Solution Approaches for the Elementary Shortest Path Problem

Transcription

1 Solution Approaches for the Elementary Shortest Path Problem Luigi Di Puglia Pugliese Francesca Guerriero Department of Electronics, Computer Science and Systems, University of Calabria, Rende, Italy. 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 the paths with minimum cost from a source node to each other node, that do not contain repeated nodes. Three different 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. The last solution strategy is an enumerative approach based on a tree structure, constructed with a cycle elimination rule. Theoretical aspects related to the innovative proposed strategies to solve the problem at hand are investigated. The practical behaviour of the defined algorithms is also evaluated by considering a meaningful number of test problems. Keywords: shortest path; negative cost cycles; dynamic programming; k shortest paths; enumerative method. 1 Introduction The Shortest Path Problem (SPP) is one of the most studied problems in network optimization [16, 23, 28]. The problem comes up in practice and arises as subproblem in many network optimization algorithms. Solution approaches for the SPP have been studied for a long time (e.g., [7, 22, 23, 29, 46, 47]). More recently, some improvements and computational study have been presented by Ahuja et al. [1], Gabow and Tarjan [31] and Goldberg [35]. The reader is referred to the works of Cherkassky et al. [11] and Gallo and Pallottino [32] for detailed surveys. In its basic formulation, the objective is to determine the minimum cost path through a network from a given source node to a destination node. Several polynomial time solution approaches have been developed in the scientific literature to Corresponding Author, ldipuglia@deis.unical.it 1

2 address the SPP in the case there are no negative cost cycles (e.g., see [17, 23, 43]). On a network with negative arc costs, but with non-negative cost cycles, the best currently known time bound O(nm) is achieved by the Bellman-Ford-Moore algorithm [7, 29, 46], where n and m denote the number of nodes and arcs in the network, respectively. If the arc costs are non-negative, implementations of Dijkstra s algorithm [23] achieve better bounds. In particular, an implementation presented by Fredman and Tarjan [30] runs in O(m + n log n) time. A trivial extension of the SPP is to find the shortest path tree rooted at a source node, by considering as destination all the nodes of the network (SPT P). This problem can be easily solved by the label-correcting methods used to address the SPP (e.g., see [11, 32]). An interesting extension of the SPP is the problem to find the first k shortest paths (kspp). A set of k shortest paths have to be determined for each node by considering the first, the second and so on up to the k-th shortest path. This problem arises in many real-life applications, e.g., in telecommunication and transportation fields. In these contexts, alternative solutions should be available when the current best is forbidden for various reasons, e.g., link interruption in telecommunication network or street closed in transportation plant. Many solution approaches have been defined to solve the kspp. The scientific literature takes into account two variants of the kspp: 1) the pure kspp and 2) the so-called loopless kspp. In the first case, a node can appear more than once in the solution paths, whereas, in the latter we are interested in finding the first k shortest paths such that they contain a node only once. A wide number of papers address the kspp (e.g., see [4, 12, 24, 25, 38, 45, 51]). Solution approaches for the loopless ksp P have been defined by Martins et al. [44] and Yen [62]. Dreyfus [24] and Yen [62] cite several additional papers on this subject going back as far as When negative cost cycles are present in the network, the SPP is not well defined, that is, a finite optimal solution does not exist. In this case, the problem is to check whether a simple cycle, whose arc costs sum up to a negative number, is present in the network. This problem is known as Negative Cost Cycle Detection Problem (N CCDP). It is worth noting that the N CCDP is polynomial. Several procedures have been defined for the N CCDP, see, e.g.,[15, 54]. The Bellman-Ford algorithm is one of the earliest and to date asymptotically fastest algorithm for the N CCDP [11, 34]. Recently, Subramani [53] has introduced a new approach for the N CCDP; the proposed solution strategy is based on exploiting the connections between the N CCDP and the problem of checking whether a system of difference constraints is feasible. It is important to point out that the scientific literature provides several works dealing with algorithms that list up all cycles in directed as well as undirected graphs in which arc costs are not considered [10, 48, 56, 57, 59, 60]. Yamada and Kinoshita [61] addressed the problem of finding all cycles in a directed graph with negative costs and they proposed a recursive algorithm. It is worth observing that all the 2

3 cited references do not solve the SPP in presence of negative cost cycles, rather they enumerate all the cycles in the network or check whether a negative cost cycle exists. The problem to find a shortest path in a network with negative cost cycles is still a very challenging problem. To the best of our knowledge, no solution approaches have been defined up to now to solve the elementary shortest path problem (ESPP). It is, of course, well known that computing shortest paths in the presence of negative cost cycles is a strongly N P-hard problem [33]. When constraints on the consumption of resource along the path are introduced, the problem is referred to as the Resource Constrained Elementary Shortest Path Problem (RCESPP). This problem has attracted the attention of many researchers in the last years. Starting with the seminal work of Beasley and Christofides [6], a variety of solution approaches have been developed to solve to optimality the RCESPP (see, e.g., [19, 39, 40]). The single-source single-destination version of the ESPP can be viewed as a particular instance of the RCESPP, where the resource consumption constraints are removed. In addition, it is possible to relax the elementarity constraints from the ESPP introducing a new resource that takes into account the number of visited nodes. Thus, the relaxed ESPP can be modeled as a RCSPP. This issue is explained in more detail in the following. In this paper, we deal with the more general case in which an elementary shortest path must be found for all nodes. In other words, we address the single-source all-destinations version of the ESPP. In order to solve to optimality the singlesource all-destinations ESPP, we apply the concepts proposed by Boland et al. [9] and Righini and Salani [49]. Beside the solution approach derived from the optimal strategies developed for the RCESPP, innovative methods are also presented. In particular, we have designed a dynamic k shortest paths algorithm, in which k is a variable and may assume an arbitrary value for each node of the network. An enumerative solution approach is also presented. In addition, an in depth analysis is carried out to evaluate the theoretical complexity of the proposed solution approaches and an extensive computational phase is conducted in order to asses the efficiency of the defined optimal strategies. In the context of the RCSPP, the single-source all-destinations SPP is applied in order to perform network reduction procedure based on the cost, to determine lower bounds on the optimal cost and to solve the dual Lagrangean problem when the resource constraints are relaxed in a Lagrangean objective function. When the RCESPP is considered, the aforementioned elements cannot be applied because of the negative cost cycles. Thus, the solution approaches for the ESPP proposed in this paper, can be used to determine lower bounds on the optimal cost and to solve the dual Lagrangean problem associated with the RCESPP. This information can be used to improve the performance of the state-of-the-art methods. In addition, it is worth observing that the ESPP can be used to model the 3

