Extended Node-Arc Formulation for the K-Edge-Disjoint Hop-Constrained Network Design Problem

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1 Extended Node-Arc Formuation for the K-Edge-Disjoint Hop-Constrained Network Design Probem Quentin Botton Université cathoique de Louvain, Louvain Schoo of Management, (Begique) Bernard Fortz Université ibre de Bruxees, Département d informatique, (Begique) bernard.fortz@ub.ac.be Keywords: Survivabiity, Network Design Probem, Hop-constraints. Abstract This paper considers the K-edge-disjoint hop-constrained Network Design Probem (HCNDP) which consists in finding a minimum cost subgraph such that there exists at east K-edge-disjoint paths between origins and destinations of demands, and such that the ength of these paths is at most equa to a given parameter L. This probem was considered in the past using ony design variabes. Here, we consider an extended node-arc formuation, introducing fow variabes to mode the paths. We conjecture that our formuation eads to the compete description of the associated poyhedron and we provide an agorithm eading to good performances in terms of computing times. 1. Introduction Our society is increasingy dependent on arge-scae, networked information systems of remarkabe scope and compexity. This dependency magnifies the far-reaching consequences of system damage from attacks and intrusions. Yet no amount of security can guarantee that systems wi not be damaged. Incorporating survivabiity capabiities into a network can mitigate the risks. Here, survivabiity is the capabiity of a system to fufi its mission in a timey manner despite intrusions, faiures, or accidents. This concept of survivabiity aows networks to remain functiona when inks are severed or nodes fai, that is, network service can be restrored in the event of catastrophic faiures. Consider an undirected graph G = (V, E), where V represents the node (vertex) set, and E the set of edges or potentia inks. To express the survivabiity conditions, we need to introduce the foowing graph-theoretic concepts. Given two distinct nodes o (the origin node of demand) and d (the destination node of demand) of V, an st-path is a sequence P = (v 0, (v s, v 1 ), v 1,..., (v 1, v ), v ), where 1, v 0, v 1,..., v are distinct nodes, v 0 = o, v = d, and (v i 1, v i ) is an edge connecting v i 1 and v i (for i = 1,..., ). A coection P 1, P 2,..., P of st-paths is caed edge-disjoint (respectivey node-disjoint) if any edge (respectivey, any node except for o and d) appears in at most one path. A subgraph H of G is caed (K-)edge-survivabe (respectivey, (K-)node-survivabe) if for any o, d V, H contains at east a prespecified number K of edge-disjoint (respectivey, node-disjoint) st-paths. Suppose that each edge e E has an instaation cost c(e) R +, then the edge-survivabe network design probem, denoted by ESNDP, consists of finding a edge-survivabe subgraph of G with minimum tota cost, where the cost of a subgraph is the sum of the costs of its edges. Simiary, the node-survivabe network-design probem, denoted by NSNDP, consists in finding a minimum-cost node-survivabe subgraph of G In genera, the survivabiity requirement is not sufficient to guarantee a cost effective routing. Indeed, the aternative routing paths may be too ong and then too much expensive to be suitabe. In consequence, in order to guarantee a eve quaity of service, further technica constraints have to be added; in particuar, one can impose a imit on the enght of the routing paths, so that if one of the paths fais, the traffic may be rerouted on the second one with a quaity of service guaranteed in terms of deay. In many works on the

2 subject, the enght of the routing path is considered as the number of inks, (aso caed hops) used in the path, and then we tak about a hop-constrained path. Survivabiity probems are generay tacked by two different rerouting techniques. The first one caed oca rerouting consists in rerouting the traffic between the extremities of the faied ink. In this study we focus on the second technique caed end-to-end rerouting. This technique considers a the demands affected by the faiure and reroute these demands from their respective origins to their respective destinations. Here we focus on end-to-end rerouting with hop constraints. Three major probems of this kind have been studied before. The hop-constrainded spanning tree probem consists in finding the minimum spanning tree such that the path between the root and any other node has no more than L hops. This probem has been argey studied by Gouveia [8], [9] in which the author proposes a muticommodity network fow formuation and discuss the use of agragian reaxation to improve the ower bound. In [10], Gouveia and Requejo proposed the hopindexed formuation using the ayered representation of the graph. In [3] Dah gives a poyhedra study of the probem when the number of hops is imited to 2. The hop-constrained path probem consists on finding between two distinguished nodes s and t a minimum cost path with no more than L hops with L fixed. In [9], Gouveia proposed vaid inequaities for the L-path poytope probem and caed these inequaities st-cut inequaities. In [4], Dah and in [6], Dah and Gouveia give a compete description of the poytope when L 3 using their jump-inequaities for which the separation probem can be soved in poynomia time. In [7], Dah, Gouveia et a. extend the previous work when L = 4. The third probem is the hop-constrained network design probem. This probem focuses on finding a minimum cost subgraph such that there are at east K edge-disjoint paths between each pair of terminas. Moreover, each path must use at most L hops. In [1], Baakrishnan et a. give a mixed integer formuation for the probem when K = 1, they derive a Lagrangean reaxation to improve the ower bound. In [12], Pirku et a. propose muticommodity network fow modes and some heuristics based on inear reaxation. In [5], Dah et a. study the probem where there are for each pair of terminas K edge-disjoint paths of maximum enght L. They prove that this probem is NP-hard even when K = 1 and L = 2. In [11], Huygens et a. study the same probem but when there is ony one commodity, or one pair of terminas. They formuate an IP mode when K = 2 and L = 3 in the space of the design variabes. Bey [2] shows that the singe demand node-disjoint enght-restricted paths is NP-compete. In this paper, we propose an extended formuation for the K-Edge-Disjoint Hop-Constrained Network Design Probem in the singe commodity case. We conjecture that our formuations give a compete description of the poytope when K = 2 and we propose a poynomia time agorithm for this case. 2. Extended Formuations In order to propose a mode for the K-Edge-Disjoint Hop-Constrained Network Design Probem in the singe commodity case, we use the ayered representation proposed by Gouveia in [10]. This representation transforms the origina graph by taking into account the hop-constraints. The idea is to dupicate a the nodes L 1 times when L is the maximum number of hops. This process creates exacty L different ayers as it is iustrated in Figure 1. From the origina non-directed graph G = (V, E) we create a directed ayered graph G = (V, A ) where V represents the set of nodes and where A is the set of directed arcs in the ayered graph. V = {V 1 V 2... V L+1 } where V 1 = {o}, V L+1 = {d} and V = V \{o}, = 2,..., L 1. The set of arcs is defined by A = {{(i, j) ij E and i V and j V +1, = 1,..., L} {(i, j) ji E and i V and j, = 1,..., L}}. Note that each path between o and d in the ayered graph is made of a maximum of L V +1 2

