- Email: [email protected]
Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem
Journal Pre-proof Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem Raad Salman, Fredrik Ekstedt, Peter Damaschke
PII: DOI: Reference:
S0167-6377(20)30017-1 https://doi.org/10.1016/j.orl.2020.01.009 OPERES 6545
To appear in:
Operations Research Letters
Received date : 25 February 2019 Revised date : 19 June 2019 Accepted date : 29 January 2020 Please cite this article as: R. Salman, F. Ekstedt and P. Damaschke, Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem, Operations Research Letters (2020), doi: https://doi.org/10.1016/j.orl.2020.01.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
Raad Salman1* [email protected] *
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Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem Fredrik Ekstedt1
Peter Damaschke2
[email protected]
[email protected]
Corresponding author. Fraunhofer-Chalmers Centre, Chalmers Science Park, 412 88 Gothenburg, Sweden 2 Department of Computer Science and Engineering, Chalmers University of Technology, 412 96 Gothenburg, Sweden 1
June 18, 2019 Abstract
The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) combines the Generalized Traveling Salesman Problem (GTSP) and the Sequential Ordering Problem (SOP). We present a novel branching technique for the GTSP which enables the extension of a powerful pruning technique. This is combined with some modifications of known bounding methods for related problems. The algorithm manages to solve problem instances with 12-26 groups within a minute, and instances with around 50 groups which are denser with precedence constraints within 24 hours. Keywords: generalized traveling salesman problem; precedence constraints; sequential ordering problem; branch-and-bound; assignment problem; minimum spanning arborescence problem
Background
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The PCGTSP combines features from the Sequential Ordering Problem (SOP) and the Generalized TSP (GTSP), both of which have been studied extensively. Precedence constraints in the SOP have been handled mostly by MILP models together with separation algorithms [1, 2]. More recently, Shobaki & Jamal [17] evaluated a branch-and-bound algorithm with a simpler bounding method where the precedence constraints are ignored, but use a powerful pruning technique where information from previous tree nodes is reused. For the GTSP there are exact algorithms based on branch-and-bound for both the symmetric and asymmetric cases, utilizing integer variable relaxation or Lagrangian relaxation schemes [4, 9, 11]. Noon & Bean [11] have shown how the GTSP can be reduced to the ATSP. Solutions to the ATSP instances produce either optimal solutions [12] or lower bounds [11] for the original GTSP instances. The PCGTSP has recently attracted interest but still remains a largely unexplored problem. Nevertheless, problems of this type often arise, for example, in industrial processes where tasks which can be performed in many different ways are to be sequenced with respect to some order which ensures the integrity of the process. Optimizing such sequences of tasks is of great importance when striving for sustainable and efficient manufacturing [18]. Heuristics with a focus on industrial applications, namely coordinate measurement machines, may be found in Salman et al. [15], together with a handful of other references.
1.1
Contribution and Outline
We present a way to extend parts of the history utilization technique from [17] to the PCGTSP by giving a branching technique and a corresponding subproblem formulation. Futhermore, three different methods for bounding the subproblems are compared.
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Journal Pre-proof Section 2 describes the PCGTSP formally and introduces the notation. Section 3 presents the new branching strategy, Section 4 introduces the various methods for computing lower bounds, and Section 5 presents our strategy for limiting branching. In Section 6 we test the algorithm on industrial and synthetic instances, and Section 7 discusses the results of the tests and gives some suggestions for future development.
Problem Description
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A PCGTSP instance P is defined by a directed graph G = (V, A) where V := {1, . . . , n} denotes the set of vertices, A ⊆ {(i, j) : i, j ∈ V, i 6= j} denotes the set of arcs, and a cost cij ≥ 0 is associated with each arc (i, j) ∈ A. Additionally we let {V1 , . . . , Vm } be a partition of V where Vp , p ∈ M , is called a group, and M := {1, . . . , m}. For each vertex i ∈ V we define g(i) to be the index of the group which contains vertex i, that is i ∈ Vg(i) . We define the precedence constraints by an acyclic digraph G0 = (M, Π) so that if (p, q) ∈ Π then group p must precede group q in a feasible tour. We assume that Π includes all relationships implied by the precedence constraints by transitivity. That is, if (p, q) ∈ Π and (q, r) ∈ Π then (p, r) ∈ Π. The PCGTSP asks to find a minimum-cost closed tour such that exactly one vertex in each group is visited and the precedence constraints are fulfilled. The tour must start and end in group V1 , therefore (1, q) ∈ Π for all q ∈ M \{1}. Note that in any feasible tour the precedence constraints apply only to the path without the last arc returning to V1 , whereas the cost of this last arc is included in the sum of arc costs.
