# A branch-and-cut algorithm for two-level survivable network design problems

TL;DR: This paper approaches the problem of designing a two-level network protected against single-edge failures by giving integer programming formulations and valid inequalities for the different versions of the problem, solving them using a branch-and-cut algorithm, and discussing computational results.

Abstract: This paper approaches the problem of designing a two-level network protected against single-edge failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs, the connection of the hubs through a backbone network (first network level), and the assignment of terminals to hubs and their connection through access networks (second network level). We consider two survivable structures in both network levels. One structure is a two-edge connected network, and the other structure is a ring. There is a limit on the number of nodes in each access network, and there are fixed costs associated with the hubs and the access and backbone links. The aim of the problem is to minimize the total cost. We give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results. Some of the new inequalities can be used also to solve other problems in the literature, like the plant cycle location problem and the hub location routing problem. HighlightsWe study two-level survivable network design problems.Two types of survivable structures are considered: rings and 2-edge connected graphs.We give an ILP formulation that models the different problem' variants.We derive strong valid inequalities and devise separation procedures for them.Some of the valid inequalities generalize other inequalities in the literature.The proposed branch-and-cut method succeeds in solving instances with up to 40 nodes.

## Summary (3 min read)

### 1. Introduction

- The authors study several two-level network design problems with survivability requirements in both levels.
- The lower level networks are called access networks and they connect the users to hubs.
- There are also studies on two-level networks with survivability requirements on the backbone network.
- They analyze worst-case performances of some heuristics for some special cases.

### 2. MIP models

- Note that this is not a restrictive assumption as one can solve the problem for different root nodes if one has not been a priori fixed.
- If an edge is used in the backbone network then, due to constraints (5) and (6), both endpoints should be hubs.
- Finally, constraints (10) and (11) are variable restrictions.
- In particular, to model the 2EC/ring and ring/ring design problems the authors add the degree constraints xðδðiÞÞr2 8 iAV : ð12Þ.
- These constraints limit to at most two the degree of a node in an access network and, together with the connectivity constraints (8), force those networks to be rings.

### 3. Valid inequalities

- This section presents several families of valid inequalities for the four variants of the 2-LSNDP.
- The feasible set of this problem is a relaxation of the 2EC/2EC network design problem obtained by dropping the constraints related with xe variables.
- The authors now present three new families of inequalities.
- There are at least rðSÞ PiA Szii distinct hubs in set V⧹S that serve nodes in S. Hence, overall at least 2 rðSÞ PiA SziiþPmk ¼ 1 zðjk : SkÞ access edges cross the cut δðSÞ.□ ð21Þ to solve the hub location routing problem.

### 4. Branch-and-cut algorithm

- The authors propose an exact branch-and-cut algorithm to solve the 2- LSNDP based on the MIP model strengthened with the valid inequalities presented in the previous section.
- The branch-and-cut approach consists of a cutting plane technique embedded into a branch-and-bound framework.
- The authors describe next the main features of the algorithm.

### 4.1. Initialization

- To start the optimization the authors solve the linear program (LP) given by the constraints (2)–(6) and variable bounds.
- To these inequalities, the authors add the degree inequalities (12) when the access networks are required to be rings, i.e., when the variant to solve is the 2EC/ring or the ring/ring.
- Similarly, to solve the design problems with required ring structure in the backbone network (i.e., ring/ 2EC and ring/ring) the authors add constraints (13).
- When the backbone network does not need to be a ring, the authors replace (13) by (7) with S¼ fig for all iAV in the initial LP.

### 4.2. Cutting plane phase

- If the optimal solution of the LP relaxation is integer, the authors check whether it is a feasible solution for the 2-LSNDP by applying the separation routines for constraints (14), (7) and (8).
- The cutting plane phase is performed only each 10 branch-and-cut nodes.
- The left hand side of all these inequalities is the same and they do not involve variables related to the backbone network.
- The authors refer to the variable values of the current fractional solution by ðxn; yn; znÞ.

