Bifurcated network design

A well studied variant of the bifurcated CNDP is the Fixed Charge Network De- sign Problem (FCNDP) that arises in the special case where at most one base capacity can be installed on each arc.

For the FCNDP Gendron and Crainic [61] describe three formulations to de- rive different Lagrangian-based relaxations that are strengthened using knapsack inequalities. They report computational results comparing different bounding procedures based on a formulation, called strong formulation, that is obtained by adding a-priori valid inequalities stating that a commodity cannot use an arc unless the fraction of capacity needed is installed on it. They report that the best results are achieved when relaxing capacity constraints, however, the end gaps between the best upper and lower bounds are very large.

Crainic et al. [46] compare two different Lagrangian relaxations obtained by dualizing capacity constraints and flow conservation constraints, respectively.

They present a detailed analysis of the resulting relaxations and compare sub- gradient and bundle methods for solving the associated Lagrangian duals on a large set of test problems corresponding to graphs with up to 30 vertices, 700 arcs and 400 commodities. Computational experiments show that bundle methods converge faster and are more robust with respect to different relaxations, prob- lem characteristics, and parameter settings. Although the best lower bounds (on average within 9% of optimality) are obtained by solving the strong formulation using a commercial integer programming solver, the proposed bounding proce- dures achieve for large scale problems very close lower bounds in a fraction of the computing time.

Crainic et al. [45] propose a tabu search meta-heuristic based on a path formu- lation that explores the space of the path-flow variables using pivot-like moves and column generation. Ghamlouche et al. [62] develop a new class of cycle- based neighborhood structures that improves the searching strategy of Crainic et al. and test such neighborhoods within a tabu-based local search. To define cycle-based neighborhoods the idea is to identify two points in the network and two paths between them that form a cycle. Then, the algorithm tries to deviate flow from one path to the other so that a different solution is obtained where previously open arcs have no flow and can be dropped. The method was further improved in Ghamlouche et al. [63] by combining the cycle-based tabu search de- scribed above with a path relinking framework [see 64]. The algorithm was tested on two sets of instances also used in Ghamlouche et al. [62] comprising FCNDP instances with up to 100 nodes 700 arcs and 400 commodities. The comparison with the cycle-based tabu search and the integer programming solver CPLEX [43] shows that the new method, on average, outperforms the cycle-based tabu search in terms of solution quality. The algorithm shows an average gap of 2.32% and 3.08% from the best solutions found by CPLEX on the two sets of instances but achieves better solutions than CPLEX on 3 instances.

Holmberg and Yuan [73] propose a branch-and-bound algorithm based on a Lagrangian heuristic. The Lagrangian relaxation is obtained by relaxing the flow

conservation constraints so that the resulting Lagrangian subproblem is decom- posable into a problem for each arc of the network. The subproblems satisfy the integrality property and thus can be solved by the greedy principle. After solving the Lagrangian subproblems a feasible solutions is obtained by solving a multicommodity flow problem over a network defined by the subproblems so- lutions. The proposed Lagrangian schema is embedded into a heuristic branch- and-bound algorithm that uses heuristic variable fixing and different dominance rules. The algorithm is compared with the integer programming solver CPLEX on a large set of problems with up to 150 nodes 1,000 arcs and 282 commodities, and obtains better solutions or shorter computing times in 52 problems out of 65. Agarwal [1] presents a heuristic algorithm for solving a multiple facility CNDP where different base capacities are available on each arc. Here, each unit of flow between two nodes is considered as a commodity, therefore the problem can be viewed as bifurcated with the restriction that flows can be splitted in integer parts. The basic approach can be thought as a neighborhood search technique based upon local improvement. Starting from an initial feasible solution the al- gorithm selects at each step a base link {a, b} and defines an associated subnet- work. The subnetwork contains the base link itself and all pairs of links {a, i} and {i, b} for each node i in the network. A corresponding subproblem is then solved to reroute the flow within the subnetwork so as to minimize the total cost of the subnetwork. The subproblem is reduced to a Multiple Choice Knapsack Prob- lem which is solved using a dynamic programming approach. The algorithm is tested on networks having different size and topologies. For networks with up to 20 vertices the algorithm is tested by comparing its heuristic solution with the lower bound achieved by a branch-and-cut algorithm. On such instances the gap between the upper and lower bound is in most cases below 5%. A set of bigger networks with up to 99 nodes 401 arcs and 4 facilities is also considered. For these instances the paper reports the worst and best solution achieved over 3 runs of the algorithm but no evaluation of the solution quality is given.

bifurcated CNDPs.

