Connectivity Upgrade Models for Survivable Network Design

Transcription

1 January 2006 McCombs Research Paper Series No. IROM Connectivity Upgrade Models for Survivable Network Design Anantaram Balakrishnan McCombs School of Business The University of Texas at Austin Prakash Mirchandani Katz Graduate School of Business University of Pittsburgh Harihara P. Natarajan School of Business Administration University of Miami This paper also can be downloaded without charge from the Social Science Research Network Electronic Paper Collection:

2 Connectivity Upgrade Models for Survivable Network Design Anantaram Balakrishnan Red Mc Combs School of Business The University of Texas at Austin Austin, TX 7872 Prakash Mirchandani Katz Graduate School of Business University of Pittsburgh Pittsburgh, PA 5260 Harihara Prasad Natarajan School of Business Administration University of Miami Coral Gables, FL 3324 June 2005 Abstract Infrastructure networks to transport material, energy, and information are critical for today s interconnected economies and communities, and network disruptions and failures can have serious economic and even catastrophic consequences. Since these networks require enormous investments, network service providers emphasize both survivability and cost-effectiveness in their network design decisions. This paper addresses the survivable network design problem, a core model to address the cost and redundancy tradeoffs facing network planners. Using a novel connectivity upgrading strategy, we develop several families of inequalities to strengthen a multicommodity flow-based formulation for the problem. By raising the linear programming lower bound, these inequalities not only provide better performance guarantees for heuristic solutions but also accelerate exact solution methods and improve heuristic performance. We propose a heuristic strategy that iteratively rounds the fractional values, starting with the linear programming solution to our strong model. Extensive computational tests confirm that the valid inequalities, cutting plane algorithm, and heuristic procedure are very effective, and their performance is robust to changes in the network dimensions and desired connectivity structure. Our solution approach generates tight lower and upper bounds that are less than 0.7% apart on average. Subject Classifications: Integer Programming: Cutting Planes and Heuristics, Networks/Graphs Area of Review: Optimization

3 . Introduction As economies and societies have become increasingly interconnected, they depend heavily on robust and resilient infrastructure networks to transport material, energy, and information efficiently and without interruption. Tight integration of geographically dispersed supply and service chains has made organizations highly vulnerable to network disruptions. For instance, disruption of computer and communication networks to a call center in Asia can lead to severe degradation of service levels to customers. Several recent examples highlight the serious ripple effects of network disruptions. In 2003, the failure of a portion of the electrical grid in the midwest left many communities in the US and Canada without power for several days, and caused several billion dollars in economic losses (Anderson and Geckil 2003). Disruptions of logistics systems due to weather, work stoppages (e.g., Burton et al. 2002), and natural disasters (e.g., Lewis 2005) have also forced companies to incur significant additional expense to keep their supply chains replenished and running. Rupture of communication links to critical service providers such as airports, hospitals, and emergency services can have grave consequences. To reduce the high economic and social costs of network disruptions, service networks must be capable of quickly rerouting traffic when links fail (e.g., Berman 200). In turn, this capability depends on the topological design of the network. In particular, the network design must contain alternate routes to permit traffic rerouting and restoration when disruptions occur. This paper addresses the survivable network design problem, a core optimization model needed to support network configuration and infrastructure decisions. The model addresses the cost versus robustness tradeoff by minimizing total network cost while incorporating survivability requirements in terms of the minimum number of alternate (edge-disjoint) paths needed between node pairs. We propose a flow-based formulation for this integer programming problem, develop structural insights to strengthen the model by upgrading the node connectivity levels, introduce several families of valid inequalities that increase the linear programming lower bound, propose an optimization-based heuristic strategy, and present computational results to demonstrate the effectiveness of the approach. Network infrastructures such as telecommunication networks and electricity distribution systems require enormous fixed investments. However, emphasizing cost minimization alone can result in sparse network designs (e.g., tree networks) that merely interconnect the terminal nodes or customers without providing any redundancies for re-routing traffic in the event of network contingencies and disruptions. At the other extreme, providing redundancy for all customers can be prohibitively expensive and unnecessary. Typically, alternate paths are needed only between

4 important customers, such as major government, business, and emergency service sites, whereas smaller customers might be willing to tolerate some service degradation. The network planner s challenge is to design a cost-effective network that provides differing levels of connectivity and protection for different customers. Optimization models play a vital role in this decision since even modest percentage cost reductions, obtained by judiciously installing redundancy, can lead to millions of dollars of savings. The survivable network design (SND) problem supports long-term and strategic decisions regarding network configuration. In the typical hierarchy of decisions, network design precedes tactical and operational decisions on capacity allocation, routing, and restoration policies. Given the set of nodes to be interconnected with their associated connectivity requirements, and available point-to-point links or edges, each with a fixed cost, the SND problem entails selecting a minimum cost network that meets the connectivity requirements. In this paper, we focus on protection against edge failures by specifying connectivity requirements in terms of the minimum number of edge-disjoint paths between pairs of nodes. Thus, if two nodes are connected by ρ edge-disjoint paths, then these two nodes can continue to communicate even if any ρ edges of the network fail. We say that a node is a critical node if it requires two or more edge-disjoint paths to other critical nodes. We refer to less important nodes that only need to be reached (via a single path) as regular nodes. We also permit optional intermediate nodes called Steiner nodes that need not be connected but can serve as junction points to minimize the network cost. Related past literature has largely focused on studying special cases of the SND problem (e.g., problems on specialized graphs or with special connectivity structures) or on developing and analyzing approximation algorithms. Grötschel et al. (995b), and Raghavan and Magnanti (997) provide comprehensive summaries of the SND literature. In recent research, Hsu and Hu (998), Luebke and Provan (2000), Huygens et al. (2004), and Winter and Zachariasen (2005) study variants of the two-connected Steiner network problem, a special case of the SND problem in which all nodes have connectivity requirement of either two or zero. For series-parallel graphs, Raghavan (2004) and Kerivin and Mahjoub (2005) establish polynomial solvability of SND problems with low or even connectivity requirement for all nodes. For survivable network design with edge- and node-disjoint paths, Jain et al. (2002) discuss approximation algorithms and establish worst-case performance guarantees. Grötschel et al. (992, 995a), Balakrishnan et al. (2004), and Magnanti and Raghavan (2005) develop various strategies to strengthen the SND problem formulation. Rajan and Atamturk (2004) address the problem of designing networks with adequate restoration capacities. As this brief review suggests, the SND model and its 2