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Branching
We have opted to branch in such a way that each branch chooses the group to visit next in the tour. This means that each node in the tree corresponds to a partial group sequence σ = (Vp1 , . . . , Vpk ) with p1 = 1. The subproblem at this node - the PCGTSP where any feasible tour is constrained to begin with the group sequence σ - is denoted by P(σ). By applying dynamic programming, the minimum-cost paths between each pair of vertices in V1 and Vpk through σ may be computed incrementally. Based on this simple observation, we define a reduced PCGTSP which could be used for computing lower bounds for P(σ). We start by defining a proper subgraph for the reduced PCGTSP by removing all inner groups of σ from the original graph.
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Definition 1. Given a PCGTSP instance P with graph G = (V, A), groups {V1 , . . . , Vm }, precedence graph Π and a fixed sequence σ of already chosen groups, we define a corresponding subgraph G(σ) = (V (σ), A(σ)), group subset M (σ), and a precedence subgraph Π(σ) according to • p ∈ M (σ) iff p ∈ M and any of Vp ∈ / σ, p = 1 or p = pk holds. • v ∈ V (σ) iff v ∈ V and g(v) ∈ M (σ).
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• (i, j) ∈ A(σ) iff (i, j) ∈ A, g(i) ∈ M (σ) and g(j) ∈ M (σ), and none or both of g(i) = 1 and g(j) = pk hold. • (p, q) ∈ Π(σ) iff (p, q) ∈ Π, p ∈ M (σ), and q ∈ M (σ). On this subgraph with corresponding groups and precedence constraints we may define a reduced PCGTSP instance as follows. Definition 2. Given a PCGTSP instance P and fixed sequence σ of already chosen groups, we define P0 (σ) as the PCGTSP instance with subgraph G(σ) = (V (σ), A(σ)), group subset M (σ), precedence subgraph Π(σ), and modified arc costs ci,j 0 (σ) according to 0 when g(i) = 1 and g(j) = pk . i,j c0 (σ) = cij otherwise. Let cmin (σ) be the cost of the shortest path through σ. Then it is fairly straightforward to prove that the following holds: Proposition 1. Cmin (P(σ)) ≥ Cmin (P0 (σ)) + cmin (σ) 2
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Proposition 1 says that computing a lower bound for P0 (σ) and adding cmin (σ) will result in a valid lower bound for P(σ). Equality in Proposition 1 is not guaranteed since the optimal P(σ) tour may have node selections for V1 and Vpk that do not match the optimal path for σ or the optimal P0 (σ) tour or both. There is an alternative way to define edge costs between the segment end groups 1 and pk which gives a reduced PCGTSP with identical optimal value as P(σ). The main reason for using the current approach is that many subproblems P0 (σ) will be identical for different nodes in the branching tree. This may be exploited to reduce computations and will be further explored in Section 5. We choose depth-first search as a branching strategy as this will rapidly deliver new upper bound estimates and is also less memory-intensive compared to breadth-first or best-first. When selecting the next branch to explore, among the possible branches at the same depth, we pick the group Vp with the largest corresponding successor set {q : (p, q) ∈ Π}. The motivation for this stems from the fact that, in an intuitive sense, a group which must precede many other groups should probably be visited early in the tour. When a group sequence with m groups is reached, the vertex choice is optimized by utilizing dynamic programming, and a special 3-opt heuristic [5] is applied. If the solution value obtained is lower than the current best upper bound, then the upper bound is updated.
Relaxation and Bounding
Similarly to the work in [17], we will opt to use a simpler bounding method where the precedence constraints are relaxed. In the following relaxed problems we assume that if (p, q) ∈ Π then all arcs (j, i) such that i ∈ Vp and j ∈ Vq are excluded from the graph. Otherwise the precedence constraints are ignored. This GTSP which is derived from a PCGTSP instance P but without precedence constraints will be referred to as the weak version of P. While additional, more sophisticated steps for eliminating edges based on the precedence constraints have been published [1, 3], these were omitted since initial experiments showed only marginal improvement. To compute bounds for the remaining GTSP we look into two relaxations that have been used for the regular ATSP, the Minumum Spanning Arborescence Problem (MSAP) and the Assignment Problem (AP). Unfortunately, the generalized versions of these problems are both NP-hard, see [10] and [7]. To overcome this, we have tried two different approaches; to either transform the GTSP to an equivalent ATSP, or to relax the vertex choice constraints which allow only one vertex to be visited in each group.
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Definition 3. For any PCGTSP instance P, let NB(P) denote the ATSP instance which arises when applying the Noon-Bean transformation [12] to the weak version of P. The modified arc costs are denoted cNB ij . Since NB(P) is an ATSP instance with n vertices with an optimal solution which is equal to that of the weak version of the original problem P, any lower bound for this problem is also valid for the original PCGTSP instance.
