# A comparison of routing sets for robust network design

## Summary (3 min read)

### 1 Introduction

- Given a graph and a set of point-to-point commodities with known demand values, the deterministic network design problem aims at installing enough capacity on the arcs of the graph so that the resulting network is able to route all commodities.
- The introduction of the uncertainty set leads to a robust optimization problem.
- Hence, the routing decisions can be seen as the recourse.
- In Section 3, the authors try to understand how good is the cost of the optimal capacity allocation provided by each of the routing sets, and they compare these costs among the different routing sets.

### 2.1 Problem formulation

- Then, the authors introduce the robust network design problem.
- Notice that in real applications, these costs are usually non-negative.
- In fact, the problem is very difficult to solve given that only deciding whether a given capacity allocation vector x supports D is coNP-complete for general polytopes D, see Chekuri et al. [14] and Gupta et al. [18].
- Frangioni et al. [16] and Poss and Raack [26] show under very strong assumptions on D that the optimal capacity allocations provided by dynamic routings are equivalent to the ones provided by static routings and affine routings, respectively, which are polynomially solvable when D has a compact formulation.

### 2.2.1 Static routing

- This framework considers a restriction on the second stage recourse known as static routing (also called oblivious routing).
- Hence, the resulting optimization problem is polynomially solvable.
- In the following, the authors review alternative routing sets F ′ that are less restrictive than static routings while not being as flexible as dynamic routings.

### 2.2.2 Covers of the uncertainty set delimited by a hyperplane

- Ben-Ameur [3] introduces the idea of covering the uncertainty set by two (or more) subsets using hyperplanes and proposes to use a routing template for each subset.
- The definition above implies that both routing templates y1 and y2 must be able to route demand vectors that lie in the hyperplane {d, αd = β} without exceeding the capacity.
- He further works on the framework in Ben-Ameur and Zotkiewicz [7].

### 2.2.3 Arbitrary covers of the uncertainty set

- Scutellà [27] introduces the idea of using conjointly two routing templates.
- She mentions that the complexity of RND(F2) is unknown.
- The framework described by F2 has been independently proposed for general robust programs by Bertsimas and Caramanis [11] (see also Bertsimas et al. [12]) where the authors propose to cover the uncertainty sets with k subsets and devise independent sets of recourse variables for each of these subsets.

### 2.2.4 Volume routings

- More recently, Ben-Ameur and Zotkiewicz [6] introduce a framework that shares the demand between two routing templates, according to thresholds hk for each k ∈ K.
- They prove that RND(FV) is an NP-hard optimization problem.
- Hence, they introduce simpler frameworks described below.

### 2.2.5 Affine routings

- Ben-Tal et al. [8] introduce Affine Adjustable Robust Counterparts restricting the recourse to be an affine function of the uncertainties.
- This framework has been compared theoretically and numerically to static and dynamic routings by Poss and Raack [26].
- In particular, the authors show that a compact formulation can be described for RND(Faff) as long as D has a compact description, generalizing the result obtained for static routing already.
- The authors point out that a major difference between Faff and the routing described in Section 2.2.1-2.2.4 is that the formers are build up using routing templates, so that it is implicitly assumed that flow conservation constraints (2) and non-negativity constraints (3) are satisfied.
- In opposition, routings in Faff are build up using ordinary vectors so that that satisfaction of (2) and (3) must be stated explicitly.

### 2.3 Contributions of this paper

- The objective of this paper is to compare opt(F ′) among the routing sets recalled in previous sections.
- The polytope introduced by Bertsimas and Sim [13], used for robust network design problems in [6, 23, 24, 20, 26], satisfies the assumption of (b) when the number of deviations allowed is integer.

### 3 Optimal costs

- The objective of this section is to compare the cost of the optimal capacity allocations obtained for RND(F ′) using different routing sets F ′.
- In Section 3.1, the authors describe the methodology used herein to obtain the desired relations.

### 3.1 Methodology

- Proving this inclusion is a very strong result, which holds only for closely related routing sets.
- Because it is not always possible to compare directly the routing sets themselves, the second approach is based on comparing the sets of all capacity allocations that support D when considering a specific routing set.
- The two approaches are formalized in the result below.
- Follows immediately from the definition of RND(F ′), also known as 1.
- Follows from the fact that opt(F ′) is the cost of the optimal solution of (11), also known as 2.

### 3.2 Routings that cover D

- Then, since the inequalities in (6) are not strict, the authors see easily that D̃1 and D̃1 are closed.
- It follows from Proposition 1 that the costs of optimal capacity allocations are always equal.
- The complexity of RND(F2) follows directly from the sufficiency condition of Theorem 1 and the fact that Ben-Ameur [3] proves RND(F2|) to be NP-hard.

### 3.3 Volume and affine routings

- In this section the authors compare volume and affine routings.
- The advantage of decreasing the flow sent on some arcs when the demand for a commodity rises allows to better combine different commodities within the available capacity.
- The problem must contain more than one commodity because dim(D) = 1 and D is orthogonal to the k-th axis.
- (15) Property (15) follows directly from the fact that any routing in FVS is a linear function.
- Thus, let us define Theorem 3 states that whenever D satisfies Assumption 1, one should not try to use the complex set of routings FV, since opt(FV) will never beat opt(FVS).

### 4 Non-comparable routings

- The authors compare opt(F2), opt(FV) and opt(Faff) for general uncertainty sets.
- To devise examples showing that Faff may yield more expensive capacity allocations than F2 and FV, the authors shall use the following result.
- One can easily extend these examples to larger graphs for which each commodity k ∈ K has at least two different paths from its source s(k) to its sink t(k).
- Edge labels from Figure 2(b) and Figure 2(c) represent optimal capacity allocations with dynamic and static routing, respectively.
- A routing f ∈ F that satisfies the capacity from Figure 3(b) is depicted on Figure 3(d) and Figure 3(e), for d1 and d2, respectively.

### 5 Concluding remarks

- This paper studies the optimal capacity allocation cost provided by robust network design models restricted to use specific routing sets.
- Affine routing, volume routing and its two simplifications, and the routings based on covers of the demand uncertainty set, also known as These routing sets are.
- The authors show that the routing set based on an arbitrary cover of the uncertainty is equivalent to the routing set that uses a separation hyperplane.
- An important characteristic of these routing sets is the complexity of the resulting network design problem.
- The general volume routings and the routing sets based on covers of the uncertainty set lead to NP-hard optimization problems.

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###### Q2. What future works have the authors mentioned in the paper "A comparison of routing sets for robust network design" ?

The authors show that the routing set based on an arbitrary cover of the uncertainty is equivalent to the routing set that uses a separation hyperplane. Moreover, while a finite linear programming formulation can be provided for the robust network design problem with dynamic routing under polyhedral uncertainty ( by considering only the extreme points of the demand polytope ), no such formulations are known for the problems that use the general volume routings or the routings based on covers of the uncertainty set.