An integer programming approach to the OSPF weight setting problem

Transcription

1 An integer programming approach to the OSPF weight setting proem Amandeep Parmar, Shair Ahmed and Joe Soko H Miton Stewart Schoo of Industria and Systems Engineering, Georgia Institute of Technoogy, 765 Ferst Drive, Atanta GA 333 Octoer 8, 6 Astract Under the Open Shortest Path First OSPF) protoco, traffic fow in an Internet Protoco IP) network is routed on the shortest paths etween each source and destination The shortest path is cacuated ased on pre-assigned weights on the network inks The OSPF weight setting proem is to determine a set of weights such that, if the fow is routed ased on the OSPF protoco, some measure of network congestion is minimized A variety of optimization approaches for this strongy NP -hard proem have een proposed However, the existing studies deveop heuristic soution methods without any quaity guarantees In this paper we propose an integer programming ased soution strategy for the OSPF weight setting proem so as to otain soutions with quaity guarantees We deveop a famiy of vaid inequaities for a mixedinteger inear programming formuation of the proem These inequaities are incorporated within a ranch-and-cut agorithm Computationa experiments using some randomy generated test proems and some proems taken from the iterature indicate that the proposed approach is ae to provide feasie soutions with significanty smaer optimaity gaps than those provided y the state-of-the-art integer programming sover CPLEX Introduction Open Shortest Path First OSPF) is the most commony used intra-domain routing protoco in IP networks Under this protoco every ink in the network is assigned a weight, and ased on these weights shortest paths etween each source and destination are computed The OSPF protoco then routes fow on shortest paths If there are mutipe shortest paths from some source to destination then the fow is spit equay among a the outgoing arcs that are on the shortest paths This equa spitting of fow is usuay referred to as the Equa-Cost Mutipath ECMP) principe The weights assigned to the inks competey determine the fow in the network and hence the oad on each ink The OSPF weight setting proem is to determine a set of weights such that, if the fow is routed ased on the OSPF protoco, network congestion is minimized A variety of optimization ased approaches for the OSPF weight setting proem have een proposed Fortz and Thorup [5] considered a version of the proem invoving a piecewise inear convex cost function to measure congestion They showed that it is NP -hard to determine an optima set of weights, and even to approximate an optima soution within a constant factor They proposed a oca search technique to get good soutions for the proem Pioro eta [8] considered the maximum oad on any ink in the network as the measure of congestion They showed that the OSPF weight setting proem with this ojective is NP - hard even for a singe-source destination pair They presented a mixed-integer inear programming MILP) formuation for the proem, and proposed some heuristic methods for its soution ased on oca search, simuated anneaing and Lagrangian reaxation Srivastava eta [] used a hyrid of the two ojectives as a measure of congestion, and suggested heuristic soution methods Srivastava eta [] and Lin and This research has een supported in part y the Office of Nava Research Award # N-5--3) and the Nationa Science Foundation Award # DMI-5766) Corresponding author: Shair Ahmed sahmed@isyegatechedu)

2 Wang [6] proposed heuristic agorithms ased on Lagrangian reaxation to determine feasie soutions for the weight setting proem Ericsson eta [] presented a genetic agorithm-ased heuristic Burio eta [3] extended the genetic agorithm proposed in [] to a memetic agorithm y adding a oca search procedure Bey and Koch [] considered a more restricted version of the proem where the aim is to determine a set of weights such that these weights define unique shortest paths for each source-destination pair Hence when the fow is routed on these paths there wi e no spitting of fows The authors proposed an integer programming approach for this restricted proem Bey [] showed that there does not exist a constant factor approximation for this restricted proem uness P =N P The existing studies on the genera OSPF weight setting proem deveop heuristic soution methods without any quaity guarantees In this paper we propose an integer programming ased soution strategy for this proem so as to otain soutions aong with quaitative guarantees We consider the MILP formuation for the OSPF weight setting proem with a maximum oad ojective presented y Pioro eta [8] Unfortunatey this MILP is extremey difficut to sove as-is y state-of-the-art sovers We deveop a famiy of vaid inequaities and incorporate these within a ranch-and-cut agorithm We provide numerica resuts on the performance of our soution methodoogy on some randomy generated test proems and some proems taken from the iterature Unike the heuristic methods the proposed integer programming ased approach is ae to provide a ound on the worst case optimaity gap for the otained soution Moreover, the soutions generated y our approach have significanty smaer optimaity gaps than those produced y the state-of-the-art MILP sover CPLEX The remainder of the paper is organized as foows In the next section, we discuss an MILP formuation of the OSPF weight setting proem In Section 3 we deveop a famiy of vaid inequaities to improve the MILP formuation In Section we descrie a ranch-and-cut impementation for the proem and present numerica resuts on some randomy generated proems and some proems taken from iterature Finay, concusions and further research issues are discussed in Section 5 A mixed integer inear programming mode In this section we present the MILP formuation of the OSPF weight setting proem from Pioro eta [8] The underying IP network is represented as a directed graph G = V, E), where the nodes represent routers and edges represent the inks etween them Each edge e E has a capacity c e We aso assume that we have a demand matrix D, where D st represents the fow to e sent from a source node s V to a destination node t V We denote V d V as the set of destination nodes Congestion in the network is measured as the maximum oad over a the edges in the network More precisey, if e denotes the tota oad fow) on edge e, then the congestion is measured as the maximum of e /c e ) over a edges e E This ojective aows the fow on an edge to exceed its capacity and hence any set of weight vectors is feasie for the proem The proem is to determine weights w e for every edge such that if the fow is routed, as per the OSPF protoco, etween each source destination pair satisfying the demand, it minimizes the congestion in the network The decision variaes and constraints of the MILP formuation are presented next Decision variaes: x t e inary variae denoting whether e is on some shortest path to destination t fe t fow on edge e for destination t w e weight on edge e d t v shortest path distance from v to destination t fd t v dummy fow variaes for spitting the fow L the maximum oad in the network