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Definition 4. For any PCGTSP instance P, let NC(P) be the ATSP instance defined on the graph GNC = (M, ANC ) where ANC = {(p, q) : p ∈ M, q ∈ M, (q, p) ∈ / Π}. For every (p, q) ∈ ANC the arc costs NC cNC pq are defined as cpq = min({cij : i ∈ Vp , j ∈ Vq }). As shown in [11], NC(P) is equivalent to the weak version of P but with the vertex choice constraints relaxed. An optimal solution to NC(P) is a lower bound on P so it follows that any lower bound on NC(P) is also valid for P.
4.1
Minimum Spanning Arborescence Problem
Relaxing the outdegree constraints in NB(P) or NC(P) will result in an MSAP instance with n or m vertices, respectively. To further tighten the bound we also add the minimum-cost incoming arc to the root vertex of the arborescence. This is commonly done when bounding the ATSP with an MSA and is known as a 1-arborescence [19].
4.2
Assignment Problem
By relaxing the subtour elimination constraints in NC(P) one obtains the cycle cover problem which asks to find pairwise vertex-disjoint directed cycles that together cover every vertex exactly once, and 3
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5
History Utilization
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minimize the total cost of the arcs in these cycles. The latter problem can be formulated as an assignment problem (AP) as follows. Create two copies M 0 and M 00 of M . The copies of every vertex p ∈ M are denoted p0 and p00 accordingly. Define a bipartite graph with bipartite sets M 0 and M 00 . For every arc (p, q) create the arcs (p0 , q 00 ) and (p00 , q 0 ), having the same cost as (p, q). This yields a one-to-one correspondence between the cycle covers in the given graph and the assignments in the bipartite graph. It should be noted that relaxing the subtour elimination constraints in Noon-Bean transformed problem, NB(P), results in a quite useless problem for the purposes of obtaining a lower bound on P. Since the Noon-Bean transformation defines zero cost cycles within each group, any problem instance where the partition V = V1 ∪ . . . ∪ Vm is not trivial will result in a very weak bound. Hence, this combination of the assignment problem and NB(P) for the bounding problem will not be considered.
In this section we describe a generalization of the history utilization pruning technique developed by Shobaki & Jamal in [17]. Since the vertex choice is never fixed during branching we are not able to extend every part of the technique, but the principal idea is still applicable. The technique revolves around mainly two ideas: (1) identical bounding subproblems in the search tree are very likely to appear, and (2) the “principle of optimality” holds. Point (2) means: Consider any subsequence in any optimal tour. Let p and q be the first and last group, respectively, in this subsequence, and let S be the set of all groups visited in between. Then the subsequence has an optimal order (i.e., with minimum costs) among all possible sequences that start in p, end in q, and visit exactly the set S in between. To describe the history utilization we first define equivalence between tree nodes based on this principle: Definition 5. For every pair (S, r), S ⊆ M , |S| > 1, and r ∈ S, we define T (S, r) to be the set of all tree nodes whose group sequences begin at V1 , traverse all groups in S (and no other groups), and end at group Vr . Any two tree nodes belonging to the same set T (S, r) are said to be equivalent. Consider a fixed unprocessed tree node N (σ) ∈ T (S, r) with partial group sequence σ = (Vp1 , . . . , Vpk ), (σ) p1 = 1, pk = r. Let Pij be a shortest path between the vertex pair i ∈ V1 to j ∈ Vpk , through σ and (σ)
(S,r)
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let c(Pij ) denote its costs. Also, assume that Pij is the shortest path from vertex i ∈ V1 to vertex j ∈ Vpk , through the groups in S, which has been discovered during the branch-and-bound search, and (S,r) (S,r) let c(Pij ) denote its cost. If no tree node in T (S, r) has been processed then c(Pij ) = ∞. (S,r)
(σ)
Proposition 2. If c(Pij ) < c(Pij ), ∀(i, j) ∈ V1 × Vpk , pk = r, then there cannot exist a solution to the PCGTSP which begins with the group sequence σ and has smaller total cost than a solution which (S,r) begins with Pij , ∀(i, j) ∈ V1 × Vpk . In other words, the tree node N (σ) can be pruned.