5 variants continue to provide a fertile and challenging research agenda. The SND problem generalizes several difficult optimization problems such as the traveling salesman and the Steiner network problem, and so is itself computationally intractable. Solving the SND problem effectively requires developing insights about the solution structure, and exploiting these insights to accelerate the solution procedure. We focus on model enhancements that can improve the performance of linear programming-based exact and heuristic solution methods; in particular, our work seeks to strengthen the model formulation by adding valid inequalities that can increase the optimal value of the problem s linear programming (LP) relaxation. Our research provides several contributions. First, we develop a new modeling framework for the SND problem, based upon a flow formulation. Most previous work on the SND problem considers a cutset formulation, with exponential constraints, that is convenient for analyzing the worst-case performance of LP relaxations and approximation algorithms (see Section 2). Our flow-based formulations are not only compact (containing a polynomial number of variables and constraints), but also well-suited for our modeling enhancements. As our reference model, we first present a flow-based formulation with O(n 2 ) commodities (where n is the number of nodes), and show how we can condense this model into an equivalent formulation with just O(n) commodities. Our second and main contribution consists of exploiting the principle of connectivity upgrading to strengthen the flow-based model. This principle relies on the observation that, although regular and Steiner nodes require connectivity levels of less than two, we can increase their effective connectivity levels depending on the topological structure of the network design solution. Such connectivity upgrades improve the LP relaxation by increasing the flows, tightening the relationship between the flow and design variables, and raising the lower bound on the number of edges in the design. We propose three broad families of inequalities, with several classes of constraints within each family, to capture these tightening opportunities, and show that these inequalities can be effective in increasing the LP value (relative to the base value). This approach of using connectivity upgrades, based on structural properties of SND solutions, to strengthen the model is new to the literature. As the third contribution, we propose an LP-based heuristic strategy that rounds up the fractional LP solution values iteratively rather than simultaneously. Finally, we implemented and tested a cutting plane algorithm, incorporating the valid inequalities, and the heuristic procedure. Our extensive computational tests on a wide range of problems with varying sizes and connectivity structures confirm the effectiveness of the valid inequalities and heuristic procedure. Our strong model formulation largely closes the integrality gap of the base model, significantly reduces the effort to solve the problem optimally, 3

6 and provides an excellent starting point for the heuristic procedure. The gap between the cost of the heuristic (or optimal) solution and the LP value of our strong model was less than 0.7% on average over 00 test problem instances for networks with up to 00 nodes. The rest of this paper is organized as follows. Section 2 formally defines the problem, and presents the basic flow-based model formulations. In Section 3, we modify the model to incorporate node connectivity upgrades, and develop and establish the validity of the various new classes of inequalities. Section 4 discusses the cutting plane procedure and LP-based heuristic strategy. Section 5 presents our computational results, and Section 6 concludes the paper. 2. Problem definition and base model formulation We define the Survivable Network Design (SND) problem over an undirected network G: (N, E) whose nodes i N = {, 2,, n} represent terminals (origins or destinations for traffic) or intermediate points, and edges (i, j) E, with positive cost c ij, represent available interconnections. Each node i has an associated non-negative, integer-valued connectivity level ρ i that reflects its relative importance, with more important nodes receiving better protection against edge failures by requiring higher connectivity levels. These connectivity levels serve to characterize the inter-node connectivity requirements in the following way: for any pair of nodes kl k l k, l N, the edges in the chosen network design must contain at least δ = min( ρ, ρ ) edgedisjoint paths between nodes k and l. This requirement ensures that nodes k and l can communicate even when any ( δkl ) network edges fail. The SND problem entails selecting the least cost set of edges satisfying the connectivity requirements. We refer to nodes having a connectivity level greater than or equal to two as critical nodes, and denote the set of all critical nodes in the network as C = { i N ρ i 2}. Observe that, since the network design must contain at least ρ i edge-disjoint paths connecting node i to every other node at the same or higher connectivity level, every critical node must be at least two-connected to every other critical node. We refer to the nodes, with connectivity requirement of one as regular nodes, and those with connectivity requirement of zero as Steiner nodes. Let R= { i N ρ i = } and S = { i N ρ i = 0} respectively denote the sets of regular and Steiner nodes of the given graph. Regular nodes are less important than critical nodes; the network design must span (or reach) regular nodes, but need not provide them additional protection in terms of alternate paths. Steiner nodes are intermediate points that the network design can 4