3 Fow conservation constraints: e:e=,t) e:e=,v) f t e D vt t V d ) fe t = e:e=t, ) v V fe t fe t = D vt v V \ {t} t V d ) e:e=v, ) f t e Mx t e e t V d 3) Constraints ) and ) ensure that tota infow is equa to tota outfow Constraints 3) force the fow to e sent on ony arcs that are chosen to e on the shortest path Here M is a arge numer that can e set to v V D vt Feasie distance ae constraints: d t u d t v + w e e = u, v) t V d ) d t v d t u + w e M x t e) e = u, v) t V d 5) x t e) Md t v d t u + w e ) e = u, v) t V d 6) Constraints ), 5) and 6) ensure that x t e is if and ony if e is on a shortest path to destination t Fow spitting constraints: f t e f t d v e : e = v, ) v, t V d 7) f t d v f t e M x t e) e : e = v, ) v, t V d 8) Constraints 7) and8) ensure that fow is equay spit amongst a outgoing arcs on the shortest path to each destination Congestion constraints: t V d f t e Lc e e 9) Constraint 9) reate the tota fow on an edge to the congestion measure L Using the variaes and constraints defined aove, a MILP mode for the OSPF weight setting proem is: min L st ), ), 3), ), 5) 6), 7), 8), 9) P ) x t e {, }, f t e, w e d t v, f t d v For a network with m edges, n nodes and k destinations, formuation P ) has 6mk + nk + m constraints and kn + m) + m + variaes, out of which mk are inary As has een discussed y Pioro eta [8], even for a sma sized network, the aove MILP is very difficut to sove using a state-of-art MILP sover such as CPLEX In the foowing section, we descrie a famiy of vaid inequaities to strengthen the MILP formuation 3 A famiy of vaid inequaities In this section we deveop vaid inequaities for simpe susystems of the OSPF weight setting proem P ) Since the susystem is a reaxation of P ) the proposed inequaities are aso vaid for P ) We consider four different susystems 3

4 3 A simpe singe node system Consider a singe node system which has n outgoing edges and fow of > units coming in OSPF requires that fow has to e equay spit amongst the edges which carry positive fow Denote f i to e fow on edge i, x i to e if and ony if edge i is used to send fow out, and f d to e a dummy variae used to ensure that we spit the fows equay The system is given y: { X = f d, f, x) R + R n + {, } n : n f i =, i= f i x i i =,, n, f i f d i =,, n, f d f i + x i ) i =,, n } The first constraint aances fow in to fow out; the second constraint reates fows to arcs chosen; and the ast two sets of constraints equate the fow on chosen arcs Let co ) denotes the convex hu of the set, we can now state the foowing resut for convex hu of X, cox ) Proposition The dimension of cox ) is given y: if n = dimcox )) = if n = n if n 3 Proof See Appendix Let us project out the variaes f i, and then aggregate the x i variaes to a genera integer variae z, ie, z = n i= x i Note that z {,, n} Under this projection, the set X is We can now define the convex hu of Y Y = {f d, z) R + Z + : f d z =, z n} Theorem The foowing inequaities define coy ): kk + )f d + z k + ) k =,, n, ) nf d + z n + ) ) The proof of the theorem omitted here) is constructed y otaining the convex hu of n points in a two dimensiona pane The aove defined vaid inequaities ) with z repaced y n i= x i are aso vaid for X and indeed are facet defining The foowing proposition states the resut Proposition The foowing inequaities are vaid for X kk + )f d + and are facet defining for k n n x i k + ) k =,, n, ) Proof Let x i =, then f d = We have kk + ) + k + ) = integer i= k+ ) ) Proposition 3 The famiy of vaid inequaities ) for X are facet defining for k n Proof See Appendix