(σ)
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If it is not possible to prune a tree node based on already explored similar nodes we can forgo solving the bounding subproblem and reuse information from an already explored tree node. This can be done (σ) when using the subproblem definition P0 (σ) since the lower bound zLB in each tree node is separated (σ) into two parts: the prefix, zprefix , which is the cost of the minimum-cost path through σ, and the suffix, zsuffix , which is a lower bound on P0 (σ). Note that we have that the suffix part of the lower bound is equal for all similar tree nodes: (σ)
(S,r)
∀N (σ) ∈ T (S, r).
zsuffix = zsuffix ,
(σ)
(S,r)
If c(Pij ) < c(Pij
) for some (i, j) ∈ V1 × Vpk then N (σ) cannot be pruned according to Proposition (S,r)
(σ)
2. However, if another node in T (S, r) has been processed and zsuffix has been stored, then zLB can be obtained in O(|V1 ||Vpk |) time by computing (σ)
zLB =
min
(σ)
(i,j)∈V1 ×Vpk
(S,r)
(c(Pij )) + zsuffix .
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Results
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The branch-and-bound algorithm was run on a set of 23 synthetic PCGTSP instances [16] and 5 industrial cases of coordinate measuring machines which are used for geometry assurance within the automotive industry. The synthetic instances were created from a library of SOP instances [14] by duplicating vertices and their edge costs, thus creating groups consisting of identical vertices, and then adding a random perturbation to the edge costs within that group. The number of groups in the synthetic instances ranges from 13 to 70, and each group contains between 1 and 9 vertices. For the industrial cases the number of groups range from 12 to 173. Each instance was run with a time limit of 24 hours on a PC with a Intel Core i7-6700k CPU and 32GB of memory. The experiments show that the lower bounds produced by the assignment problem are almost consistently stronger than those of the minimum spanning arborescence problem. This holds both when comparing the end results for each instance and when comparing the bounds at each subproblem node. Even though not all parts of the history utilization technique described in [17] are extended to the PCGTSP in this paper, the algorithm is able to solve the PCGTSP instances which correspond to the SOP instances found in TSPLIB [14] and are solved by Shobaki & Jamals algorithm in [13]. This indicates that the most important properties of this pruning technique has carried over to the PCGTSP. Furthermore, for 12 out of 18 unsolved instances the algorithm produces solutions within 10% of the optimal solution. This means that the optimality gap - the relative error when comparing with the optimal solution when it is known and otherwise the best known lower bound - is within 10%. However, while pruning with the history utilization technique is successful, it seems as if the algorithm produces very weak lower bounds. Lower bounds obtained through the best known solutions for the SOP instances from which the synthetic instances were derived are almost always stronger than the lower bounds produced by the algorithm itself. This is partly due to the depth-first search strategy which only takes the lower bound computed at the root node as the global lower bound but mainly due to the omission of precedence constraints and other relaxations during the bounding procedure.
Discussion and Future Work
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Even though the algorithm succeeds in extending some of the pruning techniques outlined in [17] to the PCGTSP, it could probably be improved by incorporating the precedence constraints in a more sophisticated way. For instance, we are considering relaxations using an assignment problem where the costs are based on feasible paths of L arcs (L fixed) from the original PCGTSP rather than single arcs.
Acknowledgment
The work was carried out in collaboration within Wingquist Laboratory and the Area of Advance Production at Chalmers within the project Smart Assembly 4.0, financed by The Swedish Foundation for Strategic Research. The support is gratefully acknowledged.
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References
[1] N. Ascheuer, M. J¨ unger, G. Reinelt, A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints, Computational Optimization and Applications 17 (2000), pp. 61-84. [2] E. Balas, M. Fischetti, W.R. Pulleyblank, The precedence-constrained asymmetric traveling salesman problem, Mathematical Programming 68 (1995), pp. 241-265. [3] L.F. Escudero, M. Guignard, K. Malik, A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships, Annals of Operations Research 50 (1994), pp. 219237. [4] M. Fischetti, J.J. Salazar Gonsalez, P. Toth, A Branch-And-Cut Algorithm for the Symmetric Generalized Traveling Salesman Problem, Operations Research 45(3) (1997), pp. 378–394. [5] L.M Gambardella and M. Dorigo, An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem, INFORMS Journal on Computing 12(3) (2000), pp 237-255. 5
Journal Pre-proof [6] L. Gouveia and M. Ruthmair, Load-dependent and precedence-based models for pickup and delivery problems, Computers & Operations Research 63 (2015), pp. 56-71. [7] G. Gutin and A. Yeo, Assignment problem based algorithms are impractical for the generalized TSP, Australasian Journal of Combinatorics 27(1) (2003), pp. 149-153.
