ILP Formulations and Optimal Solutions for the ... - Semantic Scholar

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ILP Formulations and Optimal Solutions for the RWA Problem Brigitte Jaumard(∗) , Christophe Meyer, Babacar Thiongane Department of Computer Science and Operations Research Universit´e de Montr´eal, C.P. 6128, succ. centre-ville - Montr´eal (QC) H3C 3J7, Canada (∗) Canada Research Chair - Optimization of Communication Networks Xiao Yu ´ Department of Electrical Engineering, Ecole Polytechnique de Montr´eal

Index Terms— RWA problem, mathematical programming formulation, heuristic.

while the symmetrical one is more oriented to the single bidirectional systems which is used when fiber constraints are driving the design network decision [16]. There are two classes of formulations, those with link or node variables (see [9], [13]), and those with path variables (see [11], [15], [17]). We compare the optimal values of their continuous relaxations in order to identify the formulation leading to the best upper bounds. We show in particular that several (linear and constraint aggregation) relaxations are leading to the same upper bound. We next compare these upper bounds with the lower bounds obtained by the Tabu Search heuristic of Jaumard and Hemazro [4]. We then show that the heuristic solutions of several benchmarks problems taken from [9] are indeed optimal.

I. I NTRODUCTION

II. S TATEMENT OF THE RWA P ROBLEM

The WDM (Wavelength Division Multiplexing) optical networks offer the promise of providing the high bandwidth required by the increasing communication applications. This has led to a wide interest in the RWA (Routing and Wavelength Assignment) problem defined as follows: given the physical structure of a network and the requested connections, select a suitable routing path and wavelength for each connection so that no two paths sharing a link are assigned the same wavelength. Many papers have already appeared on the RWA problem, proposing various heuristic scheme solutions under different assumptions on the traffic patterns, availability of the converters, and objectives, cf. the surveys of Dutta and Rouskas [2] and of Zang, Jue and Mukherjeee [18]. The most often studied objectives are the minimization of the number of wavelengths (called min-RWA problem), the maximization of the number of accepted connections (called max-RWA problem) and the minimization of the multiplexing costs. With respect to exact solutions, the RWA problem has been formulated as an integer programming problem but most of the times those formulations have not been used for developing solution schemes except for some rounding off procedures. We review those formulations for static traffic models, considering both symmetrical and asymmetrical traffic matrices, focusing on the max-RWA problem. The asymmetrical traffic model is supported by the classical two-fiber architecture,

We assume that the optical network is represented by a multigraph G = (V, E) with a node set V = {v1 , v2 , . . . , vn } where each node is associated with a node of the physical network, and an edge set E = {e1 , e2 , . . . , em } where each edge is associated to a link and a fiber of the physical network. The traffic corresponds to a set K of connections. Each connection is assumed to have a capacity corresponding to the capacity of a wavelength. We consider the objective of maximizing the number of connections under both symmetrical and asymmetrical traffic assumptions. When the traffic is symmetrical, connections and therefore links are bidirectional and wavelength are fullduplex. When the traffic is asymmetrical, connections and therefore links are directional and between each pair of nodes, there are usually two fibers, one for each direction. Denote by T a traffic matrix. Each element Tij defines the number of requested connections between nodes vi and vj when the matrix is symmetrical or from vi to vj when the matrix is asymmetrical. For each connection k, let sk and dk denote the source and destination nodes respectively. For each pair of source and destination, Ksd denotes the set of connections between vs and vd or from vs to vd (note that Tsd = |Ksd |). We define below the various variables that will be used in the different formulations presented in the forthcoming sections. Variables associated with a link may be used either

Abstract— We present a review of the various integer linear programming (ILP) formulations that have been proposed for the routing and wavelength assignment problem in WDM optical networks with a unified and simplified notation. We consider both symmetrical and asymmetrical traffic matrices. We propose a new formulation for symmetrical traffic. We show that all formulations proposed under asymmetrical traffic assumptions are equivalent (i.e. same optimal value for their continuous relaxations) although their number of variables and constraints differ. We propose an experimental comparison of various lower and upper bounds with the objective of minimizing the blocking rate, and show that several benchmark problems proposed by Krishnaswamy and Sivarajan (2001) can be solved exactly or with a fairly high precision.