5 For the weight setting proem, for every node that has positive fow coming in we can generate the aove famiy of cuts Let v t e a ower ound on the fow incoming to node v going to destination t v t > for a source nodes v which have fow going to destination t) and et us denote O v as the set of outgoing edges out of node v, then repeating the aove procedure we get the foowing set of cuts : kk + )fd t v + v t We now present some more vaid inequaities for X e O v x t e k + ) t v k =,, O v, v, t V d 3) Proposition Let S {,, n} and {f ),, f S ) } denote a permutation of {f i : i S} The foowing inequaities are vaid for X : S kk + )f d + x i x i + k + i)f i) k + ) ) i/ S i S i= S : S n, k n S Proof Let x i = and S = m We know that f d = and fow wi e equa to for a the arcs with x = We need to show that kk + ) km + m) m + + mm+) k + ) Note that the aove condition is equivaent to [ ] m + k + ) + km + m + m + k + k [ m + k + ) ) ) )] m + k + ) + km + m + m + k + k [ ) ] m + k + ) which is true since and m are integers This competes the vaidity proof Proposition 5 The famiy of vaid inequaities ) for X are facet defining for a S with S = Proof See Appendix The foowing proposition provides another set of facet defining vaid inequaities Proposition 6 Let S {,, n}, then the foowing inequaities are vaid for X : f d + x j + f j S + ) S 5) j S j / S Proof Let x j = and S = m We need to show that + m + m) m + ) Note that this condition is equivaent to m) which is true for a m This competes the proof Proposition 7 The famiy of vaid inequaities 5) for X are facet defining for S = and n 3 Proof See Appendix For the OSPF weight setting proem we can generate a the aove famiies of vaid inequaities, for every node that has a positive exogenous fow coming in, simiar to the famiy of inequaities 3) 5

6 Figure : A two node system 3 A two node system We now consider a two node system of a network fow proem with equa spits among chosen arcs as shown in Figure Node has n outgoing arcs with arc connecting to node which has n outgoing arcs There is a positive fow entering node and there is no exogenous fow entering node We sha use the foowing notation: Node variaes f i Fow on arc i x i Binary variae indicating if arc i is used or not f d Dummy variae to equate fow on chosen arcs Fow entering arc i we assume > ) Node variaes g i Fow on arc i y i Binary variae indicating if arc i is used or not g d Dummy variae to equate fow on chosen arcs The two node system is defined as X : { X = f d, g d, f, g, x, y) R + R n+n + {, } n +n : n f i =, i= n g i = f, i= f i x i i =,, n, g i f y i i =,, n, f i f d i =,, n, g i g d i =,, n, f d f i + x i ) i =,, n } g d g i + f y i ) i =,, n As efore the fow has to e equay spit amongst the chosen arcs In this system even though the arc from node to is not chosen, some of the outgoing arcs from node can sti have y equa to ut the fow corresponding to those arcs wi aways e zero Let X n) e defined as a singe node system with n outgoing arcs, so if n = or n = our system X can e written as X n ) or X n ) respectivey We have the foowing resut for the dimension of cox ), the convex hu of X 6

7 Proposition 8 The dimension of cox ) is given y: dimcox )) = Proof See Appendix We now present some vaid inequaities for X Proposition 9 The foowing inequaities are vaid for X : dimcox n ))) if n =, n dimcox n ))) + if n =, n + n if n =, n n + n ow x + j x j + kk + )f d + y i k + )n )g i + k + )n )g d k + ) i, k6) x + x j + kk + )f d y i + k + )g i + k + ) g j k + )g d k ) i, k 7) j j i x + y i n g i n g j + n g d i 8) j i x y i + n )n g i n )n g d i 9) Proof To prove the vaidity of the inequaities 6) we wi consider the foowing four cases Take any point in X and et x j = and y j = m Denote the difference etween the eft hand side and right hand side y LHS - RHS) i) x = and y i = We have LHS - RHS) equa to for a integer k + ) + kk + ) ) = k + ) ) ) ii) x = and y i = Since x = even though y i = we sti have g i = We have LHS - RHS) equa ) ) to k+) + iii) x = and y i = We have LHS - RHS) equa to + ) + kk + ) + k + )n ) m for a integer ) k + 3) + k + 3k + = k + 3) ) k + ) ) ) iv) x = and y i = We have LHS - RHS) equa to + ) + kk + ) + k + )) ) ) = k+) for a integer Simiar to aove to prove the vaidity for 7) we aso consider four cases Take any point in X and et xj = and y j = m i) x = and y i = We have LHS - RHS) equa to for a integer k ) + kk + ) ) = ) ) k ) + k 7