7 optionally span in order to minimize cost. We refer to regular and Steiner nodes together as noncritical nodes, and let Q = R S denote the set of all noncritical nodes in the graph. Several well-studied and difficult optimization problems arise as special cases of the SND problem. For instance, if the maximum connectivity level is one and S, then the problem reduces to the (NP-hard) Steiner tree problem. Similarly, when all nodes have the same connectivity level ρ >, we get the ρ-edge connected problem (e.g., Bienstock et al. 990, Chopra 994). In particular, if ρ = 2 and the edge costs are large (so that the solution uses as few edges as possible), the SND problem reduces to the traveling salesman problem. Since these special cases are themselves NP-hard, the general SND problem is also NP-hard. In this paper, we focus on strengthening the model formulation and effectively solving general SND problems with heterogeneous node connectivity requirements. 2. Flow-based formulation In this section, we introduce two flow-based SND formulations. We first present a fulldemand model, and then discuss improvements to reduce the size of the model without weakening its LP relaxation. The resulting star-demand model, with some auxiliary node connectivity variables, defines a base model that we later strengthen in Section 3. The flow-based formulations use binary edge-selection variables y ij, for each edge (i, j) E, to decide whether (y ij = ) or not (y ij = 0) to include edge (i, j) in the SND solution. To capture the node connectivity requirements, the full-demand model defines commodities <k, l> from each terminal (critical or regular) node k and to every other terminal node l. We set the demand for each commodity <k, l> equal to the connectivity requirement δ kl. To ensure that the SND solution routes these δ kl units of flow from node k to node l along edge-disjoint paths in the chosen design, we impose commodity-specific edge capacity constraints specifying that every selected edge can accommodate at most one unit of flow of each commodity. Let K = {<k, l>: k, l C R, k l} denote the set of all commodities. We refer to this set as the full commodity set since it contains one commodity for every pair of terminal nodes. For all edges (i, j) E and every commodity <k, l> K, let kl, f < ij > and kl, f < ji > denote the directed flow variables representing the amount of flow of commodity <k, l> from node i to node j and from node j to node i, respectively. Using the edge-selection and commodity flow variables, we can formulate the SND problem as the following mixed-integer program, which we denote as formulation [Full]. 5

8 [Full] subject to: j Z = min cy (2.) Full ij ij (, i j) E δkl if i= k, < kl, > < kl, > fij f ji = δ kl if i= l, < kl> K, (2.2) j 0 otherwise, < kl, f > y (, i j) E, < k, l > K, (2.3a), ij ij kl, ji yij kl, kl, ij f ji < f > (, i j) E, < k, l > K, (2.3b) < f >, < > 0 (, i j) E, < k, l > K, (2.4) 0 y ij (, i j) E, and (2.5) y = integer (, i j) E. (2.6) ij The objective function (2.) minimizes the total cost of the selected edges. The flow conservation constraints (2.2) ensure that, for each commodity <k, l>, we route δ kl units of flow from node k to node l. Constraints (2.3a) and (2.3b) are forcing constraints that permit commodities to flow (in either direction) on any edge (i, j) only if we include this edge in the network design. Moreover, in conjunction with the unit upper bounds (2.5) on the edge-selection variables y ij, these constraints impose a capacity of one unit for the flow of any commodity <k, l> on edge (i, j). Consequently, in any feasible solution, the δ kl flow paths for this commodity will be edge-disjoint. Constraints (2.4) to (2.6) are the non-negativity and integrality constraints. Researchers have previously focused on a cutset formulation for the SND problem (e.g., Goemans and Bertsimas 993) that uses only the binary edge-selection variables. This formulation enforces the connectivity requirements via an exponential number of constraints specifying that the number of edges chosen from any cutset must at least equal the maximum required connectivity across that cutset. The cutset model has proven useful for developing combinatorial heuristics for the SND problem and characterizing their worst-case performance. (e.g., Goemans and Bertsimas 993, Goemans et al. 994, Williamson et al. 995, Gabow et al. 998). Jain (200) showed that the LP relaxation of this model can have an integrality gap (i.e., (optimal integer value LP value) as a percentage of the optimal integer value) of up to 50%. In this paper, we employ a flow-based formulation for several reasons. First, the flow-based model contains only a polynomial number of constraints, and yet is LP-equivalent to the cutset model (by the max-flow min-cut theorem). Secondly, flow-based reformulations have proven useful to analyze, strengthen, and effectively solve related problems such as the Steiner tree problem (Wong 984) and the multi-level network design problem (Balakrishnan et al. 994). 6