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in formulations for symmetrical traffic, a link is then an edge; or for asymmetrical traffic, a link is then an arc. We have : xk = 1 if connection k is accepted and 0 otherwise, xki = 1 if connection k is going through node vi and 0 otherwise, xλk = 1 if wavelength λ supports connection k and 0 otherwise, xλke = 1 if wavelength λ supports connection k on link e and 0 otherwise, xλki = 1 if connection k is going through node vi and is supported by wavelength λ and 0 otherwise. Moreover, we will denote by ω(vi ) the set of adjacent links at node vi (non oriented networks with symmetrical traffic), and by ω + (vi ) (resp. ω − (vi )) the set of outgoing (resp. incoming) links at node vi , with ω(vi ) = ω + (vi ) ∪ ω − (vi ) (oriented networks with asymmetrical traffic). We also denote by Λ = {λ1 , λ2 , . . . , λW } the set of available wavelengths. III. RWA P ROBLEM WITH S YMMETRICAL T RAFFIC A. Link Formulation We present below a link formulation, denoted (RWA L), of the RWA problem with symmetrical traffic. X z RWA L = max xk k∈K

subject to: X X

xλke = 2xki

k ∈ K, vi ∈ V \ {sk , dk }

(1)

xλke = xki

k ∈ K, vi = sk , dk

(2)

λ∈Λ e∈ω(vi )

X X

e ∈ E, λ ∈ Λ

(3)

k∈K

(4)

k ∈ K, e ∈ E, λ ∈ Λ

(5)

2xk = xksk + xkdk

k∈K

(6)

2xλke

k ∈ K, e = {vi , vj } ∈ E,

xλke

≤1

k∈K

X

xλk = xk

λ∈Λ xλke ≤

xλk

≤ xki + xkj

xk , xλk , xλke

∈ {0, 1} xki ∈ {0, 1}

While in the previous formulation, variables were defined with respect to the links and nodes of the network, we focus here on variables associated with lightpaths, i.e., with variables xλkp such that xλkp = 1 if wavelength λ is used to support connection k on route p. Let Psd be the set of potential paths between nodes vs and vd . We define aep = 1 if edge e belongs to path p and 0 otherwise. The (PATH) formulation is as follows. X X X z PATH = max xλkp λ∈Λ k∈K p∈Psk dk

subject to: X X

aep xλkp ≤ 1

e ∈ E, λ ∈ Λ

(10)

xλkp ≤ 1

k∈K

(11)

k∈K p∈Psk dk

X

X

λ∈Λ p∈Psk dk

p ∈ Psk dk , k ∈ K, λ ∈ Λ. (12)

xλkp ∈ {0, 1}

Inequalities (10) express that at most one connection can be routed on a given edge with a given wavelength, while inequalities (11) express that, for each connection, at most one path and wavelength must be selected. C. Comparison of the Link and Path Formulations

λ∈Λ e∈ω(vi )

X

B. Path Formulation

λ∈Λ

(7)

k ∈ K, e ∈ E, λ ∈ Λ k ∈ K, vi ∈ V.

(8) (9)

Constraint (1) ensures that, if a connection k goes through vi 6= sk , dk , the wavelength continuity is satisfied in conjunction with (4) and (5): (4) imposes only one wavelength per connection and (5) forces all variables xλke to be zero except for those associated with one wavelength. Constraint (2) enforces that one wavelength on one link is assigned to connection k at the origin (destination) node if connection k is accepted. The remaining constraints express that if connection k is accepted, a path must be defined, and if it is denied, all variables xki , xλk and xλke must be equal to 0. Observe that the optimal solution of (RWA L) may contain isolated ”dummy loops”, e.g, for given λ and k, a subset of variables xλke = 1 for e ∈ E` ⊆ E, such that the edges of E` define a cycle. However, note that no isolated loop involving a source or a destination can exist. Setting the variables associated with a loop to 0 does not affect the value of the optimal solution of (RWA L).

Let us compare the formulations presented in the two previous paragraphs. In terms of variables, the (PATH) formulation is more economical than the (RWA L) one as long as the number of potential routes is limited. Otherwise, the (PATH) formulation should be solved using branch-and-price algorithms with the embedding of column generation techniques, see, e.g., Lee et al. [11] for an heuristic and Jaumard et al. [6] for an exact branch-and-price algorithm for RWA problems. The comparison in terms of upper bounds leads to the following result. Theorem 1: Let z RWA L and z PATH be the optimal values of the continuous relaxations of formulations (RWA L) and (PATH). We have: z RWA L ≥ z PATH . Proof: Let xλkp be an optimal vector of the continuous relaxation of (PATH). We define X X xk = k∈K (13) xλkp λ∈Λ p∈Psk dk

xλke

=

X

aep xλkp

k ∈ K, λ ∈ Λ, e ∈ E

(14)

k ∈ K, λ ∈ Λ

(15)

k ∈ K, vi ∈ V

(16)

p∈Psk dk

xλk =

X

xλkp

p∈Psk dk

xki =

X

X

bip xλkp

λ∈Λ p∈Psk dk

where bip = 1 if node vi belongs to path p and 0 otherwise. We now show that xk , xλk , xλke , xki define a feasible solution for the continuous relaxation of (RWA L). By definition of a

