Photon Netw Commun (2013) 26:53–64 DOI 10.1007/s11107-013-0408-0
Column generation algorithm for RSA problems in flexgrid optical networks ˙ Marc Ruiz · Michał Pióro · Mateusz Zotkiewicz · Mirosław Klinkowski · Luis Velasco
Received: 6 March 2013 / Accepted: 18 July 2013 / Published online: 8 August 2013 © Springer Science+Business Media New York 2013
Abstract Finding optimal routes and spectrum allocation in flexgrid optical networks, known as the RSA problem, is an important design problem in transport communication networks. The problem is N P-hard, and its intractability becomes profound when network instances with several tens of nodes and several hundreds of demands are to be solved to optimum. In order to deal with such instances, largescale optimization methods need to be considered. In this work, we present a column (more precisely, path) generationbased method for the RSA problem. The method is capable of finding reasonable sets of lightpaths, avoiding large sets of precomputed paths, and leading to high-quality solutions. Numerical results illustrating effectiveness of the proposed method for obtaining solutions for large RSA problem instances are presented. Keywords Integer programming · Column generation · Routing and spectrum allocation · Flexgrid optical networks
M. Ruiz (B) · L. Velasco Computers Architecture Department, Universitat Politècnica de Catalunya, c/ Jordi Girona, 31, 08034 Barcelona, Spain e-mail:
[email protected] ˙ M. Pióro · M. Zotkiewicz Institute of Telecommunications, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warszawa, Poland M. Klinkowski Department of Transmission and Optical Technologies, National Institute of Telecommunications, Szachowa 1, 04-894 Warszawa, Poland
1 Introduction Recent technologies in flexible optical networking, and in particular the flexible grid (flexgrid) technology specified in the ITU-T G.694.1 Recommendation [1], are considered as the most promising solution to deal with high volumes of data traffic that have to be carried in transport communication networks. In the flexgrid solution, the frequency spectrum of an optical fiber link is divided into narrow frequency slots. Any sequence of consecutive slots can be used as one channel, and such a channel can be switched in the network nodes to create an optical path (lightpath). The channel covers the frequency range occupied by the optical signal and the guard band required for the roll-off filters. Thus, a lightpath is determined by a routing path and a selected channel that is switched in the intermediate nodes of the route. Figure 1 illustrates with an example the above-mentioned concepts in the considered flexgrid representation. Here, two lightpaths sharing the expanded link use distinct channels in the optical spectrum, and each one of the channels uses a non-overlapped sequence of consecutive frequency slots. Note that in [1], the meaning of term slot is slightly different than in our case. However, since many works in the literature use this term in the same meaning as we do and the standardization of flexgrid technologies is still in progress, in the remainder of this paper, we consider that a slot represents a narrow segment of the frequency spectrum (as illustrated in Fig. 1). For more details on flexible optical network architectures and proofof-concept experiments, we refer the reader to [4,7,10]. In flexgrid optical networks, the problem of establish lightpaths for a set of end-to-end demands that compete for spectrum resources is called the routing and spectrum allocation (RSA) problem. RSA consists of assigning a lightpath to each connection request from a given set. Each such lightpath is routed in the network graph and assigned a contiguous
123
54
Photon Netw Commun (2013) 26:53–64
Fig. 1 Example of the considered flexgrid representation
fraction of frequency spectrum (this constraint is called spectrum contiguity) reserved on the route. Clearly, the frequencies assigned to the individual lightpaths cannot overlap on the network links. Moreover, it is commonly assumed that the same piece of the spectrum is used on all the links traversed by a lightpath (this constraint is called spectrum continuity). The RSA optimization problem is N P-hard [12]. In practice, the RSA problem is more difficult than the routing and wavelength assignment (RWA) problem related to the fixedgrid wavelength division multiplexing (WDM) networks. Note that ordinary RWA algorithms, for instance such as the one presented in Liu and Rouskas [14], are not appropriate for flexible optical networks