9 Finally, flow-based models can also accommodate commodity-dependent costs, whereas the cutset formulation only allows edge costs. 2.2 Reducing the number of commodities The full-demand commodity set K for formulation [Full] contains O(n 2 ) commodities. Consequently, if m denotes the number of edges in the given graph G, the full-demand model contains O(m n 2 ) flow variables and constraints. We now discuss a more parsimonious, but LPequivalent, model that reduces the number of commodities, and hence the number of flow variables and constraints, by a factor of n. Let ˆ ρ = max{ ρ i : i N} denote the maximum connectivity level over all the nodes of the graph, and let V = { i N: ρ = ˆ ρ} be the set of nodes with the maximum connectivity level. We refer to nodes in V as max-critical nodes. Suppose we select any max-critical node v V as the root node, and let C(v) = C\{v}. We define a star commodity set K(v) containing one commodity l corresponding to each critical and regular node l C(v) R. Commodity l originates at the root node v, terminates at node l, and has a demand of ρ l units. We refer to this demand pattern as star demand since the associated demand graph, containing edges between every origin-destination pair, has a star topology centered at the root node v. The star commodity set K(v) contains only O(n) commodities; moreover, it is a subset of the full commodity set K. Specifically, in terms of our previous notation, the star commodity set is Kv () = { < vl, > : l Cv () R}. i For any choice of root node v V, we define the star-demand formulation [Star(v)] of the SND problem analogous to formulation [Full] but restricted to the star commodity set K(v). That is, formulation [Star(v)] is the same as formulation (2.) to (2.6), but with constraints (2.2), (2.3), and (2.4) defined over the smaller commodity set K(v) instead of the full-commodity set K. The following proposition establishes the validity and effectiveness of the star-demand formulation. Proposition. For any max-critical root node v, formulation [Star(v)] is valid for the SND problem and is LP-equivalent to formulation [Full]. Proof: Let (y F, f F ) denote any feasible solution to formulation [Full]. Since K(v) K, the design vector y F permits a flow f S such that the solution (y S = y F, f S ) is feasible for formulation [Star(v)]. Conversely, given any feasible solution (y S, f S) for [Star(v)], we next show that the design vector y S is feasible for [Full]. For any terminal node l and subset of nodes P, with v P and l P, consider the cutset of edges [ PP, ] = {( i, j) E: i Pand j P= N\ P} in the original graph that 7

10 separates the root node v from node l. Since the star-demand solution (y S, f S) satisfies the connectivity requirement between nodes v and l, it must select at least δ = min{ ρ, ρ } = min{ ˆ ρ, ρ } = ρ edges in this cutset, i.e., l l (, ) [, ] ij l vl v l y ρ. A similar observation holds for i j P P any cutset separating the root node v from another terminal node k. Now, for every commodity <k, l> in the full commodity set K, consider any cutset [ P, P ] that separates node k from node l, with k P. If v P, then the cutset [ P, P ] also separates v and l, and so y S must contain at least ρ l edges in this cutset. Otherwise, if v P, then y S must include at least ρ k edges from cutset [ P, P ]. Hence, y S contains at least min {ρ k, ρ l } = δ kl edges of this cutset, implying that nodes k and l must be connected by at least δ kl edge-disjoint paths in the star-demand solution. That is, y S defines a feasible design for formulation [Full]. Further, since the preceding arguments also hold for solutions to the LP relaxations of formulations [Star(v)] and [Full], formulation [Star(v)] is LP-equivalent to formulation [Full]. 3. Strengthening the SND model formulation Since the flow-based models are LP-equivalent to the cutset formulation of the SND problem, Jain s (200) worst-case result also applies to these models. That is, the LP value of formulation [Star(v)] (and [Full]) can be as low as 50% of the optimal integer value. Our goal is to strengthen the formulation by developing and adding new valid inequalities in order to reduce the integrality gap, thereby increasing the effectiveness of LP-based solution procedures. Identifying new valid inequalities requires first developing insights on the structure of the optimal integer and LP solutions. Accordingly, we begin by exploring why the cutset and flowbased models can have a large integrality gap. We note that, since all the edge costs are positive, the optimal solution to the LP relaxation will set each edge-selection variable y ij equal to the lowest possible value satisfying the forcing constraints (2.3a) and (2.3b), namely, the maximum flow value over all commodities in either direction on this edge. So, by sending fractional flows on the edges, the LP solution can set the edge-selection variable to a fractional value; the objective function value of the LP solution then absorbs only this fraction of the total cost c ij of each edge (i, j), leading to the integrality gap. We develop three strategies and corresponding families of valid inequalities connectivity upgrade inequalities, generalized forcing constraints, and design inequalities to strengthen the flow-based model. Connectivity upgrade inequalities attempt to increase the flow on edges (and hence the y ij values) by opportunistically raising the 8

11 connectivity levels of certain regular and Steiner nodes. Generalized forcing constraints and design inequalities exploit the structure of optimal SND solutions to increase the edge-selection values beyond the maximum flow values of individual commodities on the edge. To develop these inequalities, we first introduce (in Section 3.) a base model with some additional node connectivity variables. We then discuss (in Sections 3.2, 3.3, and 3.4) the three families of inequalities, and present several classes of constraints within each family. We also demonstrate that these inequalities can improve upon the base model by strictly increasing its LP lower bound. 3. Variable node connectivity formulation We now introduce a variant of the star-demand model that treats the connectivity levels of nodes as decision variables, rather than constant values, that equal or exceed the original (given) node connectivity levels. For each node l, let w l be a variable representing the effective connectivity level of node l. (Although we require effective connectivity variables only for noncritical nodes, we also define them for critical nodes for notational convenience.) As in the star-demand model, we select a max-connectivity node v V as the root node, but now define (n ) commodities, one corresponding to every node l N\{v}. Since all commodities originate at node v, we simplify the notation by referring to the commodity from node v to node l as commodity l (instead of commodity <v, l>). Let Lv ( ) = { l N\{ v}} be the set of all commodities. Note that we now also define commodities destined to Steiner nodes in order to accommodate later model enhancements that require sending positive flows to these nodes. We refer to any commodity that terminates at a critical or noncritical node as a critical or noncritical commodity. Let C(v) = C\{v} and Q denote the sets of critical and noncritical commodities. As before, let l f ij and l f ji be the flows of commodity l from node i to node j and vice versa on edge (i, j). Using the effective connectivity and flow variables, and the previous edgeselection variables y ij, we can reformulate the SND problem as the following variable node connectivity model, which we denote as formulation [VNC(v)]: [VNC(v)] ZVNC ( v) = min cy ij ij (3.) subject to: j (, i j) E wl if i= v, l l fij f ji = wl if i= l, l L v, (3.2) j 0 otherwise, l l f + f y (, i j) E, l L( v), (3.3) () ij ji ij 9