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[8] M. Held and R.M. Karp, A Dynamic Programming Approach to Sequencing Problems, Journal of the Society for Industrial and Applied Mathematics 10:1 (1962), pp 196-210. [9] G. Laporte, H. Mercure, Y. Nobert, Generalized Travelling Salesman Problem Through n Sets Of Nodes: The Asymmetrical Case, Discrete Applied Mathematics 18 (1987), pp. 185-197. [10] YS. Myung, CH. Lee, DW. Tcha, On the generalized minimum spanning tree problem, Networks 26(4) (1995), pp. 231-241. [11] C.E. Noon and J.C. Bean, A Lagrangian Based Approach for the Asymmetric Generalized Traveling Salesman Problem, Operations Research 39(4) (1991), pp. 623-632. [12] C.E. Noon and J.C. Bean, An Efficient Transformation Of The Generalized Traveling Salesman Problem, INFOR: Information Systems and Operational Research 31(1) (1993), pp. 39-44. [13] V. Papapanagiotoua, J. Jamal, R. Montemanni, G. Shobaki, L.M. Gambardella, A Comparison of Two Exact Algorithms for the Sequential Ordering Problem, 2015 IEEE Conference on Systems, Process and Control (ICSPC), pp. 73-78. [14] G. Reinelt, TSPLIB, https://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/. [15] R. Salman, J.S. Carlson, F. Ekstedt, D. Spensieri, J. Torstensson, R. S¨ oderberg, An industrially validated CMM inspection process with sequence constraints, Procedia CIRP Volume 44 (2016), pp. 138-143. [16] R. Salman, PCGTSPLIB, https://www.fcc.chalmers.se/external/PCGTSPLIB/.
[17] G. Shobaki and J. Jamal, An exact algorithm for the sequential ordering problem and its application to switching energy minimization in compilers, Computational Optimization and Applications 61(2) (2015), pp 343-372.
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[18] R. Soderberg, K, Warmefjord, J.S. Carlson, L. Lindkvist, Toward a Digital Twin for real-time geometry assurance in individualized production, CIRP Annals Volume 66(1) (2017), pp. 137-140.
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[19] D.P. Williamson, Analysis of the Held-Karp lower bound for the asymmetric TSP, Operations Research Letters 12(2) (1992), pp 83-88.
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PII: DOI: Reference:
S0167-6377(20)30017-1 https://doi.org/10.1016/j.orl.2020.01.009 OPERES 6545
To appear in:
Operations Research Letters
Received date : 25 February 2019 Revised date : 19 June 2019 Accepted date : 29 January 2020 Please cite this article as: R. Salman, F. Ekstedt and P. Damaschke, Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem, Operations Research Letters (2020), doi: https://doi.org/10.1016/j.orl.2020.01.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
Raad Salman1* [email protected] *
lP repro of
Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem Fredrik Ekstedt1
Peter Damaschke2
[email protected]
[email protected]
Corresponding author. Fraunhofer-Chalmers Centre, Chalmers Science Park, 412 88 Gothenburg, Sweden 2 Department of Computer Science and Engineering, Chalmers University of Technology, 412 96 Gothenburg, Sweden 1
June 18, 2019 Abstract
The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) combines the Generalized Traveling Salesman Problem (GTSP) and the Sequential Ordering Problem (SOP). We present a novel branching technique for the GTSP which enables the extension of a powerful pruning technique. This is combined with some modifications of known bounding methods for related problems. The algorithm manages to solve problem instances with 12-26 groups within a minute, and instances with around 50 groups which are denser with precedence constraints within 24 hours. Keywords: generalized traveling salesman problem; precedence constraints; sequential ordering problem; branch-and-bound; assignment problem; minimum spanning arborescence problem
Background
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The PCGTSP combines features from the Sequential Ordering Problem (SOP) and the Generalized TSP (GTSP), both of which have been studied extensively. Precedence constraints in the SOP have been handled mostly by MILP models together with separation algorithms [1, 2]. More recently, Shobaki & Jamal [17] evaluated a branch-and-bound algorithm with a simpler bounding method where the precedence constraints are ignored, but use a powerful pruning technique where information from previous tree nodes is reused. For the GTSP there are exact algorithms based on branch-and-bound for both the symmetric and asymmetric cases, utilizing integer variable relaxation or Lagrangian relaxation schemes [4, 9, 11]. Noon & Bean [11] have shown how the GTSP can be reduced to the ATSP. Solutions to the ATSP instances produce either optimal solutions [12] or lower bounds [11] for the original GTSP instances. The PCGTSP has recently attracted interest but still remains a largely unexplored problem. Nevertheless, problems of this type often arise, for example, in industrial processes where tasks which can be performed in many different ways are to be sequenced with respect to some order which ensures the integrity of the process. Optimizing such sequences of tasks is of great importance when striving for sustainable and efficient manufacturing [18]. Heuristics with a focus on industrial applications, namely coordinate measurement machines, may be found in Salman et al. [15], together with a handful of other references.
1.1
Contribution and Outline
We present a way to extend parts of the history utilization technique from [17] to the PCGTSP by giving a branching technique and a corresponding subproblem formulation. Futhermore, three different methods for bounding the subproblems are compared.