4 currency conversion problem [50]. When the nodes represent currencies and the arcs the transactions with costs equal to exchange rate, the problem is to find the maximum product path, that is, the best exchange sequence. The problem can be easily transformed in the SPP and, since the graph can contain negative cost cycles, it can be viewed as an instance of the ESPP. The proposed methods are able to solve also the single-source single-destination ESPP. However, it is worth observing that well-tailored solution approaches for the single-source single-destination version can be devised but this topic is out of the scope of this work. The paper is organized as follows. In Section 2 we give some notations and definitions used in the paper. The proposed solution approaches, along with their theoretical analysis are presented in Section 3. Section 4 is devoted to the discussion of the computational results. Section 5 presents our conclusions. The appendix reports the results of the computational experiments. 2 Notations and Definitions Let G = (N, A) be a directed graph where N is the set of n nodes, whereas A denotes the set of m N N arcs. N contains also the source node s. A cost c ij is associated with each arc (i, j) A. A cycle on node i is defined as an ordered sequence of nodes C i = (i, j 1,..., j g, i), where j i j k for all pairs i, k {1,..., g} such that i k. It can also be viewed as a sequence of g + 1 arcs {(i, j 1 ),..., (j g 1, j g ), (j g, i)}. It is worth observing, as shown by Allender [2], that n!(ρ 1)! ρ!(n ρ)!. the maximum number of cycles in a complete graph is equal to C = n ρ=1 Let π si be a path from node s to node i. It is a sequence of nodes π si = (s = i 1,..., i l = i), l 2 and a corresponding sequence of l 1 arcs such that the h-th arc in the sequence is (i h, i h+1 ) A for h = 1,..., l 1. Thus, each path contains at least one arc. Let M π (j) be the multiplicity of node j in path π, that is M π (j) = {v : 0 v π, i v = j}. The path π is said to be an elementary path if M π (j) = 1, for all j π. A cycle C i on node i N is said to be a negative cost cycle (N CC) if c(c i ) = (h,k) C i c hk < 0. Let π sj be an elementary path from node s to node j such that i π sj for some i N \ {s}. Let π si = π sj {(j, i)} be the path obtained by extending path π sj to node i through arc (j, i). Since i π sj, a cycle C i = π ij {j, i} is created. Proposition 1. Given a path π sj and its sub-path π si, C i is a N CC if c(π si ) > c(π sj ) + c ji. Proof. The cost of the path π sj can be viewed as the sum of the cost of the sub-path π si and the sub-path from i to j, that is c(π sj ) = c(π si ) + c(π ij ). It follows that c(π si ) > c(π sj ) + c ji c(π si ) > c(π si ) + c(π ij ) + c ji 0 > c(π ij ) + c ji = c(c i ). This concludes the proof. 4

5 Since the directed graph G is not assumed to be acyclic and the arc costs are not constrained in sign, there may be N CCs in G. The ESPP, from the source s to all other nodes, consists in finding the paths π si, i N \ {s} such that c(π si ) is minimized and M πsi (j) = 1, j π si. It is well known that the Bellman s optimality principle is not verified for this problem, indeed sub-paths of the optimal path for some node cannot be optimal. In what follows, we formalize this property. Let us assume that π su, u N, u s, represents the path with minimum cost and no repeated nodes in a graph with N CCs. The following consideration can be formulated. Observation 1. The optimal path π su from node s to node u is composed by subpaths π si from node s to each other node i π su that are not guaranteed to be optimal for node i. Proof. We prove the result by contradiction. Let us suppose that a sub-path πsj = πsi π ij is part of the optimal solution π su, and, for the purpose of contradiction, we assume that πsi is the optimal path from s to i. It follows that c(π sj ) = c(π si )+c(π ij ). Let us suppose that another path π sj : i π sj exists to reach node j. Since π sj is different from πsj, we have that c( π sj) c(πsj ). Let us suppose that a path π ji exists. If the path π sj is extended to node i through path π ji, then we have that c( π si ) = c( π sj ) + c( π ji ). We assume that πij π ji = C i is a N CC. Since we assume that πsi is the optimal path from node s to node i, it follows that c( π si ) c(πsi ). This means that c( π sj) + c( π ji ) c(πsj ) c(π ij ), thus c( π sj ) c(πsj ) c(c i). Since c(c i ) < 0, for a sufficiently low cost c( π sj ), that is, for c( π sj ) < c(πsj ) c(c i), the relation c( π sj ) c(πsj ) c(c i) is not verified, contradicting the optimality hypothesis on path πsi. In order to better explain Observation 1, let us consider the directed graph of Figure 1. Figure 1: Directed graph example. The optimal path from node s to node 3 is πs3 = {s, 1, 2, 3} of cost 0. The cost of sub-path πs1 is equal to 1 but c(π s1 ) is not the least cost from s to 1. Indeed, the path π s1 = {s, 2, 1} is the optimal path of cost 0 to node 1. 5

6 Observation 1 suggests that it should be necessary to keep more than one path for some node. It follows that the optimal solution of the ESPP is not guaranteed to be a spanning tree. Under this respect, it is possible to devise a modified label-correcting scheme for the SPP in order to obtain valid upper bounds for the ESPP. Let y i be the label that stores the value of the dual price for each node i N and L denote the list of nodes, that have to be processed. We indicate with pred R N the vector of predecessors, that is pred i = j is the predecessor of node i in the path π si. A set F S(i) of successor nodes is associated with each node i N, that is defined as follows: F S(i) = {j : (i, j) A}. A general scheme of a label-correcting method to obtain upper bounds for the ESPP is reported in Algorithm 1. Algorithm 1 : Label Correcting Scheme 1: Step 1 (Initialization phase) 2: y s = 0; y i = +, i N \ {s}; pred i = i, i N ; L = {s}. 3: 4: Step 2 (Node selection) 5: Select a node i from the list L and remove it from L. 6: 7: Step 3 (Label extension) 8: for all j F S(i) do 9: if y j > y i + c ij then 10: if j π si then 11: y j = y i + c ij ; 12: pred j = i; 13: add node j to L if it does not already belong to it. 14: end if 15: end if 16: end for 17: 18: Step 4 (Termination check) 19: if L = then 20: STOP. y i, i N, is an upper bound on the optimal cost of the path from node s to node i. 21: else 22: Go to Step 2. 23: end if Algorithm 1 is a classical label-correcting method where the elementarity constraints are taken into account. The combination of conditions introduced in line 10 and line 19 ensures that the built paths do not contain repeated nodes and that the algorithm terminates in a finite number of iterations. Since the procedure is based on Bellman s optimality conditions, at the end of Algorithm 1, the label y i, i N, represents an upper bound on the elementary shortest path cost from node s to node i. 6

