1. Under the 2SR setting, the maximum throughput from to is 100 even though all nodes in blue color are selected as candidate SRnodes. This is due to the fact that some paths, e.g., , cannot be utilized. In the 3SR setting, however, this path can be activated if and are chosen as SRnodes. Similarly, all paths from to become available in this setting, thereby achieving the maximum thoughput 300. This type of topology is quite common in the case of interdomain routing. The severe inefficiency shown in this example originates from the misalignment of link weights setting and the objective of throughput maximization. To tackle this issue, we should steer traffic to nonshortest paths using linkstate based routing protocols, just as the well known DEFT [24], PEFT [25], and the method in [3].
Another point we want to emphasize is that the flexibility and resultant complexity of intra and intersegment routing can be converted into each other. More exactly, the maximum throughput can also be achieved using only as the SRnode by appropriately setting the values of and . The theoretical analysis to this convertion is still an open challenge.
Example 2: In Fig. 2, we want to send a unit flow from to and there is only one SRnode . Suppose all the links have unlimited capacities and identical IGP weights 1. According to the shortest path policy, only the paths (also links) and are utilized. If the weights of links and are raised to 3, the ECMP policy could be activited. If their weights are further raised to 5, only the paths and can be utilized. In other words, in such case, no matter how intersegment routing is optimized, links and can never be used. To solve this problem, we only need to introduce the adjacency segments and , i.e. setting and .
we emphasize again that in this paper we leave the intrasegment routing as an input parameter in the SR framework and focus on the intersegment routing optimization.
Iv Offline Network Throughput Maximization
In the offline network throughput maximization problem, we know all the routing requests in advance. The objective is to simultaneously maximize the throughput of all the requests. We first present the offline formulation and thereby develop an approximation algorithm.
Iva Problem Formulation
Based on the maximum concurrent flow problem [26], the offline problem can be formulated as the following LP.
s.t.  (1)  
(2)  
Similar to a typical MCF formulation, constraints (1) and (2) imply the flow conservation and capacity limitation, respectively. By associating a length with each link and a weight with each request , we can write the dual to the above LP as:
s.t.  (3)  
(4)  
Using the definitions of , an alternate way to write the dual constraint is
In the worst case, the SRnode list can be as long as and each position of can be empty or occupied by any SRnode. Thus, the number of variables in can be at most . As far as we know, no tractable generalpurpose LP solver can be directly applied to so large a problem.
IvB Approximation Algorithm
We design an FPTAS to solve the problem [26, 9]. The FPTAS is a primaldual algorithm which includes an outer loop of a primaldual update and an inner loop of mincost SRpath computation.
The algorithm to solve the problem starts by assigning a precomputed length of to all links .
The algorithm proceeds in phases. In each phase, we route units of flow from node to , for each request . A phase ends when all requests are routed.
The flow of value of request is routed from to in multiple iterations as follows. In each iteration, a mincost SRpath from to that minimizes the lefthand side of constraint (3) under current link lengths is computed.
The path is computated in Section VI. The bottleneck of this path, i.e. the maximum amount of flow that can be sent along this path, is given by
The amount of flow sent along this path in a step, denoted by , is also bounded by the remaining amount of flow for , denoted by , i.e.:
After the flow of value is sent through the SRnode list , the flow value and the link length at each link along the path are updated as follows:
1) Update the flow values as
2) Update the link lengths as
The update happens after each iteration associated with routing a portion of flow . The algorithm terminates when the dual objective function value becomes less than one.
When the algorithm terminates, dual feasibility constraints will be satisfied. However, link capacity constraint (2) in the primal solution will be violated, since we were working with the original (not the residual) link capacities at each stage. To remedy this, we scale down the traffic at each link uniformly so that the link capacity constraints are satisfied.
Theorem 1: For any specified , Algorithm 1 computes a approximation solution. If the algorithmic parameters are and , the running time is , where is the time required to compute a mincost SRpath.
Proof: See Appendix. ∎
V Online Network Throughput Maximization
In the online network throughput maximization problem, the routing requests arrive one by one without the knowledge of future arrivals. The objective is to accept as many requests as possible. We first present the online formulation and thereby develop an online primaldual algorithm.