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Journal Pre-proof Section 2 describes the PCGTSP formally and introduces the notation. Section 3 presents the new branching strategy, Section 4 introduces the various methods for computing lower bounds, and Section 5 presents our strategy for limiting branching. In Section 6 we test the algorithm on industrial and synthetic instances, and Section 7 discusses the results of the tests and gives some suggestions for future development.
Problem Description
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A PCGTSP instance P is defined by a directed graph G = (V, A) where V := {1, . . . , n} denotes the set of vertices, A ⊆ {(i, j) : i, j ∈ V, i 6= j} denotes the set of arcs, and a cost cij ≥ 0 is associated with each arc (i, j) ∈ A. Additionally we let {V1 , . . . , Vm } be a partition of V where Vp , p ∈ M , is called a group, and M := {1, . . . , m}. For each vertex i ∈ V we define g(i) to be the index of the group which contains vertex i, that is i ∈ Vg(i) . We define the precedence constraints by an acyclic digraph G0 = (M, Π) so that if (p, q) ∈ Π then group p must precede group q in a feasible tour. We assume that Π includes all relationships implied by the precedence constraints by transitivity. That is, if (p, q) ∈ Π and (q, r) ∈ Π then (p, r) ∈ Π. The PCGTSP asks to find a minimum-cost closed tour such that exactly one vertex in each group is visited and the precedence constraints are fulfilled. The tour must start and end in group V1 , therefore (1, q) ∈ Π for all q ∈ M \{1}. Note that in any feasible tour the precedence constraints apply only to the path without the last arc returning to V1 , whereas the cost of this last arc is included in the sum of arc costs.
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Branching
We have opted to branch in such a way that each branch chooses the group to visit next in the tour. This means that each node in the tree corresponds to a partial group sequence σ = (Vp1 , . . . , Vpk ) with p1 = 1. The subproblem at this node - the PCGTSP where any feasible tour is constrained to begin with the group sequence σ - is denoted by P(σ). By applying dynamic programming, the minimum-cost paths between each pair of vertices in V1 and Vpk through σ may be computed incrementally. Based on this simple observation, we define a reduced PCGTSP which could be used for computing lower bounds for P(σ). We start by defining a proper subgraph for the reduced PCGTSP by removing all inner groups of σ from the original graph.
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Definition 1. Given a PCGTSP instance P with graph G = (V, A), groups {V1 , . . . , Vm }, precedence graph Π and a fixed sequence σ of already chosen groups, we define a corresponding subgraph G(σ) = (V (σ), A(σ)), group subset M (σ), and a precedence subgraph Π(σ) according to • p ∈ M (σ) iff p ∈ M and any of Vp ∈ / σ, p = 1 or p = pk holds. • v ∈ V (σ) iff v ∈ V and g(v) ∈ M (σ).
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• (i, j) ∈ A(σ) iff (i, j) ∈ A, g(i) ∈ M (σ) and g(j) ∈ M (σ), and none or both of g(i) = 1 and g(j) = pk hold. • (p, q) ∈ Π(σ) iff (p, q) ∈ Π, p ∈ M (σ), and q ∈ M (σ). On this subgraph with corresponding groups and precedence constraints we may define a reduced PCGTSP instance as follows. Definition 2. Given a PCGTSP instance P and fixed sequence σ of already chosen groups, we define P0 (σ) as the PCGTSP instance with subgraph G(σ) = (V (σ), A(σ)), group subset M (σ), precedence subgraph Π(σ), and modified arc costs ci,j 0 (σ) according to 0 when g(i) = 1 and g(j) = pk . i,j c0 (σ) = cij otherwise. Let cmin (σ) be the cost of the shortest path through σ. Then it is fairly straightforward to prove that the following holds: Proposition 1. Cmin (P(σ)) ≥ Cmin (P0 (σ)) + cmin (σ) 2
(1)
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Proposition 1 says that computing a lower bound for P0 (σ) and adding cmin (σ) will result in a valid lower bound for P(σ). Equality in Proposition 1 is not guaranteed since the optimal P(σ) tour may have node selections for V1 and Vpk that do not match the optimal path for σ or the optimal P0 (σ) tour or both. There is an alternative way to define edge costs between the segment end groups 1 and pk which gives a reduced PCGTSP with identical optimal value as P(σ). The main reason for using the current approach is that many subproblems P0 (σ) will be identical for different nodes in the branching tree. This may be exploited to reduce computations and will be further explored in Section 5. We choose depth-first search as a branching strategy as this will rapidly deliver new upper bound estimates and is also less memory-intensive compared to breadth-first or best-first. When selecting the next branch to explore, among the possible branches at the same depth, we pick the group Vp with the largest corresponding successor set {q : (p, q) ∈ Π}. The motivation for this stems from the fact that, in an intuitive sense, a group which must precede many other groups should probably be visited early in the tour. When a group sequence with m groups is reached, the vertex choice is optimized by utilizing dynamic programming, and a special 3-opt heuristic [5] is applied. If the solution value obtained is lower than the current best upper bound, then the upper bound is updated.
