## Abstract

In this paper we analyze the computational complexity of transportation infrastructure network design problems, in the presence of a competing transportation mode. Some of these problems have previously been introduced in the literature. All problems studied have a common objective: the maximization of the number of travelers using the new network to be built. The differences between them are due to two factors. The first one is the constraints that the new network should satisfy: (1) budget constraint, (2) no-cycle constraint, (3) both constraints. The second factor is the topology of the network formed by the feasible links and stations: (1) a general network, (2) a forest. By combining these two factors, in total we analyze six problems, five of them are shown to be NP-hard, the sixth being trivial. Due to the NP-hardness of these problems, a genetic algorithm is proposed. Computational experiments show the applicability of this algorithm.

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## Acknowledgements

Mozart Menezes and Juan A. Mesa were partially supported by project MTM2015-67706-P (MINECO/FEDER,UE). Federico Perea was partially supported by the Spanish Ministry of Science, Innovation, and Universities, under projects “ OPTEP-Port Terminal Operations Optimization” (No. RTI2018-094940-B-I00) and MTM2016-74983, financed with FEDER funds, and by the *Universitat Politècnica de València* under grant SP20180164 of the program *Primeros Proyectos de Investigaciòn (PAID-06-18), Vicerrectorado de Investigaciòn, Innovaciòn y Transferencia.* All this support is gratefully acknowledged.

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## Appendix

### Appendix

### Proofs

We begin this appendix with the proof of Proposition 1.

### Proof

We prove this result by reduction from the knapsack problem. Consider the knapsack problem with item values equal to \(b_i\), and item weights equal to \(w_i\), \(i=1,\dots ,n\), and capacity \(w_{\max }\), which is formulated as:

Consider the following input data: \(N=\{v_0,v_1,\dots ,v_n\}\), \(E = \left\{ \{v_0,v_i\}:i=1,\dots ,n\right\} \), \(W= \{(v_0,v_i):i=1,\dots ,n\}\), \(C_{\max } = w_{\max }\), \(c_{0,i} = 0\), \(c_i = w_i\), \(i=1,\dots ,n\); see Fig. 2. Choose \(g_{0,i}\), \(\ell _{0,i}\), and \(u^{ALT}_{0,i}\) so that \(g_{0,i} \varphi (\ell _{0,i} - u^{ALT}_{0,i}) = b_i\) for all \(i=1,...,n\). For this, choose \(u_{0,i}^{ALT}=u\) and \(\ell _{0,i}=\ell \) for all \(i=1,\dots ,n\) so that \(\varphi (\ell - u) >0\). Note that these values exist because \(\lim \varphi _{x \rightarrow +\infty }(x) = 1\). Let \(\varphi ^* = \varphi (\ell - u).\) Then define \(g_{0,i} = b_i / \varphi ^*\).

Solving this instance of RND1 solves the knapsack problem (26). So, if there was an algorithm that could polynomially solve this RND1 instance, you would be able to solve the knapsack problem in polynomial time, which is a contradiction because the knapsack problem is NP-hard (see Garey and Johnson 1979). \(\square \)

The proof to Corollary 1 follows.

### Proof

The reader may note that, with the input data in the proof of Proposition 1, the solution to problem RND1 is the same as the solution to problem RND2, with the same data (because the underlying chosen network (*N*, *E*) is a star, and therefore any subnetwork of it is also a star). \(\square \)

### MILP model

We now detail the MILP program we designed for solving the RND1 problem in the experiments, similar to the one introduced in García-Archilla et al. (2013). The following variables are needed:

\(y_i\) is a binary variable to decide whether or not node \(v_i\) is a station of the railway network.

\(x_{ij}\) is a binary variable to decide whether or not edge (

*i*,*j*) is a link of the railway network.\(r_{pq}\) is a binary variable to decide whether or not there is a path for OD-pair (

*p*,*q*) in the railway network.\(w_{pq}\) is a binary variable to decide whether or not the railway network has a better utility than the road network for OD-pair (

*p*,*q*).\(f^{pq}_{ij}\) is a binary variable to decide whether or not OD-pair (

*p*,*q*) will use edge (*i*,*j*) from \(v_i\) to \(v_j\) in their route on the railway network.