7 3 Proposed Solution Approaches The solution approaches proposed in this paper are based on the idea to compute for each node the minimum set of paths such that the elementary shortest path is found for all nodes. Three strategies have been developed. The first one can be viewed as an application of the methods proposed to solve the RCESPP. A multi-dimensional labeling procedure is devised and the dimension of the label is updated dynamically. The second strategy is based on a labeling approach to solve the kspp. In our case, k is considered as a variable that can take different values for each node. In the last solution approach, a tree structure is defined and an enumerative procedure is devised in order to exploit the defined tree. It is worth observing that, while the first solution approach can be viewed as an extension of the algorithms for solving the RCESPP well-tailored to the ESPP, the last two strategies are proposed for the first time to solve instances of the SPP in presence of N CCs. 3.1 Dynamic Multi-Dimensional Labeling Approach The approaches, presented in this section, use the concept of critical node introduced by Kohl [42]. In particular, a node i is said to be critical if there exists a cycle C i that is a N CC. A binary variable is introduced for each critical node in order to keep track of whether the node is visited. This binary variable can be viewed as a resource associated with node i, bounded to be less than or equal to one (see [6]). Let S = {i 1, i 2,..., i S }, be the set of critical nodes and r[p], p = 1,..., S be the binary variable associated with node i p S. We denote as y i = (c(π si ), r i ) the label that contains the cost and the resource consumptions vector of the path π si from node s to node i. The resource consumptions vector r i has the following form: r i [p] = 1, p : j p π si and r i [p] = 0, p : j p π si. In other words, the vector r i keeps trace of the nodes j S that are visited along the path π si. Definition 1. Let yi 1 = (c(πsi 1 ), r1 i ) and y2 i = (c(πsi 2 ), r2 i ) be two labels associated with node i. The label yi 2 is dominated by the label yi 1 if c(πsi 1 ) c(π2 si ), r1 i [p] r2 i [p] for p = 1,..., S and at least one inequality is strict. Definition 2. A label y i = (c(π si ), r i ) associated with node i is said to be efficient or Pareto-optimal if there does not exist a label ȳ i that dominates it. Definition 3. A label y i = (c(π si ), r i ) associated with node i is said to be feasible if r i [p] 1, p = 1,..., S. A set D(i), containing all the efficient labels, is associated with each node i N. The procedure defined by Desrochers [18] can be applied to solve the ESPP. It is worth observing that the Desrochers algorithm is an extension of Algorithm 1, 7

8 without line 10, to the constrained case. In Algorithm 2, we report the steps of the label-correcting method extended to the constrained case. Algorithm 2 : Label-Correcting Scheme for the ESPP 1: Step 1 (Initialization phase) 2: y 1 s = (0, 0); D(s) = {y 1 s}; D(j) =, j N \ {s}; L = {s}. 3: 4: Step 2 (Node selection) 5: Select a node i from the list L and remove it from L. 6: 7: Step 3 (Label extention) 8: for all y ξ i D(i) do 9: for all j F S(i), j s do 10: Set: c( π sj ) = c(π ξ si ) + c ij; r j [p] = r ξ i [p], p = 1,..., S ; 11: if j S then 12: r j [ˆp] + 1 with iˆp = j. 13: end if 14: if r i [p] 1, p = 1,..., S then 15: if ȳ j = (c( π sj ), r j ) is not dominated by any label in D(j) then 16: D(j) = D(j) {ȳ j }; 17: remove from D(j) all the labels that are dominated by ȳ j ; 18: add node j to L if it does not already belong to it. 19: end if 20: end if 21: end for 22: end for 23: 24: Step 4 (Termination check) 25: if L = then 26: STOP. The label y i = minȳ D(i) {c( π) si } is associated with the optimal elementary path from s to i, i N. 27: else 28: Go to Step 2. 29: end if It is possible to devise also a label-setting algorithm by slightly modifying Algorithm 2. Indeed, the set L stores all the efficient and feasible labels and it is initialized by the label y s. At Step 2, from the list L, the lexicographically minimal label is selected. This label is used to generate feasible and efficient labels to each successor node. This procedure is the same as the one proposed by Boland et al. [9] called general label-setting algorithm. The main drawback of the procedure is that the set S of critical nodes must be known in advance. The enumerative procedure of Yamada and Kinoshita [61] could be executed to determine the entire set of negative cost cycles. The nodes belonging to each negative cost cycle can be used to initialize set S and the Algorithm 2 can be performed. However, this strategy could be inefficient, since only a sub-set of all nodes belonging to all negative cost cycles need to be considered. In addition, the procedure defined by Yamada and Kinoshita [61] is not polynomial. If the set S is not known in advance, then we can relax the elementarity constraints and introduce a surrogate resource γ that counts the number of nodes in the path to get the relaxed ESPP (ESPP R ). This idea was proposed by Feillet et al. 8