Relaxation and Bounding
Similarly to the work in [17], we will opt to use a simpler bounding method where the precedence constraints are relaxed. In the following relaxed problems we assume that if (p, q) ∈ Π then all arcs (j, i) such that i ∈ Vp and j ∈ Vq are excluded from the graph. Otherwise the precedence constraints are ignored. This GTSP which is derived from a PCGTSP instance P but without precedence constraints will be referred to as the weak version of P. While additional, more sophisticated steps for eliminating edges based on the precedence constraints have been published [1, 3], these were omitted since initial experiments showed only marginal improvement. To compute bounds for the remaining GTSP we look into two relaxations that have been used for the regular ATSP, the Minumum Spanning Arborescence Problem (MSAP) and the Assignment Problem (AP). Unfortunately, the generalized versions of these problems are both NP-hard, see [10] and [7]. To overcome this, we have tried two different approaches; to either transform the GTSP to an equivalent ATSP, or to relax the vertex choice constraints which allow only one vertex to be visited in each group.
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Definition 3. For any PCGTSP instance P, let NB(P) denote the ATSP instance which arises when applying the Noon-Bean transformation [12] to the weak version of P. The modified arc costs are denoted cNB ij . Since NB(P) is an ATSP instance with n vertices with an optimal solution which is equal to that of the weak version of the original problem P, any lower bound for this problem is also valid for the original PCGTSP instance.
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Definition 4. For any PCGTSP instance P, let NC(P) be the ATSP instance defined on the graph GNC = (M, ANC ) where ANC = {(p, q) : p ∈ M, q ∈ M, (q, p) ∈ / Π}. For every (p, q) ∈ ANC the arc costs NC cNC pq are defined as cpq = min({cij : i ∈ Vp , j ∈ Vq }). As shown in [11], NC(P) is equivalent to the weak version of P but with the vertex choice constraints relaxed. An optimal solution to NC(P) is a lower bound on P so it follows that any lower bound on NC(P) is also valid for P.
4.1
Minimum Spanning Arborescence Problem
Relaxing the outdegree constraints in NB(P) or NC(P) will result in an MSAP instance with n or m vertices, respectively. To further tighten the bound we also add the minimum-cost incoming arc to the root vertex of the arborescence. This is commonly done when bounding the ATSP with an MSA and is known as a 1-arborescence [19].
4.2
Assignment Problem
By relaxing the subtour elimination constraints in NC(P) one obtains the cycle cover problem which asks to find pairwise vertex-disjoint directed cycles that together cover every vertex exactly once, and 3
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History Utilization
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minimize the total cost of the arcs in these cycles. The latter problem can be formulated as an assignment problem (AP) as follows. Create two copies M 0 and M 00 of M . The copies of every vertex p ∈ M are denoted p0 and p00 accordingly. Define a bipartite graph with bipartite sets M 0 and M 00 . For every arc (p, q) create the arcs (p0 , q 00 ) and (p00 , q 0 ), having the same cost as (p, q). This yields a one-to-one correspondence between the cycle covers in the given graph and the assignments in the bipartite graph. It should be noted that relaxing the subtour elimination constraints in Noon-Bean transformed problem, NB(P), results in a quite useless problem for the purposes of obtaining a lower bound on P. Since the Noon-Bean transformation defines zero cost cycles within each group, any problem instance where the partition V = V1 ∪ . . . ∪ Vm is not trivial will result in a very weak bound. Hence, this combination of the assignment problem and NB(P) for the bounding problem will not be considered.