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path, we have for all p ∈ Psk dk X aep = 2bip vi ∈ V \ {sk , dk }

X

(17)

aep = bip

X

Multiplying these equalities by summing over p ∈ Psk dk and λ ∈ Λ, and using (14) and (16), leads to the constraints (1) and (2). It is easy to check that the other constraints of (RWA L) are also satisfied. To show that it is possible to have z RWA L > z PATH , let us consider the network depicted in Figure 1. Assume v

v

v 4

4

e

1

1

e

3

6

k∈K

(23)

5

e v

3

xk , xλk , xλke

∈ {0, 1}

k ∈ K, e ∈ E, λ ∈ Λ

(24)

k ∈ K, e ∈ E, λ ∈ Λ.

(25)

In order to make the comparison of the formulations of Krishnaswamy and Sivarajan [9] easier, we introduce a slight variation of formulation (KS 1), denoted by (KS 1 0 ). It is defined by the set of constraints (19), (22)-(25) and the following constraints: X xλke = xλk k ∈ K, λ ∈ Λ (20a) e∈ω + (sk )

v

2

v Fig. 1.

e

2

e e

xλk = xk

xλke ≤ xλk xλkp ,

e

(22)

λ∈Λ

(18)

vi ∈ {sk , dk }.

e∈ω(vi )

e ∈ E, λ ∈ Λ

k∈K

e∈ω(vi )

X

xλke ≤ 1

X

6

k ∈ K, λ ∈ Λ

(21a)

xλke = 0 k ∈ K, λ ∈ Λ.

(26)

xλke = xλk

e∈ω − (dk )

X

7

xλke =

e∈ω + (dk )

5

Network for the Proof of Theorem 1

that W = 1 and that K = {k1 , k2 } with (sk1 , dk1 ) = (v1 , v6 ) and (sk2 , dk2 ) = (v2 , v4 ). We have that z RWA L = 2 while z PATH ≤ 1 as paths connecting v1 and v6 and paths connecting v2 and v4 go through the ”bridge” edge e4 . IV. RWA P ROBLEM WITH A SYMMETRICAL T RAFFIC Several formulations have been proposed for asymmetrical traffic, the first one by Krishnaswamy and Sivarajan [9] that considers each connection in turn, and two others that consider set of connections grouped with respect to either their origin-destination, or their origin. We show that although they correspond to different ILP formulations with different set of feasible solutions, their optimal values are all equal.

X

e∈ω − (sk )

Note that the constraints (26) are necessary in order to prevent from isolated dummy loops involving the source or the destination of a connection. Let Ω LP (KS 1) and ΩLP (KS 10 ) be the feasible domains of the continuous relaxations of (KS 1) 0 and (KS 10 ). Denote by z KS1 and z KS1 the optimal values of the continuous relaxations of (KS 1) and (KS 1 0 ). It is easy to verify that: Theorem 2: The optimal values of the continuous relaxations of formulations (KS 1) and (KS 1 0 ) are equal: z KS 1 = 0 z KS1 , and moreover that: ΩLP (KS 10 ) ⊆ ΩLP (KS 1). B. Second Formulation of Krishnaswamy and Sivarajan Let us now examine the second formulation of Krishnaswamy and Sivarajan [9], denoted by (KS 2), in which connections are grouped with respect to their origin and λ destination of nodes. We introduce the variables ysde = P λ pair λ λ xke , ysde ∈ {0, 1}, where ysde = 1 if a connection k∈Ksd

A. First Formulation of Krishnaswamy and Sivarajan We present below the first formulation of Krishnaswamy and Sivarajan [9], denoted by (KS 1). X xk z KS 1 = max

from vs to vd uses wavelength λ on link e and 0 otherwise, and xsd ∈ {0, 1, . . . , Tsd } which is equal to the number of accepted connections from vs to vd . Note that, for a given e, λ, at most one connection can be supported. z KS 2 = max

k∈K

subject to: X

xλke =

e∈ω + (vi )

X

X

xλke

e∈ω − (vi )

k ∈ K, λ ∈ Λ, vi ∈ V \ {sk , dk }

xλke

−

X

xλke

e∈ω + (sk )

e∈ω − (sk )

X

X

e∈ω + (d

xλke − k)

e∈ω − (d

=

xλk

xλke = −xλk k)

(19)

X

xsd

(vs ,vd )∈V ×V

subject to: X

λ = ysde

e∈ω + (vi )

X

λ ysde

e∈ω − (vi )

λ ∈ Λ, (vs , vd ) ∈ V × V, vi ∈ V \ {vs , vd }

(27)

k ∈ K, λ ∈ Λ (20) k ∈ K, λ ∈ Λ

(21)

X

X

λ∈Λ e∈ω + (vs )