9 [26] and applied by Righini and Salani [49] for solving the RCESPP. The ESPP R is a RCSPP where the resource γ is bounded to be less than or equal to the number of nodes n. The optimal solution to the ESPP R may not be an elementary path. The nodes that appear more than once in the optimal paths π si, i N \ {s}, are marked as critical nodes and the RCSPP is run again until an elementary path is found. The approach to augment the set S is the same of that proposed by Boland et al. [9] and by Righini and Salani [49]. The difference is that in the latters only one path has to be found. For more detail, the reader is referred to [20]. Very recently, in [37] the authors proposed a new state-space reduction technique combined with the strategies defined in [49]. This strategy cannot be extended to the single-source all-destinations version of the ESPP. The solution approaches presented in [49, 9] can be modified in order to define well-tailored procedure for the ESPP. In particular, starting from S =, the labelsetting algorithm is executed. The search is stopped when a N CC, named C i, is detected. The set S is incremented by adding node i and the label-setting algorithm is run again. The procedure terminates when no N CC is detected. This approach is similar to the general state-space augmenting algorithm proposed in [9] with the difference that the label-setting algorithm is stopped when a negative cost cycle is detected. In this case, the surrogate resource γ is not necessary. In Algorithm 3, we describe the steps of the aforementioned procedure, whereas the truncated label-setting algorithm is depicted in Algorithm 4 Algorithm 3 : Dynamic Multi-Dimensional Labeling Approach 1: Step 1 (Initialization phase) 2: Set: ζ = 0; S ζ = 3: 4: Step 2 (Run the truncated label-setting algorithm) 5: Run Algorithm 4 with critical node restricted to S ζ and get C. 6: 7: Step 3 (Cycle detection) 8: if C then 9: S ζ+1 = S ζ {j}, j : M(j) C = 2; 10: ζ + +; 11: Go to Step 2. 12: else 13: STOP. 14: end if The defined solution approach is quite different from those proposed by Boland et al. [9] and Righini and Salani [49]. For this reason and for the sake of completeness, in the next Section we prove its correctness Correctness of the Dynamic Multi-Dimensional Labeling Approach In this section, we prove the correctness of the proposed solution approach. In particular, at the end of Algorithm 3, the label y i with the smallest cost among 9

10 Algorithm 4 : Truncated label-setting algorithm with S ζ 1: Step 1 (Initialization phase) 2: Set: y 1 s = (0, 0); D(s) = {y 1 s}; D(j) =, j N \ {s}; L = {y 1 s}; C =. 3: 4: Step 2 (Label selection) 5: Select the lexicografically minimal label y ξ i from the list L and remove it from L. 6: 7: Step 3 (Label extension) 8: for all j F S(i), j s do 9: Set: c( π sj ) = c(π ξ si ) + c ij; r j [p] = r ξ i [p], p = 1,..., Sζ ; 10: if j S ζ then 11: r j [ˆp] + 1 with iˆp = j. 12: end if 13: if j S ζ and j π ξ si and c( π sj) < c(π ξ sj ) then 14: STOP. A N CC C is detected. 15: Return C. 16: else 17: if r i [p] 1, p = 1,..., S ζ then 18: if ȳ j = (c( π sj ), r j ) is not dominated by any label in D(j) then 19: D(j) = D(j) {ȳ j }; L = L {ȳ j }; 20: remove from D(j) and L all the labels that are dominated by ȳ j. 21: end if 22: end if 23: end if 24: end for 25: 26: Step 4 (Termination check) 27: if L = then 28: STOP. The label y i = min {ȳ D(i)} {c( π) si } is associated with the optimal elementary path from s to i, i N. 29: Return C. 30: else 31: Go to Step 2. 32: end if those belonging to the set D(i) is associated with the least cost elementary path from s to i. Proposition 2. At the end of Algorithm 4, two situations can occur: 1. a N CC is detected, that is, the stop conditions of line 13 are verified; 2. if the procedure is stopped by condition of line 27, then all the efficient and feasible paths from s to each other node i N \ {s} are determined. (a) At the end of each iteration, the following conditions hold: i. D(s) = {ys} 1 = {(0, 0)}; ii. j N, if D(j) [i.e.: D(j) = {yj 1, y2 j,..., yl j }] and j s, then y ξ j, ξ = 1,..., l is a label related to a feasible path from node s to node j and D(j) is an efficient set. 10

11 (b) Upon termination of the algorithm, if D(j), j N and j s, then D(j) contains the labels of all efficient and feasible paths from node s to node j. Proof. Let us consider case 1. The conditions of line 13 derive from Proposition 1. Thus, if they are verified, then a N CC is detected. If case 1 does not occur, then all N CCs have been detected and the algorithm terminates when the list L is found empty, that is, case 2. In what follows, we prove case 2. Let us consider case 2a. Condition 2(a)i holds because, initially, D(s) = {ys} 1 = {(0, 0)} and, by the rules of the algorithm, D(s) cannot change. We prove condition 2(a)ii by induction on the iteration count. Indeed, initially, condition 2(a)ii holds, since node s is the only node for which the set D(s) is nonempty. Suppose that 2(a)ii holds for some node j at the beginning of some iteration. Let yi δ be the label removed from L. If i = s (which only occurs at the first iteration) and δ = 1, then at the end of this iteration, we have D(j) = yj 1 for all successor nodes j of s such that the corresponding path π sj is feasible, D(j) = for all other nodes j s, j / F S(i). Thus, the set of labels has the required property. If i s, then yi δ is the label of some feasible path πδ si starting from s and ending to i that is not dominated by the other paths in D(i), by the induction hypothesis. If the set D(j) changes, for some node j, such that j F S(i), as a result of the iteration, a new feasible label ȳ j = (c( π sj ), r j ) is obtained for node j. The created label is related to the feasible path π sj consisting of path πsi δ followed by the arc (i, j). Finally, note that, by the rules of the algorithm, the newly created label is added to D(j) only if it is an efficient label. This completes the induction proof of 2(a)ii. Let now consider condition 2b. Using part 2(a)ii, we have that, at each iteration, j N such that D(j), D(j) is an efficient set. Thus, the property mentioned is also satisfied when the algorithm terminates. In addition, the way in which the candidate list L is updated and the termination condition (i.e., the algorithm terminates when there are no more labels left to be scanned) guarantee that all the labels with the potential to determine a new label for at least one node are scanned during the execution of the algorithm. Proposition 3. (Correctness of Algorithm 3). At the end of Algorithm 3, the label y i with the smallest cost among those belonging to the set D(i) is associated with the least cost elementary path from s to i, i N. Proof. From Proposition 2, it follows that all the efficient and feasible labels y i, i N, are determined. Since the set D(i) is an efficient set, the path 11