Magnanti et al. [91] introduce the NLP and study two of its subproblems to derive valid inequalities. The first subproblem is a knapsack-type problem, called

single arc design problem, and is defined by the capacity constraint associated with

a single edge. The second subproblem, called three node network problem, is ob- tained by considering a 3-node network with an edge between each node pair.

A single arc design problem for each edge arises when relaxing in a Lagrangian fashion the flow conservation constraints. Therefore, the optimal value of the La- grangian dual of such relaxation provides an upper bound to the value of the LP-relaxation that can be achieved by adding all inequalities derived for single arc design problems. Studying this subproblem the authors introduce a new set of valid inequalities called residual capacity inequalities. They prove that adding residual capacity inequalities together with upper bound constraints (stating that the flow of a commodity on each edge cannot exceed the commodity demand) to the subproblem provides a complete description of the associated convex hull of feasible solutions. Studying the three node network problem they introduce the families of cut set inequalities and three-partition inequalities, expressing lower bounds on the total capacity that must be installed on two and three-sets par- titions of the network. These last two classes of inequalities, together with non negativity constraints, completely describe the convex hull of feasible solutions of the three node network problem.

In Magnanti et al. [89] the above inequalities are generalized to the Two Facil- ity NLP, where two facilities are available, the smallest one having unitary base capacity. Computational results over networks with up to 15 nodes show that the average integrality gap is around 8%. The authors report that cut-set inequalities alone are more effective than residual capacity inequalities alone in reducing the integrality gap. Moreover, they show that even adding a-priori a limited subset of cut-set inequalities significantly reduces the integrality gap at the root node.

Recently, Atamt ¨urk and Rajan [6] described a linear time algorithm for sep- arating residual capacity inequalities and present a cutting plane algorithm for

the bifurcated CNDP where these inequalities are separated exactly. Other valid inequalities, related to the Mixed-Integer Rounding inequalities, were introduced in Bienstock and G ¨unl ¨uk [23] and G ¨unl ¨uk [68] studying different variants of the bifurcated NLP.

Metric inequalities are a class of inequalities introduced by Onaga and Kakusho

[99] and Iri [75] to characterize the feasibility of (bifurcated) multicommodity flows that generalize the min-cut max-flow duality to Capacitated Multicom- modity Network Flow Problems (CMNFP). Metric inequalities can be viewed as Benders cuts associated with extreme rays of the dual polyhedron of the LP- relaxation of CMNFPs [see 42]. Therefore, these inequalities can be used to char- acterize the set of capacity vectors that can accommodate a feasible multicom- modity flow through the network. Metric inequalities give rise to an alternative formulation for the bifurcated NLP, called capacity formulation, that uses capacity variables only but requires an exponential number of constraints. Metric inequal- ities can be strengthened in different ways, e.g. by rounding arguments giving rise to the class of rounded metric inequalities [see 24] that contains as a special case the cut-set inequalities.

Bienstock et al. [24] consider a variant of the bifurcated NLP where the graph is directed and base capacities of unitary size can be installed independently on each arc. The authors describe two branch-and-cut algorithms based on two dif- ferent formulations enforced by valid inequalities. The first formulation is a stan- dard multicommodity flow formulation based on cut-set inequalities, flow-cutset inequalities and three-partition inequalities. The second formulation is a capac- ity formulation that is based on rounded metric inequalities. This formulation uses two other classes of valid inequalities called partition inequalities and total ca-

pacity inequalities. Partition inequalities are derived as a special case of rounded

metric inequalities and are used to bound the flow that must traverse a partition of the graph. Total capacity inequalities are obtained by applying the Chv´atal- Gomory procedure to partition inequalities. The two formulations are compared on two sets of instances with up to 27 nodes, 102 arcs and 702 commodities. The

results obtained show that the multicommodity formulation with the additional cuts grants, on average, better results.

The capacity formulation of the bifurcated NLP is also studied in Dahl and Stoer [48] and Avella et al. [8]. Dahl and Stoer [48] study a generalization of the NLP with additional survivability requirements and model it using the capacity formulation. They propose a cutting-plane algorithm where violated metric in- equalities are generated by solving a LP by column generation and introduce a new class of inequalities called band inequalities that exploit the Knapsack sub- structure of the problem. Avella et al. [8] introduce the new class of tight metric

inequalities that completely characterize the convex hull of the integer feasible so-

lutions of the problem. They present a branch-and-cut algorithm that looks for violated tight metric inequalities in two steps. First it looks for violated metric inequalities and then attempts to tighten them to derive tight metric inequalities. This algorithm is able to improve some of the results reported in Bienstock et al. [24].