Va Problem Formulation
Based on the maximum multicommodity flow problem [26], the online problem can be formulated as the following ILP.
s.t.  (5)  
(6)  
Similar to the offline formulation, constraints (5) and (6) imply the flow conservation and capacity limitation, respectively. We then consider the LP relaxation of this problem where is relaxed to . Note that is already implied by constraint (5). By associating a length with each link and a weight with each request , we can write the dual to the above LP as:
s.t.  (7)  
VB Online Algorithm
We design an online primaldual algorithm which includes an outer loop of a primaldual update and an inner loop of mincost SRpath computation.
The algorithm to solve the problem starts by assigning a precomputed length of zero to all links.
The algorithm proceeds in iterations and each iteration corresponds to a request. Upon the arrival of a new request , we try to route units of flow from node to , for each request .
In each iteration, a mincost SRpath from to that maximizes the righthand side of constraint (7) under current link lengths computed according to Section VI.
If the mincost value is larger than one, the request is rejected. Otherwise, the request is accepted, and the entire flow of request is routed along the mincost SRpath.
After the flow is sent, the flow value and the link length at each link along the path are updated as follows:
2) Update the link lengths as
Parameter is designed to provide a tradeoff between the competitiveness of the proposed online algorithm and the degree of violating the capacity constraint in the primal problem. That is, a smaller leads to larger network throughput as well as a larger degree of violation on the link capacity [17].
Theorem 2: Algorithm 2 is an allornothing, nonpreemptive, monotone, and competitive, more specifically competitive, online algorithm. In other words, the routing flow is competitive and it violates the link capacity constraints by .
Proof: See Appendix. ∎
Vi MinCost SRPath Computation
The key steps in the FPTAS and the online algorithm all involve the computation of the mincost SRpath for a request where the length of a link is the dual variable .
In order to accelerate the FPTAS, the auxiliary graph construction and the link lengths update are organized into independent algorithms auxiliary graph construction and mincost computation module, respectively. In the FPTAS, the auxiliary graph construction needs to be executed only once in each iteration while the mincost computation module should be executed in every step. However, it makes no difference for the online algorithm whether the two algorithms are independent because both of them are executed once in every iteration.
In the auxiliary graph construction, denote the auxiliary graph by . There are two end nodes corresponding to and for the current request , and layers of SRnodes . There are links connecting to all the SRnodes in the first layer, from each SRnode in the first layer to each SRnode in the second layer, etc, This process is repeated until all SRnodes of the last layer are connected to , as shown in Fig. 3. The node set composed of all the possibly involved SRnodes for a given request is called the Middlebox Fabric (MF).
In the mincost computation module, the main processes are as follows:
1) Execute an allpairflowsplittingcost (APFSC) computation to get the total costs between all node pairs. In particular, if the shortest path routing is employed in , the APFSC computation reduces to an allpairshortestpath computation. This, of course, can simplify the implementaiton and accelerate the algorithm. We are interested in the paths and corresponding SRnodes that are relevant for request .
2) Update the link lengths of to the total cost between the two nodes the link connects.
3) Compute the shortest path in the auxiliary graph between and . This determines the optimal segment list (SRnode list) .
Although the MF in Fig. 3 looks similar to that in [15], the computation of endtoend paths is entirely different. For the algorithm in [15], only the dual link weights are used to perform an allpair shortest path computation in each iteration. For our algorithms, the primal link weights are used to generate physical routing paths while the dual link lengths to guide flow allocation on the generated paths.
In practice, for the network operator, the MF in Fig. 3 can be automatically or even manually adapted to specific network topologies to steer the flow on a more desirable routing path while greatly reducing resource overheads.
Vii Simulation Results
Viia Simulation Settings
We use two typical networks to evaluate the proposed solutions. In the Abilene network shown in Fig. (a)a, all the 30 links are bidirectional and have equal capacities 100. In the SR network shown in Fig. (b)b, without loss of generality, we make all the 36 links unidirectional from node 1 to 21. The simulation settings are summarized in Table II
. The volume is an rough estimation to the whole network capacity and is calculated as the sum of all the link capacities.
1.0in1.0in
Strategy  Capacity  Volume  FPTAS  Online algorithm 

Abilene network  100  12 random node pairs; =20  100 random requests; =5  
SR network  100  1 node pair; =100  100 requests; =5 
The experiments on the Abilene network can be seen as blackbox tests. It shows the overall performance in realistic networks, especially the Internet backbone. The experiments on the SR network can be seen as whitebox tests. The reasons why we devise such a network are twofold. First, it essencially reflects the hierarchical characteristics of current multidomain Internet [20, 19]. Specifically, this network simulates a real interdomain network. There exist multiple available paths between a node pair within a domain and different domains are connected by edge devices which may be performance bottlenecks. Second, the optimal throughput of QSR is hard to obtained by conventional methods. For instance, it is nearly impossible to compute a 5SR setting using a general LP solver even for a network with such size, while we can easily make an estimation from this highly structured topology. In this way, the traffic distribution becomes tractable and we are able to evaluate to what extent the proposed algorithms can approximate the optimum.