8 ii) x = and y i = Since x = even though y i = we sti have g i = We have LHS - RHS) equa to ) ) + kk + ) k )) = k+) iii) x = and y i = We have LHS - RHS) equa to + ) + kk + ) ) + k + ) k + ) k ) m for a integer ) k + ) + kk + ) = iv) x = and y i = We have LHS - RHS) equa to + ) + kk + ) = for a integer k + ) ) + k + ) + +k + )m m m ) k + 3) + kk + ) + k + ) = ) ) ) k + ) k ) m k + 3) ) ) We wi prove the vaidity of the set of inequaities 8) y considering the foowing four cases Take any point in X and et x j = and y j = m i) x = and y i = We have the LHS - RHS) equa to ii) x = and y i = We have LHS - RHS) equa to iii) x = and y i = We have LHS - RHS) equa to n + n m, since m iv) x = and y i = We have LHS - RHS) equa to + n n ) m n m ) m +n m = n since Simiary to prove the vaidity of 9) we consider the foowing four cases Take any point in X and et xj = and y j = m i) x = and y i = We have eft hand side equa to ii) x = and y i = We have eft hand side equa to iii) x = and y i = We have eft hand side equa to n )n m m n ) since y i = ) iv) x = and y i = We have eft hand side equa to n )n m n )n m = 33 An aternate two node system We consider a sight variation of the aove mentioned two node system where we assume that there is positive incoming fow into oth nodes as shown in Figure Let the incoming fow into node and e and respectivey The rest of the variaes are defined simiary as in the previous section 8

9 Figure : An aternate two node system We can represent the set of feasie points as { X + = f d, g d, f, g, x, y) R + R n+n + {, } n +n : n f i =, i= n g i = f +, i= f i x i i =,, n, g i f + )y i i =,, n, f i f d i =,, n, g i g d i =,, n, f d f i + x i ) i =,, n } g d g i + + f ) y i ) i =,, n We have the foowing resut for the dimension of cox + ), the convex hu of X +, Proposition The dimension of cox + ) is given y: dimcox + )) = dimcox n )) if n =, n dimcox n )) if n =, n + n if n =, n n + n ow The proof of the aove proposition is simiar to that of Proposition 8 and is not repeated here In the foowing proposition we present some vaid inequaities for X + Proposition The foowing inequaities are vaid for X + : 9

10 x + j x j + kk + )f d + + k + ) )y i k + )n )g i + k + )n )g d k + ) + k + ) i k ) x + + n )y i n n )g i + n n )g d n i ) x + j x j + kk + )f d + k + ) g j k + ) + k + ) k ) x + n + )y i n g i n g j + n g d i 3) j i yi n gi + n g d ) Proof We wi prove the vaidity of the set of inequaities ) y considering the foowing four cases Take any point in X + and et x j = and y j = m i) x = and y i = We have LHS - RHS) equa to for a integer + kk + ) + n )k + ) m k + ) k + ) k + ) ) since m n, as y i = ) ii) x = and y i = We have LHS - RHS) equa to for a integer + kk + ) + + k + ) k + ) k + ) k + iii) x = and y i = We have LHS - RHS) equa to for a integer + ) + kk + ) ) + kk + ) iv) x = and y i = We have LHS - RHS) equa to for a integer ) ) + + k + )n ) m + m ) k + ) k + ) + k + ) k + ) since m n, as y i = ) = k + 3 ) ) + ) + kk + ) + + k + ) k + ) k + ) = k + ) )

11 Simiary to prove the vaidity of ) we consider the foowing four cases Take any point in X + and et xj = and y j = m i) x = and y i = We have LHS - RHS) equa to n n ) m n since m n ) ii) x = and y i = We have LHS - RHS) equa to n n n ) m + n n ) m n = iii) x = and y i = We have LHS - RHS) equa to + n n ) m + ) m n + n since m n ) and is since n ) iv) x = and y i = We have LHS - RHS) equa to + + n n = To prove the vaidity of ) we consider the foowing two cases Take any point in X + and et x j = and y j = m i) x = We have the LHS - RHS) equa to + kk + ) + k + ) k + ) k + ) = ) k+ ) for a integer ii) x = We have LHS - RHS) equa to + ) + kk + ) + k + ) + ) k + ) k + ) k + 3 ) ) for a integer To prove the vaidity of 3) we consider the foowing four cases Take any point in X + and et x j = and y j = m i) x = and y i = We have LHS - RHS) equa to n m + n m ii) x = and y i = We have LHS - RHS) equa to n + n m n m ) m + n m = iii) x = and y i = We have LHS - RHS) equa to n m ) iv) x = and y i = We have LHS - RHS) equa to + n + n m + m = + n n ) n m ) + ) = n + ) + n m + m ) since m + ) + n m m + ) m since n ) For the proof of vaidity of ) note that it is sufficient to show that the inequaity is vaid for the case when x = Let y i = m, then LHS - RHS) is equa to m n + n m = m ) n m ) since m n ) 3 A mutipe node system Let us now consider a set of nodes P i = {,,, P } on a path to a node i as depicted in Figure 3 assume that i > ) For each p =,, P, et n p denote the numer of outgoing arcs from node p and p ) denote the exogenous fow into node p The inary variae x p for p =,, P corresponds to arcs going from p to p +, and for p = P, corresponds to the arc from node P to i If x p = for a p =,, P, then a ower ound on the fow into node i is i + P i ) where P i ) = + n + n n n n n P P n n n P