The objective of our model is to maximize the railway trip coverage:

The constraints of our model have been grouped according to their aims:

Budget constraints,

$$\begin{aligned} \sum _{(i,j) \in E} \, c_{ij}\,x_{ij}+ \sum _{n_i\in N} c_i y_i \le C_{\max } . \end{aligned}$$(28)Edges can be used in both senses, and if a link is built, then its end nodes must be stations of the railway network:

$$\begin{aligned} x_{ij}= & \, x_{ji}, \ (i,j)\in E , \end{aligned}$$(29)$$\begin{aligned} x_{ij}\le & \, y_i, \ (i,j) \in E, \end{aligned}$$(30)$$\begin{aligned} x_{ij}\le & \, y_j, \ (i,j) \in E. \end{aligned}$$(31)Routing demand conservation constraints:

$$\begin{aligned}&\sum _{i : (i,p)\in A}\,{f}_{ip}^{pq} = 0, \ (p,q)\in W , \end{aligned}$$(32)$$\begin{aligned}&\sum _{j : (p,j)\in A}\,{f}_{pj}^{pq} = r_{pq}, \ (p,q)\in W , \end{aligned}$$(33)$$\begin{aligned}&\sum _{i: (i,q)\in A}\,{f}_{iq}^{pq} = r_{pq},\ (p,q)\in W , \end{aligned}$$(34)$$\begin{aligned}&\sum _{j : (q,j)\in A}\,{f}_{qj}^{pq} = 0,\ (p,q)\in W , \end{aligned}$$(35)$$\begin{aligned}&\sum _{i : (i,k)\in A} {f}_{ik}^{pq}\,-\,\sum _{j : (k,j)\in A} {f}_{kj}^{pq}\,=\,0, \ \forall \ k\notin \{p,q\}, \ (p,q)\in W . \end{aligned}$$(36)Location-allocation constraints:

$$\begin{aligned} {f}_{ij}^{pq} +r_{pq}-1\,\le \, \, x_{ij}, \ (i,j)\in A,\ (p,q)\in W . \end{aligned}$$(37)Disutility of the railway network

$$\begin{aligned} u_{pq}\,= & \, \,\sum _{(i,j)\in A}d_{ij}{f}_{ij}^{pq}+ M(1 - r_{pq}) + t_s \left( \left( \sum _{(i,j) \in A} f^{pq}_{ij}\right) - 1\right) \nonumber \\&+ \gamma (|zone_p - zone_q| + 1) . \end{aligned}$$(38)Splitting demand constraints:

$$\begin{aligned} (u_{pq} - u^{ALT}_{pq}) - M(1-w_{pq}) \le 0, \end{aligned}$$(39)where

*M*is a real number sufficiently large.

Constraint (28) states that construction costs cannot exceed the budget, \(C_{\max }\). Constraints (29) allow the constructed links to be used in both directions. Constraint (30) and (31) impose that, if a link is built, then its corresponding end nodes should have a station. Constraints (32)– (36) are flow conservation constraints for variables *f*. Note that if \(r_{pq} = 0\), there will be no flow from *p*–*q* via the railway network. Constraints (37) force that demands are only allocated through public arcs if the corresponding edges are built. Constraints (38) define the utility of each OD-pair in the railway network, which depends on the riding time, the number of stops, and the journey price. In these experiments, we took \(t_s = 0.5\) and \(\gamma = 1\). This implies that the stop time at stations is 0.5, and that the ticket price is 1, 2, or 3, if the number of zone changes is 0, 1, or 2, respectively. Note that the utility of (*p*, *q*) is a large enough constant *M* if there is no path from *p* to *q* in the railway network. Constraints (39) impose for each OD-pair (*p*, *q*) that, if their utility using the road network is better than their utility using the railway network, then \(w_{pq} = 0\), and therefore this OD-pair is not covered.

Besides these constraints, we also added to our model the following cuts:

The first one imposes that no edge can be used if its corresponding link is not built. The second one imposes that an OD-pair without a path in the railway network cannot be covered. Previous experience showed that these two cuts significantly reduced the computational times.

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Perea, F., Menezes, M.B.C., Mesa, J.A. *et al.* Transportation infrastructure network design in the presence of modal competition: computational complexity classification and a genetic algorithm.
*TOP* **28, **442–474 (2020). https://doi.org/10.1007/s11750-019-00537-x

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### Keywords

- Networks/graphs
- Transportation
- Computational complexity
- Genetic algorithms

### Mathematics Subject Classification

- 90B06