12 π si = arg min {y D(i)} {c(π si )} has the cost as low as possible among all the feasible paths from s to i, that is π si is the optimal solution i N Complexity Analysis As far as the complexity analysis of the proposed dynamic multi-dimensional labeling approach for the ESPP is concerned, let D be the set of efficient and feasible labels with the highest cardinality. The complexity of Algorithm 3 is O(n 2 n τ=1 τ22τ ), in the worst case. This result is formally stated in the following lemma. Lemma 1. In the worst case, the complexity of Algorithm 3 is O(4 n n 3 ). Proof. Let D be the maximum number of labels at a node. The operations executed in lines 9-23 of Algorithm 4 are S D, that is, the test of dominance. Since the F S can contain at most n 1 elements, the total number of operations executed in lines 8-24 is n S D. Since the for-loop of line 8 is invoked n D times, that is an upper bound on the total number of generated labels, thus Algorithm 4 takes O(n 2 S D 2 ) operations. In the worst case, the number of efficient labels stored at each node is equal to 2 S. It follows that Algorithm 4 takes O(n 2 S 4 S ) operations. Starting from S =, the set of critical nodes is incremented by one at each iteration. In the worst case, set S contains all nodes. In other words, the operations executed by Algorithm 3 are n n n 2 n 4 n = 4 9 n2 [4 n (3n 1) + 1]. Consequently, the complexity of Algorithm 3 is O(4 n n 3 ), in the worst case. In order to better explain how Algorithm 3 works, let us consider the example of Figure 2. In Table 1, we report set S, the labels at the last iteration of Algorithm 4, the last selected label and the detected cycle C. The labels reported in Table 1 have the following form: < π(y) >. The superscript reported next to the labels indicates that the related label is dominated. Figure 2: Toy example. At the first iteration (see column 1 st iteration of Table 1), the label < s, 2, 5, 4( 8) > is selected. When we try to extend the label to node 2 a N CC is detected. Indeed, path π s2 = {s, 2, 5, 4} {2} has a cost equal to 5 that is less than the cost of path π s2 = {s, 2} associated with label < s, 2( 4) >. The dummy resource is introduced to node 2. At the 2 nd iteration, the N CC {1, 2, 3, 1} is detected and node 1 is inserted 12

13 1 st iteration 2 nd iteration 3 rd iteration S {2} {2, 1} s < s(0) > < s(0, 0) > < s(0, 0, 0) > 1 < s, 1(2) > < s, 1(2, 0) > < s, 1(2, 0, 1) > < s, 2, 1( 1, 1) > < s, 2, 1( 1, 1, 1) > 2 < s, 2( 4) > < s, 2( 4, 1) > D < s, 2( 4, 1, 0) > < s, 1, 2( 5, 1) > < s, 1, 2( 5, 1, 1) > 3 < s, 2, 3( 2) > < s, 2, 3( 2, 1) > D < s, 2, 3( 2, 1, 0) > < s, 1, 2, 3( 3, 1) > < s, 1, 2, 3( 3, 1, 1) > 4 < s, 2, 5, 4( 8) > < s, 2, 3, 1, 4(4, 1) > D < s, 2, 3, 1, 4(4, 1, 1) > D < s, 2, 5, 4( 8, 1) > D < s, 2, 5, 4( 8, 1, 0) > < s, 1, 4(7, 0) > < s, 1, 4(7, 0, 1) > < s, 1, 2, 5, 4( 9, 1) > < s, 1, 2, 5, 4( 9, 1, 1) > 5 < s, 2, 5( 6) > < s, 2, 5( 6, 1) > < s, 2, 5( 6, 1, 0) > < s, 1, 2, 5( 7, 1) > < s, 1, 2, 5( 7, 1, 1) > 6 < s, 2, 5, 6( 4) > < s, 2, 5, 6( 4, 1) > D < s, 2, 5, 6( 4, 1, 0) > D < s, 2, 5, 4, 6( 5, 1) > D < s, 2, 5, 4, 6( 5, 1, 0) > < s, 1, 2, 5, 4, 6( 6, 1) > < s, 1, 2, 5, 4, 6( 6, 1, 1) > < s, 1, 2, 5, 6( 5, 1, 1) > D < s, 1, 4, 6(10, 0, 1) > last extracted label < s, 2, 5, 4( 8) > < s, 1, 2, 3( 3, 1) > < s, 1, 4(7, 0, 1) > C {2, 5, 4, 2} {1, 2, 3, 1} Table 1: Labels associated with each node at the end of the corresponding iteration of Algorithm 4. in the set S. The introduction of the resource at node 2 and to node 1 avoids the generation of N CCs and the optimal solution is found at the 3 rd iteration. 3.2 Labeling Approach based on the k Shortest Paths Problem In this Section we describe a labeling approach in which the first k paths are determined for each node i N. The aim of the procedure is to determine the smallest set of paths for each node i. Under this respect, k is a variable and it can take different values for each node. The main idea is to start with k = 1 for each node, whenever a N CC is detected, k is incremented for a specific node and the labeling algorithm is run again. When no N CC is detected, from the set of paths associated with each node, the optimal solution is chosen. It is worth observing that when the kspp is solved, the paths are ranked by nondecreasing cost. In our context, it is necessary to store equivalent paths. This is necessary because paths with the same cost can have a different structure, that is, they can contain different nodes. Let K i be the number of best paths from node s to node i. The set Π i contains the K i paths ordered by nondecreasing cost, that is c(πsi k ) c(πk+1 si ), k = 1,..., K i 1 and if c(πsi k ) = c(πk+1 si ), then πsi k πk+1 si. In other words, Π i [k] = πsi k is the k-th shortest path from node s to node i. Definition 4. Set Π i, i N is said to be path-elementary, if each path π k si Π i, k = 1,..., K i does not contain repeated nodes. A vector y i R K i is associated with each node i that stores the costs of the paths belonging to the set Π i, that is, y i [k] represents the cost of path πsi k Π i. 13