### 4.2.1. Separation of inequalities (7)

- To separate constraints (7) the authors use an exact and polynomial procedure presented in [20], and inspired by the known separation algorithm for the subtour elimination constraints for the travelling salesman problem.
- The procedure consists of solving max-fow/min-cut problems on an appropriately defined support graph.
- Next the authors determine a min-cut set S V 0 with iAS and 0=2S, and finally they check the violation of inequality (7) for that set.

### 4.2.3. Separation of inequalities (14)

- Inequalities (14) can be separated in polynomial time.
- If the inequality for this choice of S is not violated, then there exists no violated inequality (14) for edge fi; jg.

### 4.2.4. Separation of inequalities (15)

- Constraints (15) are heuristically separated using a procedure presented in [11] that works as follows.
- The authors look for sets of nodes fv1;…; vpg that determine odd cycles in G0.
- The set F is formed by taking an odd number of edges from δðV0Þ with fractional values yne41=2.
- Then, the corresponding inequality is checked for violation.

### 4.2.5. Separation of inequalities (16)

- The bridge inequalities (16) can be separated exactly in polynomial time with the following algorithm.
- If the capacity of this cut is less than 2yne , the corresponding inequality (16) is violated.
- Therefore the authors use the following strategy to reduce the number of min-cut computations.
- Note that, with these changes, the separation procedure for inequalities (16) becomes heuristic.

### 4.2.6. Separation of inequalities (18)

- The capacity of the edges in E0 is set to the positive value considered for their definition.
- Therefore, again the separation problem can be solved exactly by performing a min-cut computation for each pair of nodes.

### 4.2.7. Separation of inequalities (22)

- To separate constraints (22) the authors use a heuristic procedure based on the separation algorithm for (20) described in [27].
- The edges in E0 are the edges in E, with capacity xne ; the edges connecting s with each iAV , with capacity 2ðznjiþ1=qÞ; and the edges connecting each iAV with t, with capacity 2znii.

### 4.2.8. Separation of inequalities (24)

- This section describes a heuristic separation for (24) based on an exact and polynomial-time separation for a closely related family of inequalities.
- These inequalities are the ones obtained by not considering the rounding up operator in (24), i.e. yðEðSÞÞþyðδðSÞÞþxðδðSÞÞZ.
- This separation algorithm can be adapted to deal with other inequalities (23) with m¼1, like for example when r(S) is defined as in (21).

### 4.3. Branching strategy

- The branching strategy the authors devised prioritizes the set of variables zjj.
- This is done because fixing the set of hubs contributes to determine other features of the solution.
- Branching on the other variables is done only when all variables zjj are integral.

### 5. Computational results

- The authors used CPLEX 12.5 as mixed integer linear programming solver.
- For a given number of nodes n and fix hub cost setting f j, the hardest instances are those with tighter capacity.
- To assess the influence of the different sets of valid inequalities proposed in this work, the authors compare four versions of the branchand-cut algorithm that differ by the sets of valid inequalities considered.
- The addition of constraints (18) reduces the gaps and the running times, this being more evident for Class II instances.
- This happens because the violation of constraints (17) is checked inside the separation routines for (8) and (18), and the authors are certain to find a violated constraint of the first type when any of the two latter is violated.

### 6. Conclusions

- This paper addresses a two-level survivable network design problem in which location, assignment, and network design tasks are jointly tackled.
- The authors present an integer programming formulation and valid inequalities for the problem.
- This formulation models, with minor modifications, the four possible combinations of the two singleedge protected topologies considered (rings and two-edge connected networks).
- The authors have designed an exact branch- and-cut algorithm and they have tested its performance on two sets of instances with different degree of difficulty.
- The results are satisfactory since they show that instances with up to 40 nodes, which is already a large size for this kind of problems, are solved to optimality in a reasonable amount of time.

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##### References

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### "A branch-and-cut algorithm for two-..." refers background or methods in this paper

...Klincewicz [18] reviews design problems that involve location of hubs....

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...To separate constraints (7), (8), (14), and (18), we follow approaches presented in [20] and [27]....

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...Separation of inequalities (7) To separate constraints (7) we use an exact and polynomial procedure presented in [20], and inspired by the known separation algorithm for the subtour elimination constraints for the travelling salesman problem....

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