12 P i P i O i P i Figure 3: Path to i Thus, the foowing inequaity is vaid f di i + P i ) ) + P i ) x p P i, z i p P i where f di is the fow on any used arc from i and z i is the numer of out-going used arcs We can now appy the idea in ) to inearize the aove inequaity and otain the foowing resut Proposition The foowing inequaities are vaid for the mutipe node system defined aove: kk + )f di + i + P i )) x j kk + )P i ) x p k + ) i + P i )) kk + )P i ) P i j O i p P i where O i is the set of outgoing arcs from i k =,, n i 5) The proof of the vaidity omitted here) of 5) is constructed y otaining the convex hu of n points in a two dimensiona pane, exacty simiar to that of ) 35 Separation In order to use the vaid inequaities in a ranch and cut fashion we need to separate them, ie, find out which inequaities are vioated y a given fractiona soution Lets ook at the famiy of inequaities given y ) Given a fractiona soution x, f d ), et us define hk) as hk) = kk + ) f d + n x i k + ) 6) In order to find a vioated inequaity of the form ) we need to find a k n such that hk) < From the definition of hk), it is cear that it is a convex quadratic function in k since f d > ) Let k e the point where hk) reaches its minimum We know from simpe cacuus that k = fd Now if hk ) then we know that hk) k, otherwise we can find the two roots k and k of hk) and hk) < for a the integers etween k, k ) Aove is true ony if oth the roots of hk) are rea, if oth the roots are not rea then hk) < k Simiary if one of them is rea, say k, then we know if k < k, then hk) < for a k > k and if k > k, then hk) < for a k < k Based on this information can check for what vaues of k [, n ], the vaid inequaity is vioated This gives a constant time separation routine for famiy of vaid inequaities ) We can use the same procedure as descried for famiy of vaid inequaities ) and ) for the case when S = This is the case for which we know that the inequaities are indeed facet defining It is not cear how to separate efficienty for sets S with size stricty igger than i=

13 The famiy of vaid inequaities descried for the two node susystems as descried in Propositions 9 and can e easiy separated using the procedure descried aove However it is not cear how to separate the inequaities 5) descried for the mutipe node system A ranch-and-cut impementation We incorporated the cuts derived in the previous section within a ranch-and-cut agorithm to sove the OSPF weight setting proem We used CPLEX 9 caae iraries to impement the ranch and cut agorithm Cuts were added using the cut caack function as they were vioated Computations were done on a Inte Pentium Zeon machine with GHz speed and GB RAM running Linux kerne 7 Test Proems We experimented using some proems taken from iterature and some randomy generated proems To generate random networks we first fixed the numer of nodes and numer of demand pairs, and then an edge etween two nodes was randomy added with a proaiity of 5 If the random graph otained was not connected, more edges were added to ensure connectedness Capacities and demands were aso randomy generated Tae provides the numer of nodes, edges, demand pairs and numer of inary variaes in the MILP modes for the proems considered The proem names starting with n are randomy generated The proems Pioro7, Piorow and Piorow are taken from [8] The proem snh is otained from [9] The remaining proems, ie, graph and cadata are are proprietary teecommunication networks Pro Nodes Edges Demand pairs Binary Vars n5d 5 6 nd5 5 nd nd 336 nd nd n5d n5d n5d n5d n5d n5d graph 8 68 cadata snh Pioro7 7 Piorow Piorow Tae : Size of the proems Cuts We identified singe node susystems X ) and added the corresponding vaid inequaities It is easiy seen that the cuts ) are a specia case of 5) when the path ength is zero ie, P = ) There are exponentiay many cuts of type 5) and it is not cear how to separate these efficienty Moreover, the cuts ecome weak, in genera, as we increase the ength of the the path P So in our ranch-and-cut procedure 3