14 Algorithm 5 : Truncated labeling algorithm for dynamic kspp 1: Step 1 (Initialization phase) 2: Set: y s[1] = 0; y i [k] = +, k = 1,..., K i, i N \ {s}; π 1 ss = {s}, Π s = {π 1 ss}; Π i =, i N \ {s}; L = {s}; C =. 3: 4: Step 2 (Node selection) 5: Select a node i from L and delete it from L. 6: 7: Step 3 (Label extension) 8: for all j F S(i), j s do 9: if y k i +, k {1,..., K i} AND k : j π k si AND k : j π k si, y i [ k] + c ij < y j [1] then 10: STOP. A cycle C = π k si {(i, j)} is detected. 11: Return C. 12: else 13: for all k = 1,..., K i : y i [k] +, j π k si do 14: for all ξ = 1,..., K j do 15: if y i [k] + c ij < y j [ξ] OR (y i [k] + c ij = y j [ξ] AND π k si {(i, j)} = πξ sj ) then 16: y j [δ + 1] = y j [δ], δ = ξ,..., K j ; 17: π δ+1 sj = π δ sj, δ = ξ,..., K j; 18: y j [ξ] = y i [k] + c ij ; 19: π ξ sj = πk si {j}; 20: add node j to L if it does not already belong to it. 21: BREAK 22: end if 23: end for 24: end for 25: end if 26: end for 27: 28: Step 4 (termination check) 29: if L = then 30: STOP. y i [1] is the cost of the optimal elementary path π 1 si. 31: Return C. 32: else 33: Go to Step 2. 34: end if Every time a N CC C is detected, the number of paths that have to be found for each node i C is increased, that is, K i = K i + 1, i C, and the labeling algorithm is executed again. When no N CC is detected, y i [1] is the cost of the optimal elementary shortest path from node s to node i N. The proposed solution approach iteratively solves the kspp for a fixed value of K i, i N. The steps of the labeling procedure are depicted in Algorithm 5. In the proposed strategy, Algorithm 5 is iteratively run after that K i, i N, is coherently updated. The dynamic labeling solution strategy is reported in Algorithm Correctness of the Dynamic Labeling Approach In this section, we prove the correctness of the proposed solution approach. In particular, at the end of Algorithm 5, πsi 1 is the least cost elementary path from s 14

15 Algorithm 6 : Dynamic Labeling Approach 1: Step 1 (Initialization phase) 2: Set: ζ = 0; K s = 1; K ζ i = 1 i N \ {s}. 3: 4: Step 2 (Run the truncated labeling algorithm) 5: Run Algorithm 5 with K ζ i, i N and get C. 6: 7: Step 3 (Cycle detection) 8: if C then 9: K ζ+1 i + 1, i C, K ζ+1 i = K ζ i, i N \ C; 10: ζ + +; 11: Go to Step 2. 12: else 13: STOP. 14: end if to i. Proposition 4. At the end of Algorithm 5 we have that: 1. a N CC is detected, that is, conditions of line 9 are verified; 2. if condition of line 29 is verified, then all the first K i elementary paths are determined for each node i N, that is, sets Π i, i N, are path-elementary. (a) At the end of each iteration, the following conditions hold: i. y s [1] = 0; ii. i N \ {s}, if y i [k] +, then πsi k is an elementary path. (b) Upon termination of the algorithm, Π i, i N, is path-elementary. Proof. Let us consider case 1: The first part of conditions in line 9, that is, yi k +, k {1,..., K i }, checks whether all the K i paths to node i are determined. The second part derives from Proposition 1, thus if it is verified for some k, a N CC is found and the algorithm terminates. It follows that the presence of a N CC is verified only if the entire set of paths from node s to the considered node i are determined. If case 1 does not occur, then all N CCs have been detected and the algorithm terminates when the list L is found empty, that is, case 2. Case 2: Let us consider case 2a. Condition 2(a)i holds because, initially, y s [1] = 0 and, by the rules of the algorithm, y s [1] cannot change. We prove condition 2(a)ii by induction on the iteration count. Indeed, initially, condition 2(a)ii holds, since node s is the only node for which the cost is not equal to +. Suppose that 2(a)ii holds for some node j at the beginning of some iteration. Let i be the node removed from L. If i = s (which only occurs at the first iteration), then at the end of this iteration, we have y j [1] + for all successor nodes j of s and the corresponding path 15

16 π sj = {(s, j)} is elementary, y j [k] = +, k 1 and y j [k] = + for all other nodes j s, j / F S(i). Thus, the set of paths has the required property. If i s, then y i [k] is the cost of the k-th path πsi k starting from s and ending to i that does not contain repeated nodes, by the induction hypothesis. If y j [k] changes, for some node j F S(i) and k, then a new path is obtained for node j. The created path πsj k consists of path πk si followed by the arc (i, j). Finally, note that, by the rules of the algorithm, the newly created path is elementary because j πsi k. This completes the induction proof of 2(a)ii. Let now consider condition 2b. Using part 2(a)ii, we have that, at each iteration, πsj k, j N such that y j[k] + is an elementary path. Thus, the property mentioned is also satisfied when the algorithm terminates. In addition, the way in which the candidate list L is updated and the termination condition (i.e., the algorithm terminates when there are no more nodes left to be scanned) guarantee that all the nodes with the potential to determine a new path for at least one node are scanned during the execution of the algorithm. Proposition 5. (Correctness of Algorithm 6). At the end of Algorithm 6, πsi 1 is the optimal path from node s to node i, i N. Proof. The algorithm terminates when no cycle is detected. From Proposition 4, we know that if no cycles are detected, then Algorithm 5 provides a path-elementary set Π i, i N. Since the paths are ordered in nondecreasing order of the cost, the first path of each set represents the least cost path. In addition, being Π i pathelementary, πsi 1 is the optimal path Complexity Analysis The complexity of Algorithm 6 is derived in the following lemma. Lemma 2. In the worst case, the complexity of the proposed solution approach is O(n 9 C7 ). Proof. Let K be the highest number of paths that have to be found. The operations executed in lines are 2K + 3. Since the forloops of line 13 and 14 take K 2, the number of iterations in lines is 2K 3 + 3K 2. The F S contains at most n 1 elements. The for-loop of line 8 is invoked nk times. Consequently, Algorithm 5 takes O(n 2 (2K 4 + 3K 3 )). Since each node can be part of each cycle, the iterations executed by Algorithm 6 are n C in the worst case. This means that the proposed approach executes n 2 ( )+n 2 ( )+...+n 2 (2 n C 4 +3 n C 3 ) = 2 15 (n C) 6 (6n C + 15) (n C) 4 (n C + 1) (n C) 3 (10(n C) 2 1) operations, thus the complexity of Algorithm 6 is O(n 9 C7 ). 16