λ = ysde

X

X

λ = xsd ysde

λ∈Λ e∈ω − (vd )

(vs , vd ) ∈ V × V

(28)

4

X

X

λ ysde =

X

X

If xk1
0. Moreover only the length of the shortest paths matters. Denote by wsd the length of a shortest path from vs to vd . Using the variables xsd defined in Section IV-B, we get X xsd z PCC = max (vs ,vd )∈V ×V

e∈ω + (vd )

ysde ∈ {0, 1, . . . , W }

B. Shortest Path Upper Bound: z PCC

subject to: X

(53)

wsd xsd ≤ mW

(vs ,vd )∈V ×V

xsd ∈ {0, 1, . . . , Tsd },

(vs , vd ) ∈ V × V.

(54)

This is a special case of the knapsack problem, that can be solved in polynomial time by sorting the wsd in nondecreasing order and setting xsd to 1 until the constraint (53) becomes violated. This upper bound is never better than z PATH - AGGR , but is much more efficient to compute.

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VI. C OMPUTATIONAL EXPERIENCE We compare the values of the various upper bounds discussed in the previous sections together with the best known solution on two data sets of the literature taken either from Jaumard and Hemazro [4] or from Jaumard and Yu [7]. We consider two optical networks widely used in the literature, the NSF and the EON networks. The NSF network is a network with 14 nodes and 21 optical links, described in, e.g., [10], while the EON network is a network with 20 nodes and 39 optical links, described in, e.g., [12]. We used the asymmetrical traffic matrices of Krishnaswamy [8]. They correspond to 268 connections for the NSF instance, and 374 for the EON one. For the experimental results with symmetrical traffic, we modify those matrices and use max{Tsd , Tds } for the number of connections between a pair of source and destination nodes (vs , vd ). The resulting symmetrical matrices lead to 191 connections for the NSF instance and to 270 for the EON one. All data are available in [3]. The comparison of the various upper bounds proposed in the previous sections is summarized in Tables I and II. For the symmetrical traffic instances, we observe that the z PCC upper bound improves the linear programming bound z RWA L , and is in turn improved significantly by the z PATH - AGGR and z PATH - AGGR bounds. It must also be noted that the z PATH - AGGR bound is always equal to z PATH in practice, see [14] [5] for a proof that these two bounds are indeed equal. Moreover, these last two bounds are very often equal to z PATH - AGGR , with the observation that the optimal vectors associated with z PATH - AGGR and z PATH are often almost integer. At last, solving (PATH - AGGR) is easy as a very limited of nodes are developed in the branch-andbound search tree of CPLEX - MIP. For both symmetrical and asymmetrical traffic instances, we indicate the gap between the best lower bound and the best upper bound, with several heuristic solutions proved to be indeed optimal. Not only it shows that the Tabu Search of [4][7] is very efficient, but also that it is now possible to solve some RWA problems optimally. W

z RWA

10 12 14 16 18 20 22 24

126 147 168 189 191 191 191 191

W

z RWA

10 12 14 16 18 20 22 24

219 246 262 269 270 270 270 270

L

z PCC 123 137 151 165 179 191 191 191

L

z PCC 197 223 245 264 270 270 270 270

NSF network z PATH - AGGR z PATH - AGGR 5 = z PATH 5 115 115 130 129 143 143 153 153 161 161 169 169 177 177 185 185 EON network z PATH - AGGR 7 z PATH - AGGR PATH 7 =z 176 176 194 194 212 212 225 225 237 237 249 249 256 256 262 262

5

z 114[7] 125[7] 137[7] 146[4] 158[4] 167[7] 177[4] 185[4]

7

z 168[7] 185[7] 204[4] 220[4] 235[4] 247[4] 254[4] 262[4]

gap (%) 0.8 3.1 4.2 4.6 1.9 1.2 0.0 0.0 gap (%) 4.5 4.6 3.8 2.2 0.8 0.8 0.8 0.0

TABLE I C OMPARISON OF L OWER AND U PPER B OUNDS - S YMMETRICAL T RAFFIC

W 10 12 14 16 18 20 22 24

z KS 1 198 218 238 258 267 268 268 268

network z gap 188[7] 5.3% 212[4] 2.8% 235[7] 1.2% 253[4] 1.9% 265[4] 0.7% 268[4] 0.0% 268[4] 0.0% 268[4] 0.0%

NSF

z KS 1 285 317 337 350 362 370 374 374

network z gap 278[4] 2.5% 306[4] 3.5% 329[4] 2.4% 350[7] 0.0% 362[7] 0.0% 370[4] 0.0% 374[4] 0.0% 374[4] 0.0%

EON

TABLE II C OMPARISON OF L OWER AND U PPER B OUNDS - A SYMMETRICAL T RAFFIC

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