3 Figure 1: Basic Network and its ayered representation when L = 4 arcs (hops). Linked to this ayered representation we have the foowing mode: min c ij Uij (1) L subject to j:(j,i) A U ji j: U +1 ij = K if (i = o) and ( = 0) K if (i = d) and ( = L) 0 ese, i N, L, (2) U ij {0, 1}, (i, j) A, L. (3) In this mode Uij represents the quantity of information send on arc (i, j) when arc (i, j) is the th hop in the path. The parameter c ij represents the cost to send one unit of information on arc (i, j). (1) represents the objective fonction which minimizes the tota cost when (2) are simpy the fow conservation constraints. To mode the disjoint paths we send from the origin node o a quantity equa to the number of disjoint paths K and we add the constraint (3) which is an upper bound on the fow variabes. When K = 1 this formuation is simiar to the one proposed by Dah et a. in [7]. Note that this formuation has an integra inear reaxation as it is a network fow probem. When K > 1, a fow of vaue K in the ayered graph corresponds to a soution of our probem if and ony if the transposition of the soution in the origina graph uses each arc at most once. This remark eads to add a new constraint in our mode: Uij 1, (i, j) A (4) L Our numerica experiments ead us to conjecture that the addition of (4) does not break integraity of the formuation, as the inear reaxation was integra for a the instances we tested. As a first step towards a proof, we propose a Lagrangean reaxation approach to sove the probem. We are currenty working on a proof that this agorithm converges in poynomia time to a feasibe soution of the probem whith a Lagrangean cost equa to the origina cost, therefore proving the integraity of our formuation. 3

4 3. Agorithm Our agorithm is based on a Lagrangean reaxation (caed here LR mode) based on the reaxation of constraints (4) with mutipiers β ij. The Lagrangean subprobem can be written as: min (c ij + β ij )Uij β ij (5) L subject to j:(j,i) A U ji j: U +1 ij = K if (i = o) and ( = 0) K if (i = d) and ( = L) 0 ese, i N, L, (6) U ij 1, (i, j) A, L, (7) U ij 0, (i, j) A, L. (8) The Lagrangean subprobem computes K-edge disjoint shortest paths in the ayered network. This work can be done by using e.g. the agorithm of Suurbae [13]. The Lagrangean mutipiers β are increased when an arc in the origina network is used more than once. If for some (i, j) A, (4) is not satisfied, we compute the ength of the shortest path with at most L hops not using this arc, and we increase β ij by the difference between this ength and the ength of the shortest path in the current soution. For K = 2, we conjecture that this approach converges in poynomia time to an optima soution of our probem. 4. Future work We are currenty working on the proof of convergence of our agotithm for K = 2 and for any vaue of L, and the proof of the integraity of the extended formuations. We wi aso extend this work to the muticommodity case. Because the size of the mode grows rapidy when the number of commodities increases, we wi experiment Benders decomposition on this mode. References [1] Baakrishnan, A., K. Atinkemer, Using a hop-constrained mode to generate aternative communication network design, ORSA J Comput, vo. 4, pp , [2] Bey, A., Node-disjoint enght-restricted paths, PhD TU Berin, [3] Dah, G., The 2-hop spanning tree probem, Oper Res Lett, vo. 23, pp , [4] Dah, G., Notes on poyhedra associated with hop-constrained wak poytopes, Oper Res Lett, vo. 25, pp , [5] Dah, G., B. Johannessen, The 2-path network design, Networks, vo. 43(3), pp , [6] Dah, G., L. Gouveia, On the directed hop-constrained shortest path probem, Oper Res Lett, vo. 32, pp , [7] Dah, G., N. Fodnes, L. Gouveia, A note on hop-constrained wak poytopes, Oper Res Lett, vo. 32, pp , [8] Gouveia, L., Muticommodity fow modes for spanning trees with hop constraints, European Journa of Operationa Research, vo. 95, pp ,

5 [9] Gouveia, L., Using variabe redefinition for computing ower bounds for minimum spanning and steiner trees with hop constraints, INFORMS J Comput, vo. 10, pp , [10] Gouveia, L., C. Requejo, A new agrangian reaxation approach for the hop-constrained minimum spanning tree probem, European Journa of Operationa Research, vo. 132, pp , [11] Huygens, D., A. R. Mahjoub, P. Pesneau, Two edge-disjoint hop-constrained paths and poyhedra, SIAM J. Discrete Math., vo. 18(2), pp , [12] Pirku, H., S. Soni, New formuations and soution procedures for the hop constrained network design probem, European Journa of Operationa Research, vo. 148, pp , [13] Suurbae, J. W., Disjoint paths in a network, Networks, vo. 4, pp ,