14 we ony use the cuts for which the path ength P For the famiy of inequaities ) we oserved after doing computationa experiments that the inequaities ecome weaker as the size of set S increases, so we just use vaid inequaities for S =, The famiy of vaid inequaities 5) are facet defining for ony S = Proposition 7), so these are the ony ones that are used in our ranch-and-cut procedure The vaid inequaities that can not e separated efficienty as discussed in Section 35 are added to the CPLEX cut poo We aso identified two node susystems, as shown in Figure, in our OSPF mode and used the cuts 6),7),8) and 9) Simiary we identified susystems as shown in Figure and added the vaid inequaities ),),),3) and ) A these famiy of inequaities are poynomia in numer and can e easiy separated as discussed in Section 35 Inequaities that coud not e separated efficienty were added directy to the CPLEX cut poo The rest of them were added, using the cut caack function of CPLEX, as they were vioated At any given fractiona node a vioated vaid inequaities are added through cut caack) 3 Heuristics We impemented some asic heuristic methods to start CPLEX with a good initia soution Loca search We know that any set of weights on edges) is feasie for the OSPF weight setting proem So we impemented a weight adjustment heuristic in which we start with a set of weights a edge weights are set to ) and after routing the fow ased on these weights we adjust the weights, increasing for the edges with arger oad and decreasing for edges with smaer oad, in hope of decreasing the maximum oad in the network This procedure is repeated a fixed numer of times and the est soution otained over a the iterations is stored IP heuristic We reaized that the proem with ojective as the sum of the oads rather than maximum deviation is much easier to minimize And since the feasie region remains the same this IP provides a feasie soution to our proem with ojective as minimizing the maximum deviation So in the heuristic stage we soved an IP with ojective as sum of oads on each arc and in order to get etter starting soution, we aso added the maximum oad variae with certain penaty) to our ojective The more we increased the penaty on the maximum oad variae, harder the proem ecame to sove If the penaty on the maximum oad variae is much arger then the proem asicay reduces to our origina proem We did some computationa experiments to determine the est penaty Since we did not want so spend too much time on the heuristic IP, so we put a time imit of seconds on this procedure and coect the est soution otained We aso kept track of the est soution otained in terms of the maximum oad amongst a the feasie soutions otained We got the est soution from oth the heuristics and started our weight setting proem with the one with etter ojective function vaue It was oserved that the IP heuristic outperformed the oca search heuristic most of the times An advantage of getting a good starting soution is that the MILP mode does not have any capacity constraints on the fow that can e routed on any edge, and good heuristic soutions provide an upper ound on the capacity This faciitates generation of genera fow cover type of cuts which hep in the ranch-and-cut procedure Symmetry The integer programming formuation for the weight setting proem has a ot of symmetry issues Given a feasie soution with certain oad, another feasie soution with the same ojective can e otained y changing a x variae from to and keeping the fow zero In order to get rid of this issue we added ower ounding inequaities, which impy that if the arc is chosen then there has to e a certain positive fow on that arc These vaid are easy to otain for the arcs emanating from the nodes with positive exogenous

15 incoming fow For the simpe singe node system we can add the foowing inequaities f i i n x i For the other arcs we find a path from the cosest source node and add an inequaity simiar to the one otained for the mutipe node system This does not competey remove the symmetry proem ut adding these inequaities hep in the computations 5 Resuts In Tae we provide the optimaity gap of the est feasie soution returned after 8 seconds The reason we chose to stop our computations after 8 seconds is ecause after around 8 seconds the gaps did not change must at a when the proem is given to CPLEX, so either the proem woud e soved to optimaity efore that time or it woud e soved at a The first coumn is the name of the proem The second coumn provides optimaity gaps for the soutions returned y the defaut CPLEX sover The third coumn provides optimaity gaps for the soutions for our ranch-and-cut procedure when a the aove defined cuts are added The fourth coumn presents resuts after adding the heuristic methods and adding a the cuts otained in the previous sections for penaty of 5 on the maximum oad ojectivesee discussion in Section 3) The fifth and the sixth coumns present resuts when penaty is set to and respectivey Heuristic + Cuts Pro CPLEX Cuts Penaty5) Penaty) Penaty) n5d % % % % % nd5 % % % % % nd6 6% 6% % 735% 35% nd 3% 39% 8% 8% % nd 963% 89% 89% % 987% nd5 397% 95% 3% 99% 77% n5d5 85% 337% 6% 7% % n5d6 7% 3333% 68% 789% 36% n5d7 7983% 3536% 387% 5% 783% n5d8 57% 367% 337% 8% % n5d9 538% 833% 833% % % n5d 765% 63% 688% 385% 3% graph 76% % % % % cadata - 8% 8% 83% 8% snh - 358% 9% 99% 99% Pioro7 57% 36% % % % Piorow - 63% 63% 63% % Piorow 9836% 558% 5% 88% % Average Gap 355% 756% 83% 9% 55% Computed over the proems for which CPLEX found a feasie soution Tae : Percentage optimaity gap after 8 secs As is evident from Tae the OSPF weight setting MILP is quite difficut to sove as-is Within the aotted 8 second time imit, the defaut CPLEX sover produced soutions with an average optimaity gap of 357%, and for three of the proems, it was not even ae to find a feasie soution When cuts are used in the ranch and ound procedure we see that there is a significant reduction in the optimaity gaps for the proems considered as seen in coumn 3 When the proposed ranch-and-cut scheme is used with a heuristic to hot start it produced feasie soutions with optimaity gaps significanty reduced from those of defaut CPLEX The penaty setting of for the heuristic procedure seems to e the est in terms of computationa resuts as in Tae We can see from these resuts that the cuts and heuristic methods are oth equay important for getting good soutions with smaer optimaity gaps 5

