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On the shortest path problem with negative cost cycles

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Abstract

In this paper, the elementary single-source all-destinations shortest path problem is considered. Given a directed graph, containing negative cost cycles, the aim is to find paths with minimum cost from a source node to each other node, that do not contain repeated nodes. Two solution strategies are proposed to solve the problem under investigation and their theoretical properties are investigated. The first is a dynamic programming approach, the second method is based on the solution of the k shortest paths problem, where k is considered as a variable. Theoretical aspects related to the innovative proposed strategies to solve the problem at hand are investigated. The practical behaviour of the defined algorithms is evaluated by considering random generated networks and instances derived from vehicle routing benchmark test problems.

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Acknowledgments

The authors would like to thank Professor Francesco Scarcello for his valuable comments and helpful suggestions related to the complexity analysis of the proposed solution approaches. They also wish to thank the editor and the anonymous referees having contributed to improve the quality and readability of the paper, with their constructive suggestions and comments.

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Authors and Affiliations

  1. Department of Mechanical, Energy and Management Engineering, University of Calabria, 87036, Rende, CS, Italy

    Luigi Di Puglia Pugliese & Francesca Guerriero

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  1. Luigi Di Puglia Pugliese
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  2. Francesca Guerriero
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Correspondence to Luigi Di Puglia Pugliese.

Appendix

Appendix

In this appendix, we show how the proposed Algorithm 1 and 4 work by considering the instance of Figure 5.

Fig. 5
figure 5

Graph example

Full size image

1.1 Algorithm 1

In Table 6, we report set S, the labels at the end of Algorithm 2, the last selected label and the detected cycle C. The labels reported in Table 6 have the following form: \(<\pi (y)>\). The superscript reported next to the labels indicates that the related label is dominated.

Table 6 Labels associated with each node at the end of Algorithm 2 at each iteration of Algorithm 1
Full size table

At the first iteration of Algorithm 1 (see column \(1^{st}\) iteration of Table 6), the label \(<s,2,5,4(-8)>\) is selected. When we try to extend the label to node 2 a \( NCC \) is detected. Indeed, path \(\pi _{s2} = \{s,2,5,4\} \cup \{2\}\) has a cost equal to \(-5\) that is less than the cost of path \(\pi _{s2}=\{s,2\}\) associated with label \(<s,2(-4)>\). The dummy resource is introduced to node 2. At the \(2^{nd}\) iteration, the \( NCC \,\{1,2,3,1\}\) is detected and node 1 is inserted in the set S. The introduction of the resource at node 2 and 1 avoids the generation of \( NCC s\) and the optimal solution is found at the \(3^{rd}\) iteration.

1.2 Algorithm 4

In Table 7, we report the paths and the related cost (\(<\pi _{si}^{k} (y_i[k])>\)) at the end of each iteration of Algorithm 4 when solving the ESPP on the network of Figure 5. At the \(1^{st}\) iteration, node 3 is selected. Since all the conditions of line 9 are verified, Algorithm 3 terminates and the cycle \(\{1,2,3,1\}\) is returned. The value of \(K_i, i=1,2,3\) is incremented of one and Algorithm 3 is executed again. Node 4 is selected (see \(2^{nd}\) iteration). When path \(\pi _{s4}^1=\{s,1,2,5,4\}\) is extended to node 2, conditions of line 9 are satisfied, thus the NCC \(\{2,5,4,2\}\) is detected and Algorithm 3 is stopped. The number of paths that need to be found for nodes 2, 5 and 4 is increased. After other three iterations, see \(3^{rd}\), \(4^{th}\) and \(5^{th}\) iteration of Table 7, condition of line 29 is verified. Since \(C = \emptyset \), Algorithm 4 terminates and \(y_i[1], \forall i \in \mathcal {N}\), is the optimal cost.

Table 7 Paths and costs that are found for each node at the end of Algorithm 3 in each iteration of Algorithm 4
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Di Puglia Pugliese, L., Guerriero, F. On the shortest path problem with negative cost cycles. Comput Optim Appl 63, 559–583 (2016). https://doi.org/10.1007/s10589-015-9773-1

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