In the following, we conduct the simulations from two perspectives. From the algorithimic perspective, we want to validate and analyze the effects of parameters and . From the SR perspective, since we have already parameterized the SRnode number, the segment number and the multipath policy for intrasegment routing in our model, we will not validate all of the parameters or variables in this paper due to space limitation. Instead, we are more interesting in whether would virtually influence the routing performance and resource consumption. This is because lots of literature claim that it is unprofitable to set in real networks.
Generally speaking, a larger (and hence ) will lead to a larger thoughput while accompanied by heavier computation overheads in the offline setting and severer bandwidth constraints violations in the online setting. How to achieve a tradeoff is closely relevant to the network topology and thus is not the focus of this paper. Since we aim to highlight the computation efficiency of the proposed algorithms, we let unless otherwise specified.
To evaluate the FPTAS, for the Abilene network, each of the 12 nodes randomly selects another node to send a request with size ; for the SR network, all the requests aggregate to one request from node 1 to 21 with size .
To evaluate the online algorithm, we randomly generate 100 requests with equal size . These requests enter the network one by one in a nonpreemptive manner. That is, once a request enters the network, it will stay for ever. For the Abilene network, the traffic spreads across amost the whole network. For the SR network, the traffic is injected into the network at node 1 and is finally absorbed at node 21.
ViiB Fptas
In this section, we validate the parameters and in terms of the routing performance metric and the computation cost metric. The normalized computation time is defined as the ratio of real computation time to the real computation time when (a commonly used setting in literature). More precisely, the algorithms execute within a few seconds for all the instances considered in our simulation.
Fig. 5 shows how influences the throughput as well as the computation overheads. For the Abilene network, the throughput reaches the optimum when and further increasing of does not bring any improvent. For the SR network, the throughput gradually increases with until it reaches the maximum 4.53 when . It can be easily seen from Fig. (b)b that the theoretical optimal throughput is when . All the 5 parallel paths, e.g. (1,2,6,7,11,12,16,17,21), from node 1 to 21 are fully utilized. On the other hand, the optimum can only be achieved when , more exactly . In fact, our algorithms can even support a rapid computation for a very large , say , with only small additional overheads than the that just reaches the optimum.
Considering that the Abilene network can reach a satisfactory throughput when while is the best setting for the SR network, we use these settings of for the validation of and .
Notably, the computation overheads also concern with the topologies. The computation time increases almost simultaneously with the throughput in the SR network, while there is only a slow increasing in the Abilene network.
Fig. 6 shows that the effects of are quite similar in the two networks. When becomes smaller, grows linearly while the computation time grows exponentially. Obviously, is a good choice to reach a tradeoff between routing performance and computation cost. For the SR network, the setting leads to a throughput when , which is fairly close to the optimum .
ViiC Online Algorithm
In this section, we validate the parameters and in terms of the routing performance metric acceptance ratio and the resource cost metric violation ratio. The acceptance ratio is defined as the ratio of accepted number of requests to the total request number. The violation ratio is defined as the maximum ratio of the real flow amount on a link to its capacity over all links.
Similar to the offline scenario, as seen from Fig. 7, has significant influences on the routing performance as well as the resource consumption. However, unlike the offline scenario, approaching the optimum needs an even larger in the online setting. For the Abilene network, the acceptance ratio reaches the optimum when . For the SR network, the acceptance ratio gradually increases with until reaches the optimum when .
As for the violation ratio, the two networks have a similar trend. The violation ratio gradually increases and reaches almost stable when surpasses some value.
As shown in Fig. 8, the effects of are also quite similar in the two networks. When becomes smaller, grows linearly while the computation time grows exponentially.
Similar to the effects of imposed on the FPTAS, there is also a tradeoff between routing performance and resource consumption when choosing an appropriate . It is virtually meaningless to compare the performance between the two neworks, because the acceptance ratio can be raised by reducing the request sizes or enlarging the link capacities.
Viii Discussion
The proposed SR framework is flexible enough to be extended in the following ways.