12 w ρ l L() v, (3.4) f, f 0 (, i j) E, l L( v), (3.5) l l ij l l ji 0 y ij (, i j) E, and (3.6) y = integer (, i j) E. (3.7) ij Constraints (3.2) are flow conservation equations, similar to those of formulation [Star(v)], except that the demand for each commodity l is variable and equals the effective connectivity level w l instead of the original connectivity level ρ l. Model [VNC(v)] combines the previous separate forcing constraints (2.3a) and (2.3b) into a single forcing constraint (3.3), exploiting the observation that, in the optimal SND solution, no commodity will simultaneously flow in both directions on any edge. Constraints (3.4) specify that the effective connectivity level w l of each node l must equal or exceed the original connectivity requirement ρ l. With positive edge costs, we can show that an optimal (LP or integer) solution to [VNC(v)] sets the effective connectivity level w l for each node l equal to the original connectivity level ρ l. So, this model is LP-equivalent to the previous flow-based models and the cutset formulation. We next discuss three families of inequalities to strengthen this model. 3.2 Upgrading the node connectivity levels The effective connectivity level w l of any node l serves as the demand for commodity l from the root node v to node l. If we can raise or upgrade this connectivity level without affecting the optimal integer solution, we can increase the edge flows, thereby potentially increasing the values of the edge-selection variables and hence the LP value. In this section, we discuss valid inequalities to increase the effective connectivity levels of regular and Steiner nodes. As background for this discussion, we partition the optimal SND solution into the following two subgraphs: (i) the backbone network which provides the required multiple edge-disjoint paths interconnecting the critical nodes, and (ii) the access network, consisting of one or more trees, that connects the backbone network to regular nodes not spanned by the backbone. Note that, although we require only single-connectivity for regular nodes, the optimal solution might nevertheless include some regular nodes in the backbone network in order to minimize cost; similarly, the backbone and access networks might span some Steiner nodes. As we explain next, these two observations underlie the node upgrade inequalities that we develop in this section. Let us first consider a regular node r that is spanned by the backbone network of an optimal SND solution. Although this node has a required connectivity level ρ r of only one, it has an 0

13 effective connectivity level w r of at least two in the optimal solution since the backbone network contains two or more edge-disjoint paths from node v to node r. Therefore, the same solution is feasible, and hence optimal, for a modified problem in which we treat node r as a critical node with a connectivity level of two. In other words, we can upgrade to two the effective connectivity requirement of any node r belonging to the backbone network of the optimal design. A similar connectivity upgrading argument applies to Steiner nodes. If the SND solution spans a Steiner node s, we can upgrade its effective connectivity requirement to one; and, if node s also belongs to the backbone network, we can further increase its connectivity level to two. We refer to this tactic of increasing the connectivity level of a noncritical node to two (or more) as criticalizing that node. To capture the connectivity upgrade opportunities, we define two additional sets of variables for every noncritical node q a node-selection variable u q and a criticality variable t q. The binary node-selection variable u q equals one if the SND solution spans node q, and zero otherwise. Although we do not require node-selection variables for regular nodes (since the SND solution must necessarily span these nodes), we nevertheless define u q, for all q R, for notational convenience, and set these variables equal to one. For each noncritical node q Q, the criticality variable t q indicates if the SND solution criticalizes this node. This variable takes the value two if node q belongs to the backbone network, one if node q belongs only to the access network, and zero otherwise. These definitions imply the following bounding constraints for the node-selection and criticality variables: 0 u q q S, (3.8a) u q = q R, and (3.8b) uq tq 2uq q Q. (3.8c) Moreover, since the value of the criticality variable is a lower bound on the effective connectivity level, we can add the following effective connectivity constraint: w q t q Q. (3.8d) q To strengthen the base model, we must identify and exploit opportunities to increase the values of the node-selection, criticality, and effective connectivity variables in the LP solution. (With just constraints (3.8a) to (3.8d), these variables will be at their lower bounds in the optimal solution, i.e., u = t = w = 0 for all q S, and u = t = w = for all q R; we need additional q q q q q q constraints to raise their values.) Increasing the effective connectivity level w q tightens the LP relaxation by increasing the flow of commodity q, thereby possibly increasing the values of some