17 In Table 2, we report the paths and the related cost (< πsi k (y i[k]) >) at the end of each iteration of Algorithm 6 when solving the ESPP on the network of Figure 2. At the 1 st iteration, node 3 is selected. Since all the conditions of line 9 are verified, Algorithm 5 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 5 is executed again. Node 4 is selected (see 2 nd iteration). When path πs4 1 = {s, 1, 2, 5, 4} is extended to node 2, conditions of line 9 are satisfied, thus the N CC {2, 5, 4, 2} is detected and Algorithm 5 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 2, condition of line 29 is verified. Since C =, Algorithm 6 terminates and y i [1], i N, is the optimal cost. 17

18 18 1 st iteration 2 nd iteration 3 rd iteration 4 th iteration 5 th iteration K i y i K i y i K i y i K i y i K i y i s 1 < s(0) > 1 < s(0) > 1 < s(0) > 1 < s(0) > 1 < s(0) > 1 1 < s, 1(2) > 2 < s, 2, 3, 1( 1) > 2 < s, 2, 3, 1( 1) > 2 < s, 2, 3, 1( 1) > 2 < s, 2, 3, 1( 1) > < s, 1(2) > < s, 1(2) > < s, 1(2) > < s, 1(2) > 2 1 < s, 1, 2( 5) > 2 < s, 1, 2( 5) > 3 < s, 1, 2( 5) > 4 < s, 1, 2( 5) > 5 < s, 1, 2( 5) > < s, 2( 4) > < s, 2( 4) > < s, 2( 4) > < s, 2( 4) > < s, 1, 4, 2(10) > < s, 1, 4, 2(10) > < s, 1, 4, 2(10) > < s, 1, 2, 3( 3) > 2 < s, 1, 2, 3( 3) > 2 < s, 1, 2, 3( 3) > 2 < s, 1, 2, 3( 3) > 2 < s, 1, 2, 3( 3) > < s, 2, 3( 2) > < s, 2, 3( 2) > < s, 2, 3( 2) > < s, 2, 3( 2) > 4 1 < s, 1, 4(7) > 1 < s, 1, 2, 5, 4( 9) > 2 < s, 1, 2, 5, 4( 9) > 3 < s, 1, 2, 5, 4( 9) > 4 < s, 1, 2, 5, 4( 9) > < s, 2, 5, 4( 8) > < s, 2, 5, 4( 8) > < s, 2, 5, 4( 8) > < s, 2, 3, 1, 4(4) > < s, 2, 3, 1, 4(4) > < s, 1, 4(7) > 5 1 < s, 1, 2, 5( 7) > 1 < s, 1, 2, 5( 7) > 2 < s, 1, 2, 5( 7) > 3 < s, 1, 2, 5( 7) > 4 < s, 1, 2, 5( 7) > < s, 2, 5( 6) > < s, 2, 5( 6) > < s, 2, 5( 6) > + < s, 1, 4, 2, 5(8) > < s, 1, 2, 5, 4, 6( 6) > 1 < s, 1, 2, 5, 4, 6( 6) > 1 < s, 1, 2, 5, 4, 6( 6) > 1 < s, 1, 2, 5, 4, 6( 6) > last extracted node C {1, 2, 3, 1} {2, 5, 4, 2} {2, 5, 4, 2} {2, 5, 4, 2} Table 2: Paths and costs that are found for each node at the end of Algorithm 5.

19 3.3 Enumerative Strategy The proposed approach looks for the optimal solution through a tree structure. Each node is associated with a sub-graph of G excluding the root node that represents the graph G. The nodes at a certain level are associated with sub-graphs in which at least one N CC is removed. Given a father node, all child nodes are associated with sub-graphs in which the same N CCs are removed. Figure 3 shows the structure of the aforementioned tree. Figure 3: The Figure represents the tree structure of a graph with two negative cost cycles, that is < u, v, u > and < j, u, v, j >. Each node is associated with a graph. The root node is the graph G(N, A). Under each node is reported the detected cycle, whereas, the set of arcs is reported over each node. From both graphs G 1 and G 2 the cycle < u, v, u > is removed. Both cycles < u, v, u > and < j, u, v, j > are removed from graphs G 3, G 4 and G 5. In what follows we formalize the search process in the tree. Let G(N, A) be a graph associated with a generic node of the tree. The truncated labeling approach, that is, Algorithm 5 with K i = 1, i N, is run on G(N, A). If a N CC C i is detected, then δ = C i 1 sub-graphs are generated starting from G(N, A) by removing the δ-th arc of C i. In other words, the graph G δ (N, A δ ), δ = 1,..., C i 1 are generated where A δ = A \ {(u δ, v δ )}, with (u δ, v δ ) C i. Since at least one arc of the detected N CC C i is removed, the related sub-graph does not contain this cycle. When the truncated labeling algorithm is run on this sub-graph, if a N CC C i, for some node i N is detected, we are sure that C i C i. This means that at a certain level σ of the tree exactly σ N CCs have been detected. Consequently, in all the sub-graphs at level σ, at least σ N CCs are removed (the same arc can be part of many N CCs). 19

20 3.3.1 Algorithm Let y ζ R N be the vector that stores the best solutions for each node i N, determined so far, and y R N be the vector of the optimal solutions for each node i N. The value ζ is the iterations counter. At the last iteration ζ, y ζ = y, that is y i is the cost of the optimal path from node s to node i. Let y i (G) be the cost of the path from node s to node i obtained applying the truncated labeling algorithm on graph G. If a N CC C j is detected when the truncated labeling algorithm is executed on G, then we have that y i (G) is a valid upper bound for all nodes i N, where N = {u N : v π su, v C j }. This means that y i (G), i N can be used to initialize the cost of node i for the sub-graph G δ. In other words, y i (G δ ) = y i (G), i N. In addition, if y i (G) < y ζ i for some i N, then y ζ i = y i(g). Let L be the list containing the nodes of the tree that have to be processed. The steps of the proposed enumerative approach are reported in Algorithm 7. Algorithm 7 : Enumerative Algorithm 1: Step 1 (Initialization phase) 2: Set: ζ = 0; y ζ [s] = 0; y ζ [i] = +, i N \ {s}; L 0 = {G}, G = (N, A); y G [s] = 0; y G [i] = +, i N. 3: 4: Step 2 (Selection phase) 5: Select a graph Ḡ = (N, Ā) from Lζ and remove it from L ζ. 6: 7: Step 3 (Search process) 8: Apply the truncated labeling algorithm to Ḡ. 9: if a N CC C j is detected then 10: Generate δ = C j 1 sub-graphs G δ = (N, Ā {(u δ, v δ )}) and add them to L ζ ; 11: set: y G δ [i] = y Ḡ [i], i N. 12: if y Ḡ [i] < yζ [i] for some i N then 13: y ζ [i] = y Ḡ [i]. 14: end if 15: y ζ+1 = y ζ ; 16: L ζ+1 = L ζ ; 17: ζ + +; 18: go to Step 4. 19: else 20: if y Ḡ [i] < yζ [i] for some i N then 21: y ζ [i] = y Ḡ [i]. 22: end if 23: y ζ+1 = y ζ ; 24: L ζ+1 = L ζ ; 25: ζ + +; 26: go to Step 4 27: end if 28: 29: Step 4 (Termination check) 30: if L ζ = then 31: STOP. y ζ 1 [i] is the cost of the optimal path from node s to node i. 32: else 33: Go to Step 2. 34: end if 20