Segment multicast: The proposed framework as a whole can be extended to a novel routing paradigm segment multicast. By doing this, the SRpath becomes a pseudo directed steiner tree [18]. Since the computation of a directed steiner tree is NPhard, we can invoke an approximation algorithm in the mincost SRpath computation model. Of course, this may introduce some implementation issues and protocol overheads on encoding a multicast tree to packet header in a source routing manner.
Intrasegment routing: The intrasegment routing should be fully investigated, including link weights setting and intrasegment routing policy [7]. For instance, the intrasegment routing module can be replaced with other linkstate routing policies or even a centralized mincost MCF routing module. In the current SR architecture, the intrasegment routing is fixed and the only optimization space left lies in the intersegment routing.
SRnode selection and placement: The topologyadaptive and trafficaware SRnode selection and placement methods should be further considered. Specifically, each layer of SRnodes in the MF structure shown in Fig. 3 can be independently specified for each request. In the worst case, e.g. the SR network, only if all the intermediate nodes are employed will the throughput be maximized. Therefore, how to improve the routing performance while keeping as small overall costs as possible is also a major challenge.
Combining with SFC: How to combine the research methods and results of SR and SFC, just as indicated in Section II, has both theory value and practical significance. From the standpoint of network operator, for instance, there is a strong need to incorporate a realistic SRnode cost model to the framework, while this may have been well solved in the context of SFC [17, 18].
Ix Conclusion
In this paper, we propose a flexible SR model and its formulation where segment number, SRnode number, intrasegment routing policy are all parameterized. The model leads to a highly extensible framework to design and evaluate algorithms that can be adapted to various network topologies and traffic matrices. For both offline and online settings, we develop primaldual algorithms with provable worst case performance bounds. The advantage of computation efficiency of the algorithms over existing methods is so great that it enables quantitative evaluation of various SR parameters and algorithmic parameters on various types of network topologies.
Appendix
Ixa Proofs for Algorithm 1
Lemma 1: When the FPTAS terminates, the primal solution needs to be scaled by a factor of at most to ensure primal feasibility (i.e., satisfying link capacity constraints).
Proof: Serialize all the steps of all the iterations of all the phases into steps. Define the flow scaling factor of link as:
According to the update rule of , we have:
Using the Taylor Formula, the inequality holds. Setting and , we have:
Hence, the lemma is proven.
∎
Lemma 2: At the end of phases in the FPTAS, we have
Proof: Define
Define
We now sum over all iterations during phase to obtain:
Since , we have:
Using the initial value , we have for
The last step uses the assumption that . The procedure stops at the first phase for which
which implies that
∎
Proof of Theorem 1: The analysis of the algorithm proceeds similar to [26].
1) Approximation ratio: Let represent the ratio of the dual to the primal solution. Then we have
Substituting the bound on from Lemma 2, we have
Setting leads to .
Equating the desired approximation factor to this ratio and solving for , we get the value of stated in the theorem.
2) Running time: Using weakduality from linear programming theory, we have
Then the number of phases is upper bounded by
Note that the number of phases derived above is under the assumption . The case can be recast as by scaling the link capacities and/or request sizes using the same technique in Section 5.3 of [26]. Then, the number of phases is at most . We omit the details here.
Since each link length has an initial value of and a final length less than , the number of steps exceeds the number of iterations by at most . Considering that each phase contains iterations, the total number of steps is at most
Multiplying the above expression by , i.e. the running time of each step, proves the theorem. ∎
IxB Proofs for Algorithm 2
Proof of Theorem 2: The online algorithm is by nature an approximation algorithm, and the performance guarantee can be proved in three steps as in [8].
1) Dual feasibility: We first show that the dual variables and generated in each step by the algorithm are feasible.
Let denote the intermediate node that minimizes .
Setting makes hold for all SRnode lists. The subsequent increase in will always maintain the inequality since does not change.
2) Competitive ratio: First, we give an upper bound of . Suppose there are paths from node to and the flow amount on path is , then
Denote by the number of links of path , then
Thus, the SRfunction for request is
During the step where request is accepted, the increase in the primal function is:
and the increase in the dual function is:
Therefore, the competitive ratio can be calculated as:
3) Primal feasibility: We now show that the solution is almost primal feasible.
Denote the link price after request has been accepted and processed by , and the utilization of link as
First, we prove an lower bound of :
We use the induction method. According to the update rule of , we have:
The last inequality follows from:
and
Second, we prove an upper bound of :
Denote . After request is accepted, the mincost value . Then:
According to the update rule of and , we have:
Combining the lower bound and the upper bound, we have:
∎
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