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Partition inequalities for survivable network design using directed p-cycles ¨ Alper Atamturk, University of California at Berkeley. Deepak Rajan, IBM T.J Watson Research Center. Keywords: Survivable network design, partition inequalities, directed p-cycles, mixed-integer programming. 1. Overview We study the design of capacitated survivable networks using directed p-cycles (SNP). We ﬁrst introduce this methodology for designing survivable networks, and present a mixed-integer programming formulation. Next we develop valid partition inequalities for this problem, and prove that they are facets to the polyhedron of SNP under mild conditions. We study the separation problems for these inequalities and present polynomial time algorithms (exact/heuristic, as the case may be). We compare these inequalities with those developed for survivable network design using global rerouting. Finally, we present computational results that illustrate the effectiveness of the new inequalities when incorporated in a branch-and-cut framework. 2. Introduction and Related Literature Given a directed network, ﬂow costs, capacity installation costs for each edge, and a set of commodities (given as origin-destination pairs and corresponding demands), the capacitated network design problem (NDP) is to route the commodities so that the net ﬂow on any arc is at most the capacity installed on that edge and all demands are met, at minimum total cost. Network design problems become signiﬁcantly more difﬁcult when the networks have to be designed to survive failures. As a simultaneous failure of multiple network elements occurs very infrequently, here we focus on single-edge failures. A network is said to be survivable if sufﬁcient capacity exists on the edges of the network so that disrupted ﬂow can be rerouted in the event of an edge failure. To do so, requires installing spare capacity to the network. Various approaches that attempt to minimize the capacity requirement of the survivable networks have been developed. The most capacity-efﬁcient survivable networks can be designed by formulating the problem as a capacitated network design problem (NDP) for every failure scenario, linked by common capacity variables across the scenarios [2]. An optimal solution to such a scenario formulation may call for rerouting the ﬂow of com- modities unaffected by the failure. Because it is not practical to manipulate unaffected ﬂow while restoring affected ﬂows, this approach, also referred to as global rerouting (GNP), is not popular. However, GNP still serves an important purpose as it provides a lower bound on the capacity requirement of a survivable net- work. Methodologies that are implemented in practice usually involve some form of local rerouting, either by enforcing a ring-like topology (dedicated protection) on the network, or by shared local protection schemes. We study a hybrid approach for designing survivable networks, where we use directed p-cycles for shared protection of disrupted ﬂow and do not impose any topology on the network. The idea of hybrid networks was ﬁrst introduced in [5]. In this approach, one imposes no restrictions on no-failure routing, but uses failure- ﬂow patterns for all failure ﬂows. Using predeﬁned undirected cycles as failure-ﬂow patterns has been shown to be capacity-efﬁcient, and with fast reconﬁguration times. In [7] the authors present computational experiments comparing various frameworks for designing survivable networks, and noted that using directed p-cycles yields networks with comparable capacity requirements as global rerouting. In [8] the authors present polyhedral inequalities for designing survivable network using directed cycles (SDC). 3. Motivation A directed p-cycle is a logical construct on a graph used to reroute ﬂows disrupted by edge failures. In any solution to SNP, this is done by reserving sufﬁcient fractional capacity on directed p-cycles on the network. To see exactly how this works, we present a small example that also emphasizes the distinction between directed cycles and directed p-cycles, and shows why using directed cycles yields a more conservative framework. A directed p-cycle provides coverage for ﬂows in the reverse direction for the arc on the p-cycle, see arc (ba) in Figure 1. A directed p-cycle also provides one recovery path for the ﬂow on a chord, see arcs (cd) and (dc) in Figure 1. Directed cycles (in SDC) provide recovery paths only for the arcs on the cycle, and not for the chords. In the example in Figure 1, a directed cycle can be used only to reroute the disrupted ﬂow on arc (ba), and not on arcs (cd) or (dc). b c a d Figure 1: A directed p-cycle Next we compare the capacity-efﬁciency of the frameworks GNP, SDC and SNP