16 5 Concusions We proposed an integer programming approach to otain provay good soutions to the OSPF weight setting proem The key contriution is to strengthen a mixed-integer inear programming formuation of the proem using cutting panes and integrate these cuts with oca search heuristics within a ranch-andcut agorithm Computationa resuts indicate that the proposed method performs significanty etter than straight-forward use of the commercia sover CPLEX Even though we are sti not ae to sove most of the proems to optimaity within a reasonae time imit, there is evidence that the proposed methodoogy heped to get significanty tighter ounds Integrating the proposed scheme with more sophisticated heuristic schemes in the iterature may provide significant additiona enefits References [] A Bey and T Koch Integer programming approaches to access and ackone IP-network panning preprint ZIB ZR--, [] A Bey On the approximaiity of the minimum congestion unspittae shortest path routing proem Proceedings of th Conference on Integer Programming and Cominatoria Optimization IPCO 5), 5 [3] LS Burio, MGC Resende, CC Reeiro and M Thorup A memetic agorithm for OSPF routing Proceedings of the 6th INFORMS Teecom, pp 87-88, [] M Ericsson, M Resende and P Pardoas A genetic agorithm for weight setting proem in OSPF routing Journa of Cominatoria Optimization, pp , vo 6, [5] B Fortz and M Thorup Increasing internet capacity using oca search Computationa Optimization and Appications, pp 3-8, vo 9, [6] F Lin and J Wang Minimax open shortest path first routing agorithms in networks supporting the smds services Proceedings of IEEE Internationa Conference on Communications, pp , vo, 993 [7] GL Nemhauser and LA Wosey Integer and Cominatoria Optimization Wiey-Interscience Series in Discrete Mathematics and Optimization, John Wiey & Sons, New York, 989 [8] M Pioro, A Szentsi, J Harmatos, A Juttner, P Gajownicczek and S Kozdrowski On open shortest path first reated network optimization proemsperformance Evauation, pp -3, vo 8), [9] H Sakauchi, Y Nichimura and S Hasegawa A sef-heaing network with an economica spare channe assignment Proceedings of IEEE Goa Teecommunications Conference, pp 38-3, 99 [] S Srivastava, G Agarwa, D Medhi and M Pioro Determining feasie ink weight systems under various ojectives for OSPF networks IEEE etransactions on Network and Service Management, vo ), 5 6

17 Appendix Proposition The dimension of cox ) is given y: if n = dimcox )) = if n = n if n 3 Proof For n = there is ony one feasie point and for n = there are ony three feasie points it can e easiy verified that these points are affiney independent), so the resut easiy foows for n =, For n 3, ook at the foowing set of n + points in X ; x x x n f f f n f d n n n n n n n n n n n n To see that these points are affiney independent, et λ,, λ n+ e the mutipiers Then we have the foowing set of equaities, λ i + j i λ n+j + λ n+ = i =,, n 7) λ i + i n+ i= ) λ n+j + n j i ) λ i + λ n+j + n j n ) λ n+ = i =,, n 8) n ) λ n+ = 9) n λ i = 3) Sutracting 7) from 3) and 8) from 9) yieds j i λ j + λ n+i = and j i λ j + n )λ n+i = respectivey Sutracting the aove two we get λ n+i = i Pugging ack vaues of λ n+i, we get λ i = i, which yieds λ n+ = Hence the points are affiney independent and the resut foows Proposition 3 The famiy of vaid inequaities ) for X are facet defining for k n Proof Look at a the points in X that satisfy ) at equaity It is easy to see that these points are such that x i = i I, x i = otherwise, where I {,, n} and I = k or k + Let λx = λ for a the points x which satisfy ) at equaity, then we know that λ, λ ) shoud satisfy the foowing set of equaities; λ i + λ n+i + k k λ n+ = λ I {,, n}, I = k 3) λ i + λ n+i + k + k + λ n+ = λ I {,, n}, I = k + 3) 7

18 From 3), taking appropriate sets I and sutracting we get λ i + k λ n+i = λ j + k λ n+j Simiary from 3) we get λ i + k+ λ n+i = λ j + k+ λ n+j From these two equaities we get λ i = λ j and λ n+i = λ n+j Pugging it ack into 3) and 3) we get two equaities with four unknowns we denote λ i = λ i and λ n+i = λ n+ i) A genera soution to aove equaities 33) and 3)is given y: λ λ n+ λ n+ = kk + ) µ + λ k + ) Hence the genera soution to 3) and 3) can e written as: λ λ n λ n+ = µ + λ n λ n+ kk + ) λ k + ) kλ + λ n+ + k λ n+ = λ 33) k + )λ + λ n+ + k + λ n+ = λ 3) µ The first vector is the vector defining the vaid inequaity and the second vector is the one defining the equaity in the description of X Hence the resut foows from Theorem 36 in [7] We aso require k n to get adequate numer of affiney independent points Proposition 5 The famiy of vaid inequaities ) for X are facet defining for a S with S = Proof We prove the facet defining property when S = Let S = {j}, then the inequaity ) ecomes, µ kk + )f d + i j x i x j + k + )f j k + ) j, k n 35) To prove the facet defining property, ook at a the points in X that satisfy 35) at equaity The points are; x j =, x i = i I, ow, I {,, j, j +,, n}, I = k or k + x j =, x i = i I, ow, I {,, j, j +,, n}, I = k or k + Let λx = λ for a the points x which satisfy 35) at equaity, then we know that λ, λ ) shoud satisfy the foowing set of equaities; λ i + λ n+i + k k λ n+ = λ I {,, n}, I = k 36) λ i + λ n+i + k + k + λ n+ = λ I {,, n}, I = k + 37) λ i + λ j + λ n+i + k + λ i + λ j + λ n+i + k + k + λ n+j + k + λ n+j + 8 k + λ n+ = λ I {,, n}, I = k 38) k + λ n+ = λ I {,, n}, I = k + 39)