14 edge-selection variables. Raising the values of the t and u variables not only increases the w values (via constraints (3.8c) and (3.8d)) but also permits us to formulate tighter forcing constraints relating the edge-selection variables to the flow values. Observe that the connectivity upgrading opportunities are contingent upon the network design solution. That is, we can upgrade the connectivity of a regular (Steiner) node only if the solution includes this node in the backbone (backbone or access) network. We will use the flow values of critical and noncritical commodities in any given solution to identify noncritical nodes belonging to the backbone network and Steiner nodes spanned by the access network. For instance, a regular or Steiner node belongs to the backbone network of an optimal SND solution if a critical commodity flows through this node. Such observations motivate the following valid inequalities Steiner node selection constraints If the SND solution selects any edge (s, j) incident to a Steiner node s, then this node must belong to either the backbone or access network. In this case, we can set the node-selection variable for node s to one using the following Steiner node selection constraints: u s y s S,( s, j) E. (3.9) sj l l Note that we can alternatively impose the constraints u f + f, for all commodities l L(v), s sj js specifying that the design must span Steiner node s if any commodity flows through this node. However, since y f + f, constraint (3.9) is at least as strong as (and far fewer than) these l l sj sj js commodity-wise constraints. We refer to the formulation obtained by adding to formulation [VNC(v)] the bounding and effective connectivity constraints (3.8a) to (3.8d) and the Steiner node selection constraints (3.9) as the base model. We can show that, with positive edge costs, the base model is LP-equivalent to formulation [VNC(v)]. We refer to the optimal value of the LP relaxation for the base model as the base LP value. In the following discussions, we say that a class of inequalities tightens the base model if adding these inequalities to the base model can strictly increase the LP value Edge-flow criticalizing constraints If a critical commodity flows on any edge (q, j) incident to a noncritical node q, then this node must necessarily belong to the backbone network. In this case, the following edge-flow criticalizing constraints set the criticality variables t q to two: c c t ( f + f ) + u q Q, c C( v),( q, j) E. (3.0) q qj jq q 2

15 Proposition 2. The edge-flow criticalizing inequalities (3.0) are valid for the SND problem, and tighten the base model. c c Proof. If no critical commodity flows through a noncritical node q Q, then ( f + f ) is zero, and so constraints (3.0) reduce to (3.8c). Suppose a critical commodity c flows through node q. c c Then, constraints (3.3) and (3.6) together imply that ( f + f ) is at most one, and so the righthand side of (3.0) is at most two. (Note that setting t q equal to this right-hand side value does not violate constraints (3.8c) since u q =.) Since node q must belong to the backbone network, it is at least two-connected to the root node, and so constraints (3.0) are valid. jq qj jq qj Figure shows that adding the constraints (3.0) strengthens the base model. The SND example (Figure a) contains two regular nodes, two level-2 critical nodes (i.e., critical nodes with connectivity requirement of two), and five unit cost edges. Figure b depicts the base LP solution, with an optimal value of 3. Adding the edge-flow criticalizing constraints (3.0) for the two regular nodes tightens the base model by increasing the LP value to 4 (Figure c). INSERT FIGURE ABOUT HERE For the example in Figure, the optimal integer value is also 4, i.e., the edge-flow criticalizing inequalities (3.0) are very effective since they close the integrality gap. We can also show that constraints (3.0) can close the integrality gap for SND instances with Steiner nodes Aggregate-flow criticalizing constraints Constraints (3.0) upgrade the connectivity of noncritical nodes q by considering the flow of each critical commodity on individual edges incident to node q. We can also set the criticality variable t q based on the total flow of any critical commodity c entering (or leaving) node q. In c particular, if the proportion f ρ jq jq c of commodity c s demand flowing into node q is (, ) positive, then we can criticalize node q. The following aggregate-flow criticalizing inequalities incorporate this principle: t f ρ + u q Q, c C( v). (3.) c q jq c q ( jq, ) E Proposition 3. The aggregate-flow criticalizing inequalities (3.) are valid for the SND problem, and tighten the base model. c Proof. If no critical commodity flows through the noncritical node q Q, then f ρ ( jq, ) jq c is zero, and so constraint (3.) reduces to (3.8c). Otherwise, if a critical commodity c flows 3

16 c through node q, then positive costs imply that f ρ jq jq c is at most one in an optimal solution. Therefore, the right-hand side of (3.) is at most two. Since node q must belong to the backbone network in this case, we can increase t q to two, and so constraint (3.) is valid. (, ) Figure 2 shows that inequality (3.) can strictly increase the LP value, even if the model contains the edge-flow criticalizing constraints (3.0). The network for this example (Figure 2a) contains one regular node and six level-2 critical nodes. The base LP solution costs 7 (Figure 2b); with constraints (3.0), the LP value increases to 7.2 (Figure 2c). Adding the aggregateflow criticalizing inequalities (3.), further increases the LP value to 7.33 (Figure 2d). For this example, the optimal integer solution value is 8. By increasing the number of critical node triplets (similar to the three critical nodes on the left) but preserving the remaining structure of the example, we can reduce the LP gap arbitrarily close to zero. We can develop an analogous example to show that constraints (3.) are also effective for SND instances with Steiner nodes. INSERT FIGURE 2 ABOUT HERE Aggregate-flow connectivity upgrade constraints The criticalizing constraints (3.0) and (3.) upgrade the effective connectivity levels w q by increasing the values of the criticality variable t q. Since t q 2, constraint (3.8d) raises w q to at most two. We can also directly increase w q (possibly above two) through the following two classes of aggregate-flow connectivity upgrade constraints. First, if h q denotes the maximum total flow into node q, over all commodities l L( v) \{ q}, in a solution, then we can raise the demand w q at node q to h q since the edges of the network already accommodate h q units of flow from the root node to node q. Hence, the following connectivity upgrade constraints are valid: w q f q Q, l L( v)\{ q}. (3.2a) ( jq, ) E l jq Note that since we define (3.2a) for all commodities l L( v) \{ q} rather than just the critical commodities, an upgraded regular or Steiner commodity flowing through node q could determine the value of w q. The following second class of aggregate-flow connectivity upgrade constraints considers the total flow of only the critical commodities through node q. c wq f jq 2 + uq q Q, c C( v). (3.2b) ( jq, ) E 4