using a small computational study. In Figure 2, we report the ratio of installed capacity for the solutions provided by GNP, SDC, and SNP to the capacity requirements of NDP. We see that survivable networks produced by SDC require 80% more capacity than NDP, whereas SNP requires only about a 45% increase. Since GNP, which provisions the lowest possible capacity for survivable networks, requires about 30% more capacity than NDP, we see that SNP requires only an additional 18% capacity, whereas SDC provisions 38% excess capacity over GNP. 2.50 GNP SNP SDC 2.00 Ratio 1.50 1.00 5 6 7 8 9 10 11 12 13 14 15 Problem Size Figure 2: Comparing capacity efﬁciency of the survivability models Finally we present a theoretical argument supporting the capacity-efﬁciency of SNP. The relative strength of methodologies can also be stated in terms of the set of feasible capacities. For any non-empty 2-partition 2 (A, B) of G, let [AB] be the set of edges with one end in A, the other in B. Let dA be the net demand with source node in A and destination node in B. To illustrate our claim about the relative strength of SDC and SNP when compared to GNP, we consider the partition inequality xe ≥ f (η), (1) e∈[AB] where η = dA , and x is the capacity variable for each edge. For an “apples-to-apples” comparison, we consider the strongest (facet-deﬁning) inequality of the form (1) for each framework. Inequality (1) is the cut-set inequality for NDP [1] when f (η) = η. For GNP [3], (1) is the same as the cardinality-k cut-set inequality when f (η) = n ∗ η/(n − 1) , where n = |AB|. For SNP, we show that (1) is a special case of the subset-Q inequalities, with f (η) = n ∗ η/(n − 1) , which is the same for GNP. This comparison indicates that the difference between capacity-efﬁciency of SNP and GNP is quite small. However, for SDC [8], inequality (1) is the cut-set inequality with f (η) = 2η. Therefore SDC yields a network with signiﬁcantly larger capacity installation than SNP does. 4. Contributions The focus of this paper is a polyhedral study of the survivable network design problem using directed p-cycles. We present classes of strong valid inequalities for SNP. Interestingly, some of the inequalities presented in this paper can also be derived as metric inequalities [6]. We compare our results with polyhedral results for global rerouting given in [3, 4], and generalize some of the results therein. The theoretical results in this work also provide valuable insight on why using directed p-cycles yields highly capacity-efﬁcient networks, and characterize when SNP yields almost as capacity-efﬁcient networks as GNP. Finally, we incorporate the partition inequalities in a branch-and-cut framework to solve survivable network design problems using directed p-cycles. We present the results of these computational experiments, which show that the inequalities often reduce computation times by an order of magnitude. References u [1] Atamt¨ rk, A., “On capacitated network design cut-set polyhedra,” Mathematical Programming, 92, pp. 425–437, 2002. o a [2] Alevras, D., M. Gr¨ tschel, and R. Wess¨ ly, “Cost-efﬁcient network synthesis from leased lines,” Annals of Operations Research, 76, pp. 1–20, 1998. [3] Balakrishnan, A., T. L. Magnanti, J. S. Sokol, and Y. Wang, “Spare-capacity assignment for line- restoration using a single-facility type,” Operations Research, 50, pp. 617–635, 2002. [4] Bienstock, D. and G. Muratore, “Strong inequalities for capacitated survivable network design prob- lems,” Mathematical Programming, 89, pp. 127–147, 2000. [5] Grover, W. D. and R. G. Martens, “Optimized design of ring-mesh hybrid networks,” Proceedings of IEEE/VDE Design of Reliable Communication Networks 2000, pp. 291–297, 2000. [6] Rajan, D., “Designing capacitated survivable networks: Polyhedral analysis and algorithms,” University of California at Berkeley, Berkeley, USA, 2004, u [7] Rajan, D. and A. Atamt¨ rk, “Survivable Network Design: Routing of ﬂows and slacks,” Telecommuni- cations Network Design and Management, eds. Anandalingam, G. and S. Raghavan, Kluwer Academic Publishers, pp. 65–81, 2002. u [8] Rajan, D. and A. Atamt¨ rk, “A Directed Cycle based Column-and-Cut Generation Method for Capaci- tated Survivable Network Design,” Networks, 43, pp. 201–211, 2004. 3

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valid inequalities, survivable network, network design, connectivity requirements, survivable networks, network design problem, connected subgraph, the network, network design problems, discrete optimization, combinatorial optimization problems, operations research, edge set, mathematical programming, valid inequality

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posted: | 5/2/2010 |

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