as the elastic spectrum allocation in flexible optical networks is different from channel assignment in WDM networks. In the former, contrary to the latter, the channel width is not rigidly defined but can be tailored to the actual width of the transmitted signal. Several alternative mixed-integer programming (MIP) formulations of the RSA problem can be found in the literature [3,12,19,20]. Among them, the formulation proposed in Velasco et al. [19] seems to be the most attractive. In this formulation, both the continuity and contiguity constraints are removed from the MIP by using a set of precomputed lightpaths. Although this formulation is superior to others in terms of the computational time and the size of network instances handled, realistic instances are still difficult to solve due to the large set of the involved integer path-flow variables. To make large instances of RSA tractable by MIP formulations, decomposition methods must be applied. Column generation (CG) is one of such decomposition techniques. It allows for reducing the amount of variables (referred to as columns) in the LP-based problem formulations [2,13]. In
123
CG, the problem is initialized with a small list of admissible columns which is then extended. A key element of CG is to formulate and solve a pricing problem, which concerns finding new columns that, when added to the problem, can improve the objective function value. Applications of CG for solving design problems in communication networks can be found in Pióro et al. [17]. In the context of network flow problems, CG is referred to as path generation (PG) since the variables to find and add are admissible path-flows. In RSA problems, a technique based on CG must provide lightpaths containing continuous and contiguous spectrum channels. In this paper, we present a novel lightpath generation algorithm (referred to as CGA) for the MIP formulations of RSA. This PG algorithm is adapted to optical paths with a specific and unique piece of assigned spectrum. The proposed method is applied for a RSA problem similar to the one described in Klinkowski et al. [11]. The problem aims at minimizing the total amount of unserved bit-rate for a given traffic matrix. The final total amount of unserved bit-rate (caused by the lack of spectrum resources) results either from rejecting entire demands or from rejecting only a part of the requested demanded bit-rate maintaining a minimum served bit-rate. Since network operators usually aim at providing connectivity service to a wide variety of clients (to increase revenues), the former case is strongly penalized. Presented models are developed for a network operating with a single modulation format. Further extensions corresponding to adaptive transponders, i.e., making use of multiple modulation formats in accordance with lightpath characteristics, are left for further study. To the best of our knowledge, the presented work is among the first concerning application of CG to RSA problems in
Photon Netw Commun (2013) 26:53–64
flexgrid optical networks. Note that CG has been widely studied in the context of RWA, for example, in Jaumard et al. [9]. However, the RWA problem formulations proposed in Jaumard et al. [9] are not applicable to RSA since, as already mentioned, the flexible spectrum allocation in EON differs with the fixed-grid channel assignment in WDM. The rest of the paper is organized as follows. In Sect. 2, we present the RSA problem dealt with in the paper. Then, in Sect. 3, a lightpath generation algorithm for the linear relaxation of the introduced RSA problem is described, followed by Sect. 4 presenting the main solution approach to RSA. Effectiveness of the proposed optimization approach is discussed in Sect. 5 using the results of a numerical study of small-to-large size network instances derived from a wellknow realistic optical network. Finally, Sect. 6 concludes the paper and sketches directions for the future work.