17 Proposition 4. The aggregate-flow connectivity upgrade inequalities (3.2b) are valid for the SND problem, and tighten the base model. Proof. If no critical commodity flows through node q, then the first term of the right-hand side is zero, and so constraints (3.2b) are implied by the bounding and effective connectivity constraints (3.8c) an d(3.8d). When a critical commodity c flows through node q, thus permitting us to criticalize this node, we consider two cases. If the total flow g c q = f of commodity c through node q is less than or equal to two, then the first term in the right-hand side of (3.2b) is less than or equal to one, and so constraint (3.2b) raises the effective connectivity level to at most two, a valid requirement. If less than or equal to must contain demand of up to c g q. However, since ( jq, ) c g q is greater than two, then the right-hand side of (3.2b) is c g q units of commodity c enter node q, the solution c g q edge-disjoint paths from the root node v to node q, and so, we can satisfy a c g q units of commodity q. Therefore, constraint (3.2b) is again valid. c jq The example in Figure 3 shows that constraints (3.2b) can be effective even if the model contains the edge-flow and aggregate-flow criticalizing constraints (3.0) and (3.). The network for this example contains one regular node, six level-2 critical nodes, and six level-4 critical nodes. Adding constraints (3.0) and (3.) to the base model raises the LP value from 29 to 29.2; adding constraints (3.2b) further raises the LP value to (Figure 3b). INSERT FIGURE 3 ABOUT HERE We refer to constraints (3.0) to (3.2b) collectively as the connectivity upgrade constraints. Although all of these constraints seek to increase the criticality variables and effective connectivity levels, none of them systematically dominates the others. For instance, for the example in Figure, the edge-flow criticalizing constraints (3.0) are tighter than the aggregateflow criticalizing constraints (3.). Moreover, since (3.) and (3.2b) are equivalent if ˆ ρ = 2, constraints (3.0) also dominate the aggregate-flow connectivity upgrade constraints (3.2b) in this example. On the other hand, the example in Figure 2 shows that constraints (3.) are stronger than (3.0), and the example in Figure 3 shows that constraints (3.2b) dominate both (3.0) and (3.). Also, if the total flow of some commodity l L( v) is at least two, then (3.2a) is tighter than (3.0), (3.) and (3.2b). 5

18 Having examined how to strengthen the flow-based SND formulation by increasing the effective connectivity levels, we next discuss strategies to strengthen the forcing constraints that relate the edge-selection variables to the flow variables. 3.3 Generalized forcing constraints The forcing constraints (3.3) specify that the total flow of any single commodity in both directions on an edge must not exceed the value of the edge-selection variable for this edge. In some other network design contexts, we can strengthen these constraints by considering opposing flows of two different commodities on the edge. For instance, for the uncapacitated and multilevel network design problems (Balakrishnan et al. 989, Balakrishnan et al. 994), the forcing constraints generalize to f + f y for all pairs of commodities l and l ' that share the same l l' ij ji ij origin or destination. That is, these two commodities are not permitted to simultaneously flow in opposite directions on the edge. However, we can verify that this two-commodity generalization does not directly apply to the SND problem even though all the commodities have the same origin. Nevertheless, we can use the criticality variables t q to identify circumstances under which we can strengthen the forcing constraints (3.3) by considering opposing flows of commodity pairs. In this section, we develop three new classes of such strong generalized forcing constraints for the SND problem. These inequalities rely on the following result regarding optimal flows to noncritical nodes that do not belong to the backbone network. Lemma 5. The SND problem has an optimal solution that satisfies the following properties: (i) for every noncritical node q that does not belong to the backbone network, any backbone edge (i, j) of the solution carries at most half a unit of commodity q; and, (ii) on every edge of the access network, all commodities must flow in the same direction. Proof. The result is trivially true for Steiner nodes q that the SND solution does not span. Let node q be a Steiner or regular node belonging only to the access network. Since edge costs are positive, node q must belong to a tree that is rooted at a node b(q) of the backbone network. Since node b(q) belongs to the backbone network, the solution contains at least two edge-disjoint paths from the root node to b(q). Therefore, we can send half a unit of flow of commodity q on q q each of these two paths, thus satisfying the constraint f + f 2 for any edge (i, j) of the backbone network. And, within each rooted tree of the access network, all commodities must flow in the same direction on each edge. ij ji 6