In this section we describe a generalization of the history utilization pruning technique developed by Shobaki & Jamal in [17]. Since the vertex choice is never fixed during branching we are not able to extend every part of the technique, but the principal idea is still applicable. The technique revolves around mainly two ideas: (1) identical bounding subproblems in the search tree are very likely to appear, and (2) the “principle of optimality” holds. Point (2) means: Consider any subsequence in any optimal tour. Let p and q be the first and last group, respectively, in this subsequence, and let S be the set of all groups visited in between. Then the subsequence has an optimal order (i.e., with minimum costs) among all possible sequences that start in p, end in q, and visit exactly the set S in between. To describe the history utilization we first define equivalence between tree nodes based on this principle: Definition 5. For every pair (S, r), S ⊆ M , |S| > 1, and r ∈ S, we define T (S, r) to be the set of all tree nodes whose group sequences begin at V1 , traverse all groups in S (and no other groups), and end at group Vr . Any two tree nodes belonging to the same set T (S, r) are said to be equivalent. Consider a fixed unprocessed tree node N (σ) ∈ T (S, r) with partial group sequence σ = (Vp1 , . . . , Vpk ), (σ) p1 = 1, pk = r. Let Pij be a shortest path between the vertex pair i ∈ V1 to j ∈ Vpk , through σ and (σ)
(S,r)
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let c(Pij ) denote its costs. Also, assume that Pij is the shortest path from vertex i ∈ V1 to vertex j ∈ Vpk , through the groups in S, which has been discovered during the branch-and-bound search, and (S,r) (S,r) let c(Pij ) denote its cost. If no tree node in T (S, r) has been processed then c(Pij ) = ∞. (S,r)
(σ)
Proposition 2. If c(Pij ) < c(Pij ), ∀(i, j) ∈ V1 × Vpk , pk = r, then there cannot exist a solution to the PCGTSP which begins with the group sequence σ and has smaller total cost than a solution which (S,r) begins with Pij , ∀(i, j) ∈ V1 × Vpk . In other words, the tree node N (σ) can be pruned.
(σ)
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If it is not possible to prune a tree node based on already explored similar nodes we can forgo solving the bounding subproblem and reuse information from an already explored tree node. This can be done (σ) when using the subproblem definition P0 (σ) since the lower bound zLB in each tree node is separated (σ) into two parts: the prefix, zprefix , which is the cost of the minimum-cost path through σ, and the suffix, zsuffix , which is a lower bound on P0 (σ). Note that we have that the suffix part of the lower bound is equal for all similar tree nodes: (σ)
(S,r)
∀N (σ) ∈ T (S, r).
zsuffix = zsuffix ,
(σ)
(S,r)
If c(Pij ) < c(Pij
) for some (i, j) ∈ V1 × Vpk then N (σ) cannot be pruned according to Proposition (S,r)
(σ)
2. However, if another node in T (S, r) has been processed and zsuffix has been stored, then zLB can be obtained in O(|V1 ||Vpk |) time by computing (σ)
zLB =
min
(σ)
(i,j)∈V1 ×Vpk
(S,r)
(c(Pij )) + zsuffix .
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Results
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The branch-and-bound algorithm was run on a set of 23 synthetic PCGTSP instances [16] and 5 industrial cases of coordinate measuring machines which are used for geometry assurance within the automotive industry. The synthetic instances were created from a library of SOP instances [14] by duplicating vertices and their edge costs, thus creating groups consisting of identical vertices, and then adding a random perturbation to the edge costs within that group. The number of groups in the synthetic instances ranges from 13 to 70, and each group contains between 1 and 9 vertices. For the industrial cases the number of groups range from 12 to 173. Each instance was run with a time limit of 24 hours on a PC with a Intel Core i7-6700k CPU and 32GB of memory. The experiments show that the lower bounds produced by the assignment problem are almost consistently stronger than those of the minimum spanning arborescence problem. This holds both when comparing the end results for each instance and when comparing the bounds at each subproblem node. Even though not all parts of the history utilization technique described in [17] are extended to the PCGTSP in this paper, the algorithm is able to solve the PCGTSP instances which correspond to the SOP instances found in TSPLIB [14] and are solved by Shobaki & Jamals algorithm in [13]. This indicates that the most important properties of this pruning technique has carried over to the PCGTSP. Furthermore, for 12 out of 18 unsolved instances the algorithm produces solutions within 10% of the optimal solution. This means that the optimality gap - the relative error when comparing with the optimal solution when it is known and otherwise the best known lower bound - is within 10%. However, while pruning with the history utilization technique is successful, it seems as if the algorithm produces very weak lower bounds. Lower bounds obtained through the best known solutions for the SOP instances from which the synthetic instances were derived are almost always stronger than the lower bounds produced by the algorithm itself. This is partly due to the depth-first search strategy which only takes the lower bound computed at the root node as the global lower bound but mainly due to the omission of precedence constraints and other relaxations during the bounding procedure.
Discussion and Future Work
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Even though the algorithm succeeds in extending some of the pruning techniques outlined in [17] to the PCGTSP, it could probably be improved by incorporating the precedence constraints in a more sophisticated way. For instance, we are considering relaxations using an assignment problem where the costs are based on feasible paths of L arcs (L fixed) from the original PCGTSP rather than single arcs.
Acknowledgment
The work was carried out in collaboration within Wingquist Laboratory and the Area of Advance Production at Chalmers within the project Smart Assembly 4.0, financed by The Swedish Foundation for Strategic Research. The support is gratefully acknowledged.
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References
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