2 Problem statement 2.1 Problem definition and notation In the paper, a network is represented by a graph G = (V, E, D) where V is the set of nodes, E is the set of links, and D is the set of (traffic) demands. For the problems considered in the paper, links and demands can be undirected as well as directed. In the sequel, we will assume the undirected case. Let V |2| denote the set of all two-element subsets of the set of nodes V. Each link e ∈ E is represented by its end nodes s(e) and t (e) with {s(e), t (e)} ∈ V |2| . It is assumed that the capacity of each link is divided into a set S = {s1 , s2 , . . . , s S } of slots (see Table 1). The slots of a link can be used by connections in the form of channels. Channel c of capacity n is a set of contiguous (i.e., consecutive) slots of the form c = {si , si+1 , . . . , si+n−1 } for some i between 1 and S − (n − 1). In the sequel, C will denote the set of all possible channels and n(c), c ∈ C, the number of slots in channel c. Obviously, the number of all possible channels (for all n = 1, 2, . . . , S) is equal to . |C| = S(S+1) 2 Each demand d ∈ D is represented by its end nodes s(d) and t (d) with {s(d), t (d)} ∈ V |2| and is characterized by a minimum demand bit-rate and a maximum demand bit-rate, h(d) and H (d), respectively. The interpretation is that the demand, if realized at all, must be assigned to a bandwidth (i.e., a portion of the spectrum, in terms of the number of slots) able to serve the bit-rate between h(d) and H (d). We use n(d) and N (d) to denote, respectively, these bandwidth limits. Consequently, the set C(d)(C(d) ⊆ C) of channels that can be used for demand d ∈ D is defined as follows: c ∈ C(d) if and only if n(d) ≤ n(c) ≤ N (d), which means that the paths allowable for demand d ∈ D can use only those channels whose capacity n(c) is sufficient to carry the
55
minimum demand bit-rate h(d) and is not too large in terms of the maximum bit-rate H (d). Note that |C(d)| =
(2(S + 1)− N (d)−n(d))(N (d)−n(d)+1) . 2 (1)
Each demand d ∈ D is assigned to a pre-defined set P(d) of allowable lightpaths (each using a selected channel) connecting its end nodes. The set of links composing the route of lightpath p ∈ P(d) (path in graph G connecting s(d) and t (d)) is denoted by E( p), and the channel is assigned to this lightpath by c( p) ∈ C(d), The RSA problem studied in this paper consists of determining, for each demand d ∈ D, a single route over a given flexgrid optical network, p(d) ∈ P(d), together with its assigned channel (spectrum allocation) c( p(d)) ∈ C(d) to minimize the number of rejected demands (primary objective) and the amount of unserved bit-rate (secondary objective). The served bit-rate of each demand is a value between h(d) and H (d). If h(d) cannot be satisfied, then the demand is rejected. In other words, we are to find a single lightpath (i.e., a route and its channel assignment) over a flexgrid optical network for every transported demand minimizing the traffic objective subject to the following constraints: – Spectrum contiguity: the channel consists of a contiguous (adjacent) subset of slots – Spectrum continuity: the channel should be the same for each link on the routing path – Channel capacity: the channel should serve a bit-rate between h(d) and H (d) – Slot occupancy: each slot in each link can be allocated to at most one lightpath. 2.2 Formulation of the RSA problem The list of objects, parameters, and problem variables required for the formal statement of the RSA problem described in Sect. 2.1 is given in Table 1. The mixed-integer formulation of the problem can be written down in the following way: h(d)X d + Yd (2a) P(P) : min F = A d∈D
Xd +
d∈D
xd p = 1 d ∈ D
(2b)
xd p ≤ 1 e ∈ E, s ∈ S
(2c)
g( p)xd p = H (d) d ∈ D
(2d)
X d ∈ {0, 1}, Yd ∈ R d ∈ D
(2e)
xd p ∈ {0, 1} d ∈ D, p ∈ P(d).
(2f)
p∈P (d)
d∈D p∈Q(d,e,s)
Yd +
p∈P (d)
123
56
Photon Netw Commun (2013) 26:53–64
Table 1 Notation Set of links, E = |E |
E S
Set of spectrum slots, S = |S |
b(n)
Bit-rate carried in a channel consisting of n slots
C
Set of channels, C =
S(S+1) 2
n(c)
Number of slots used by channel c ∈ C
S (c)
Set of (contiguous) slots composing channel c ∈ C
D
Set of demands, D = |D|
h(d)
Minimum bit-rate of demand d ∈ D
n(d)
Number of slots required to carry h(d), d ∈ D
H (d)
Maximum bit-rate of demand d ∈ D
N (d)
g(d, n) =
Number of slots required to carry H (d), d ∈ D H (d), b(n),
C (d) ⊆ C
if n = N (d) if n(d) ≤ n < N (d)
Bit-rate of demand d ∈ D carried on n slots Set of channels allowable for demand d ∈ D, c ∈ C (d) ⇔ n(d) ≤ n(c) ≤ N (d)
P (d) P=
d∈D
Set of lightpaths allowable for demand d ∈ D P (d)
Q(d, e, s) ⊆ P (d) E ( p)
d( p)
Demand d ∈ D realized by lightpath p ∈ P
c( p)
Channel occupied by lightpath p ∈ P , c( p) ∈ C (d)
n( p) = n(c( p))
Number of slots occupied by lightpath p ∈ P
S ( p) = S (c( p))
Set of slots used by lightpath p ∈ P
g( p) = g(d( p), n( p))
Bit-rate carried on lightpath p ∈ P
A
A weight for objective function (A 1)
Xd , d ∈ D
Binary variables, X d = 1 when d is not realized at all
xdp , d ∈ D, p ∈ P (d)
Binary variables, xdp = 1 when lightpath p carries its demand d Continuous variables, unserved bit-rate with respect to H (d).