21 3.3.2 Correctness of the Enumerative Algorithm The correctness of the method follows from the search procedure. As mentioned above, the nodes represent sub-graphs for which the detected cycle in the father graph is excluded. Each sub-graph is defined by removing at most one arc of the detected cycle. In the sub-graphs of the last level all the cycles are removed, thus for each node i we have a path for each sub-graph that is an elementary path. The path with the smallest cost among those obtained in the sub-graphs is the optimal one Complexity Analysis Let M be the maximum cardinality among those of the N CCs in the graph, that is M = n. We know that the maximum number of N CCs in the graph is C. In addition, for each level σ of the tree exactly σ N CCs have been removed from the graphs at level σ. Thus the maximum number of levels in the tree is C. The complexity of the proposed approach is O(n 2 C τ=0 nτ ). This result is formally stated in the following lemma. Lemma 3. In the worst case, the complexity of the proposed method for the ESPP is O(n 2 C τ=0 nτ ). Proof. In the worst case, the original graph associated with the first node of the tree is solved and a N CC is detected. The branching procedure is applied and δ = n 1 sub-graphs are generated. When each of these sub-graph is solved, another N CC is detected and n 1 graphs of the next level are generated. This means that the number of graphs that have to be solved at level σ is (n 1) σ n σ. Since we know that the levels are bounded above by C, thus the total number of node is C τ=0 nτ. In addition, the complexity to solve each graph associated with a node with Algorithm 5 where K = 1 takes O(n 2 ) (see Lemma 2). It follows that the worst case complexity of the proposed method is O(n 2 C τ=0 nτ ). In Table 3, we show the tree structure explored by Algorithm 7 when solving the network of Figure 2. We report the graph associated with each node of the tree. In Table 4, we report for each iteration, the selected node/graph, the cost associated with each node after Algorithm 5 with K i = 1 is run, the label y ζ and the detected cycle. 21

22 G 4 (N, A \ {(1, 2); (2, 5)}) G 1 (N, A \ {(1, 2)}) G 5 (N, A \ {(1, 2); (5, 4)}) G 6 (N, A \ {(1, 2); (4, 2)}) G 7 (N, A \ {(2, 3); (2, 5)}) G 0 (N, A) G 2 (N, A \ {(2, 3)}) G 8 (N, A \ {(2, 3); (5, 4)}) G 9 (N, A \ {(2, 3); (4, 2)}) G 10 (N, A \ {(3, 1); (2, 5)}) G 3 (N, A \ {(3, 1)}) G 11 (N, A \ {(3, 1); (5, 4)}) G 12 (N, A \ {(3, 1); (4, 2)}) Table 3: Graph associated with the nodes of the tree structure. 22

23 23 y(g) ζ Selected node C 0 G 0 (N, A) < 1, 2, 3, 1 > 1 G 1 (N, A \ {(1, 2)}) < 2, 5, 4, 2 > 2 G 2 (N, A \ {(2, 3)}) < 2, 5, 4, 2 > 3 G 3 (N, A \ {(3, 1)}) < 2, 5, 4, 2 > 4 G 4 (N, A \ {(1, 2); (2, 5)}) G 5 (N, A \ {(1, 2); (5, 4)}) G 6 (N, A \ {(1, 2); (4, 2)}) G 7 (N, A \ {(2, 3); (2, 5)}) G 8 (N, A \ {(2, 3); (5, 4)}) G 9 (N, A \ {(2, 3); (4, 2)}) G 10 (N, A \ {(3, 1); (2, 5)}) G 11 (N, A \ {(3, 1); (5, 4)}) G 12 (N, A \ {(3, 1); (4, 2)}) y ζ Table 4: Information available at the end of each iteration. Under column y(g) we report, for each node i N \ {s}, the cost as output of Algorithm 5 with K i = 1. Column y ζ shows the potential optimal cost for each node i N \ {s}. C indicates the cycle returned by Algorithm 5.

24 4 Computational Experiments The aim of this Section is to evaluate the behaviour of the proposed solution approaches and to compare them in terms of computational cost. The experiments have been carried out on instances generated by ourselves. Indeed, the ESPP is not addressed by the scientific literature, thus no benchmark instances are available. In the next Section, we present the considered instances and how they were generated. 4.1 Test Problems The computational results are carried out on two groups of test problems. The first one is composed of random networks, the instances belonging to the second group are derived from Vehicle Routing Problem (VRP) benchmark test problems. The test problems of the first group (i.e., fully random networks) have been generated randomly by using the Netgen generator of Klinglman et al. [41]. Table 5 shows the characteristics of the considered fully random networks, where the number of nodes, the number of arcs and the density, that is the ratio between the number of arcs and the number of nodes are highlighted. The cost c ij, (i, j) A is randomly generated from the interval [0, 100]. test nodes arcs density R R R R R R R R R R R R R R R Table 5: Characteristic of the fully random networks. For each network, a given number of instances has been generated, in such a way that at least a fixed number of negative cost cycles is present. The procedure used to build the test problems is detailed in what follows. Let #c be the number of negative cost cycles and let leng be the number of arcs belonging to a given cycle. For each 24