19 Lemma 5, in conjunction with the node-selection and the criticality variables, permits us to strengthen the forcing constraints for each edge depending on whether the edge belongs to the backbone or access network. We next present two classes of generalized forcing constraints noncritical commodity-pair and mixed commodity-pair forcing inequalities corresponding to two possible combinations of commodities flowing on an edge. We then discuss a third class of forcing constraints for edges that are incident to noncritical nodes. Later, we show that these three classes of inequalities tighten the base model Noncritical commodity-pair forcing constraints Lemma 5 shows that two noncritical commodities q and q ' can flow in opposite directions on any edge (i, j) only if both nodes q and q ' belong to the backbone network. Since the value of the criticality variable for nodes in the backbone network is two, and Steiner nodes spanned by the SND solution have a node-selection value of one, we can model this bi-directional flow condition using the following noncritical commodity-pair forcing constraints: t ' q + tq' u q q q + uq' fij + f ji yij + qq, ' Qq, q',( i, j) E. (3.3) 2 2 Proposition 6. The noncritical commodity-pair forcing constraints (3.4) are valid for the SND problem. Proof. If the SND solution does not contain edge (i, j), or if q or q ' (or both) is a Steiner node that the solution does not span, then the appropriate variables in constraint (3.3) are zero, and the inequality is implied by the other constraints in the model. So, suppose the SND solution spans both q and q ', and includes edge (i, j). If edge (i, j) belongs to the backbone network, we consider three cases depending on whether the backbone network spans nodes q and/or q '. If the backbone network spans both nodes, then t = t ' = 2, and so (3.3) follows from the forcing q q constraints (3.3) and the edge-selection upper bound constraints (3.6) corresponding to commodities q and q ' and edge (i, j). If the backbone network spans only one of q and q ', then {( t + t ) 2} =. In this case, inequality (3.3) follows from Lemma 5 and constraints (3.3). q q ' 2 Finally, if the backbone network spans neither q nor q ', then t = t ' =, and Lemma 5 implies q q inequality (3.3). Suppose edge (i, j) belongs to one of the access trees. If q and/or q does not belong to the tree containing edge (i, j), then the flow of the corresponding commodity on edge (i, j) must be zero. On the other hand, if both q and q ' belong to this tree, then Lemma 5 implies 7

20 that commodities q and these cases, inequality (3.3) is valid. q ' cannot flow in opposite directions on edge (i, j). Therefore, in each of We can further strengthen constraints (3.3) for edges that must necessarily belong to the backbone network, if chosen, such as any edge (i, j) connecting two critical nodes i and j. For such edges, the following inequality is stronger than inequality (3.3): t ' ' q + tq' u q q q q q + uq' fij + f ji + fij + f ji yij + qq, ' Qq, q', i, j C,( i, j) E. (3.4) 2 2 The validity of (3.4) stems by noting (from Lemma 5) that, for any backbone edge, the total flow of the two noncritical commodities q and q ' on this edge cannot exceed two units,.5 units, or one unit if the backbone network contains both, only one, or neither of the nodes q and q ' Mixed commodity-pair forcing constraints Now consider a noncritical commodity q Q and a critical commodity c C(v). Any edge (i, j) in the backbone network can carry at most one unit of the critical commodity c. And, by Lemma 5, no more than one or half unit of commodity q can flow on the edge depending on whether or not node q belongs to the backbone network. Edges in the access network do not carry any critical flows, and must accommodate at most one unit of the regular commodity. These observations motivate the following mixed commodity-pair forcing constraints: t q q c c q fij + f ji + fij + f ji yij + q Q, c C( v),( i, j) E. (3.5) 2 These constraints specify that any selected edge can carry at most two units of flow if the design criticalizes node q, and at most.5 units otherwise, establishing the following result. Proposition 7. The mixed commodity-pair forcing inequalities (3.5) are valid for the SND problem Noncritical-incident forcing constraints Consider any edge ( q, j) E incident to a noncritical node q Q. This edge can be part of the backbone network only if node q also belongs to the backbone network and is therefore criticalized. In this case, two commodities can have unit flows in opposite directions on edge (q, j). Otherwise, if node q is not criticalized, this edge can only belong to the access network, and so cannot carry flows in opposite directions (by Lemma 5). The following noncriticalincident forcing constraints encode these properties: 8

21 f + f y + t u q Q,,' l l Q\{}, q l l', and ( q, j) E. (3.6) l l' qj jq qj q q Observe that, for constraints (3.6), we only consider pairs of noncritical commodities; the previous constraints in the model imply the analogous constraints when either commodity is a critical commodity. Proposition 8. The noncritical-incident forcing constraints (3.6) are valid for the SND problem. Proof. If ( q, j ) is a backbone edge, then node q is spanned by the backbone network, and so t q is two. The validity of inequality (3.6) then follows from inequalities (3.3) and (3.6). If ( q, j ) is an access edge, then from Lemma 5 all the commodities flowing on this edge must either flow into or out of node q. That is, f + f y, validating inequality (3.6) in this case. l l' jq qj qj We next show that the preceding three classes of generalized forcing constraints can strengthen the base model, even with constraints (3.0) to (3.2b). Proposition 9. The forcing constraints (3.3) to (3.6) can tighten the base model. Proof. Figure 4a displays an SND problem instance containing one Steiner node, two regular nodes, and three level-2 critical nodes. Figure 4b shows the optimal LP solution for the base model; this solution costs Adding the connectivity upgrade constraints (3.0) to (3.2b) increases the LP value to (see Figure 4c). Figure 4d shows the LP solution after adding the noncritical commodity-pair forcing constraints (3.3) and the mixed commodity-pair forcing constraints (3.5). This solution costs Adding the noncritical-incident forcing constraints (3.6) for just the regular nodes improves the LP value to 7.983; adding constraints (3.6) for all noncritical nodes gives the optimal SND solution, with cost 8 (Figure 4e). INSERT FIGURE 4 ABOUT HERE 3.4 Design constraints We now discuss other families of inequalities that impose variable lower bounds on the values of the edge-selection and node connectivity variables. First, we introduce two classes of constraints interior regular node constraints and an aggregate design constraint that respectively specify the minimum number of edges incident to certain regular nodes and in the overall design. Later, we present inequalities that impose lower bounds on the total connectivity upgrades over all the noncritical nodes of the network. 9