Yd , d ∈ D
Objective function (2a) minimizes the total amount of unserved bitrate. By using the weight factor A, the unserved bit-rate from all refused demands (i.e., X d = 1) prevails over the unserved bitrate from accepted ones (xd p = 1), thus giving priority to the minimization of demand rejection. Constraint (2b) either assigns a lightpath to a demand (i.e., p∈P (d) xd p = 1 and X d = 0) or rejects the demand (i.e., p∈P (d) x d p = 0 and X d = 1). Moreover, in the first case, it selects exactly one lightpath to be assigned for the demand. Constraint (2c) makes sure that occupancy of the slots is not violated, ensuring that the number of active lightpaths (i.e., lightpaths with xd p = 1) sharing a specific slot is at most equal to 1. Constraint (2d) sets the variable Yd equal to the unserved, with respect to H (d), bit-rate of demand d ∈ D. Finally, constraints (2e) and (2f) define the type of variables (R denotes the set of real numbers). Observe that formulation (2) is based on path-flows, and therefore, it assumes a given set of allowable lightpaths P. Clearly, the formulation is non-compact as it requires an
123
Set of all allowable lightpaths Set of lightpaths for d ∈ D using slot s ∈ S on link e ∈ E Set of links traversed by lightpath p ∈ P
exponential number of variables xd p , d ∈ D, p ∈ P(d), in order to consider all possible lightpaths. The problem ˆ with the set Pˆ containing all possible lightpaths will P(P) ˆ be referred to simply as P. The set P is defined by the ˆ and for each d ∈ D, p ∈ condition: Pˆ = d∈D P(d) ˆ P(d) if, and only if, the E( p) defines an elementary (i.e., loop-less) path in graph G between s(d) and t (d), and c( p) ∈ C(d). Certainly, in the paper, we are looking for solutions of P that can possibly utilize any path in a network.
3 Lightpath generation algorithm Below we consider the linear relaxation L(P) of the mixedinteger programming formulation P(P) given in (2), and its solution via CG. The linear relaxation of problem P (with the full list of allowable paths) will be denoted by L. For the considered problem, CG actually means generation of
Photon Netw Commun (2013) 26:53–64
57
lightpaths that will actually be performed through the constraint generation for the problem dual to L.
Let P be a given list of admissible paths. The linear relaxation of the mixed-integer formulation P(P), call the primal problem below, is as follows: h(d)X d + Yd (3a) L(P) : min F = A d∈D
[λd ] X d +
[πes ≥ 0]
d∈D
xd p = 1 d ∈ D
(3b)
xd p ≤ 1 e ∈ E, s ∈ S
(3c)
g( p)xd p = H (d) d ∈ D
(3d)
X d ∈ R+ , Yd ∈ R d ∈ D
(3e)
xd p ∈ R+ d ∈ D, p ∈ P(d)
(3f)
p∈P (d)
d∈D p∈Q(d,e,s)
[γd ] Yd +
p∈P (d)
where R+ denotes the set of non-negative real numbers. To derive the problem dual to L(P), we use the dual variables specified on the left-hand sides of constraints (3b)–(3d). We first formulate the Lagrangian function (see [13] for more details): L(x, X, Y ; λ, π, γ ) γd H (d) + λd − πes = d∈D
+
d∈D
e∈E s∈S
(Ah(d) − λd )X d
d∈D
+
πes (5c)
d ∈ D, p ∈ P(d) ≥ 0 e ∈ E, s ∈ S.
(5d)
Note that dual variables γd , d ∈ D have disappeared, as only γd = 1, d ∈ D are allowed in the domain of the dual function (4). Problem D(P) is a linear programming problem with the set of feasible solutions defined by the (dual) polyhedron (5b)–(5d) in the dual space R D+E·S . We will denote this polyhedron by T (P). It is well known (see [13]) that the optimal objective of the dual problem, W ∗ (P), is equal to the optimal objective, F ∗ (P), of the primal problem. Each optimal vertex solution of T (P) identifies an optimal vertex solution of the primal problem L(P). Hence, any optimal vertex solution of D would yield an optimal vertex solution of L. 3.2 Dual pricing Clearly, problem D can be solved through constraint generation (see for example [17]), namely through generating constraints (5c). Let T denote the solution polyhedron for problem D. Now suppose we are given a set P = d∈D P(d) of allowable lightpaths (we may assume that P(d) = ∅, d ∈ D, although even this is not necessary to assure feasibility of D(P)) and let (λ∗ , π ∗ ) be an optimal solution of D(P). If for ˆ some d ∈ D, there existed a lightpath p(d) ∈ P(d)\P(d) with ∗ πes − g( p(d)) < λ∗d e∈E ( p(d)) s∈S ( p(d))
(1 − γd )Yd
d∈D
+
e∈E ( p) s∈S ( p)
πes
3.1 Linear relaxation of problem P and its dual
λd + g( p) ≤
(−λd − g( p) +
d∈D p∈P (d)
then the inequality (5c) corresponding to p(d), i.e., πes λd + g( p(d))