19 It is easy to see that λ i = λ & λ n+i = λ n+ i j Pugging it ack into 36)-39) we get the foowing four inequaities with six unknownsdenote λ i = λ, λ n+i = λ n+ i j); kλ + λ n+ + k λ n+ = λ ) k + )λ + λ n+ + k + λ n+ = λ ) kλ + λ j + k k + λ n+ + k + λ n+j + k + λ n+ = λ ) k + ) k + )λ + λ j + λ n+ + k + k + λ n+j + k + λ n+ = λ 3) A genera soution to aove equaities )-3) is given y: λ λ j λ n+ λ n+j = µ + λ n+ kk + ) λ k + ) The first vector is the vector defining the vaid inequaity 35) and the second vector is the vector defining the equaity in the description of X Hence the resut foows from Theorem 36 in [7] We aso require k n to get adequate numer of affiney independent points Proposition 7 The famiy of vaid inequaities 5) for X are facet defining for S =, n 3 Proof We prove the facet defining property for the case when S = Let S = {j}, then the inequaity ecomes, f d + x j + i j µ f i ) Now ook at a the points that satisfy ) at equaity These points are precisey, For each k, x k =, x i = i k For each k j, x j =, x k =, x i = i j, k For each k, j, x j =, x k =, x =, x i = i j, k, Let λx = λ for a the points x which satisfy ) at equaity, then we know that λ, λ ) shoud satisfy the foowing set of equaities; λ k + λ n+k + λ n+ = λ k 5) λ k + λ j + λ n+k + λ n+j + λ n+ = λ k j 6) λ k + λ j + λ + 3 λ n+k + 3 λ n+j + 3 λ n+ + 3 λ n+ = λ k,, k j 7) We can easiy derive from 5)-7) that λ k = λ & λ n+k = λ n+ k, k j Pugging the vaues ack into 5)-7) we get four equations with six unknowns denote λ k = λ & λ n+k = λ n+ k j): λ + λ n+ + λ n+ = λ 8) λ j + λ n+j + λ n+ = λ 9) λ + λ j + λ n+ + λ n+j + λ n+ = λ 5) λ + λ j + 3 λ n+ + 3 λ n+j + 3 λ n+ = λ 5) 9

20 A genera soution to 8)-5) is given y : λ λ j λ n+ λ n+j = λ n+ λ µ + The first vector is the vector defining the vaid inequaity in question and the second vector is the vector defining the equaity in the description of X Hence the resut foows from Theorem 36 in [7] Proposition 8 The dimension of cox ) is given y: dimcox )) = µ dimcox n ))) if n =, n dimcox n ))) + if n =, n + n if n =, n n + n ow Proof Case n = The system reduces to X n ) since the arc outgoing from node has to e chosen and it serves as an exogenous fow entering node Hence it just reduces to a singe node system with positive incoming fow and n eaving arcs Case n = If n = we can easiy check that there are ony four points in X a of which are affiney independent, which proves the resut for this case For n 3, first note that since n = a the feasie soutions satisfy g = g d, hence the rank of the equaity set of convex hu of X is at east 3, hence the dimension of convex hu is at most n = n + But we have n + affiney independent points in X, n + are the same as defined in Proposition aong with y, g and g d ) The ast point comes from the fact that when x = we can have oth y = and y = and these points are affiney independent, which competes n + points we needed to prove the dimension Case n =, n We know from Proposition that for the case of n =, the equaity set for convex hu of X n ) is of rank 35-) The same equaity set must e present in convex hu of X, since X contains a the constraints of X n ) So the rank of the equaity set of convex hu of X must e at east 3 + comes from the equaity for second node) Hence the dimension of convex hu of X is at most 5 + n + = + n We now show 3 + n affiney independent points in X x x f f f d y y y n g g g n g d The matrix aove gives us n + 3 points, it can e easiy verified that the points are affiney independent Case n 3, n In this case we have the rank of equaity set of convex hu of X is at east, hence dimension of convex hu of X is at most n + + n + = n + n We now show n + n + affiney independent points in x

21 x x x n f f f n f d y y y n g g g n g d n n n n n n n n n n n n We have n + n + points aove, which can e easiy seen to e affiney independent This competes the proof