1 Introduction
Given a node, arc complete directed weighted graph , the Asymmetric Traveling Salesman Problem [1] (ATSP) consists in finding a partial subgraph of which forms a Hamiltonian circuit of minimum cost. This NPhard problem is one of the most studied by the Operation Research community. It has various practical applications such as vehicle routing problems of logistics, microchips production optimization or even scheduling.
The symmetric TSP is well handled by linear programming techniques
[1]. However, such methods suffer from the addition of side constraints and asymmetric cost matrix, whereas constraint programming models do not. Since the real world is not symmetric and industrial application often involve constraints such as time windows, precedences, loading capacities and several other constraints, improving the CP general model for solving the ATSP leads to make CP more competitive on real world routing problems. Recent improvements on cost based relaxations [4] had a very strong impact on the ability of CP technologies to solve the TSP. In this paper, we investigate how the graph structure can contribute to the resolution process, in order to tackle larger instances. For this purpose, we developed usual and original filtering algorithms using classical graph structures, such as strongly connected components or dominators. We analyzed their behavior both from a quantitative (time complexity) and a qualitative (consistency level) point of view. Also, we experimentally show that such implied propagators bring robustness to hard instances, and highlight the fact that the graph structure can significantly improve the behavior of search heuristics. Our main contribution is both a theoretical and an experimental study which lead to a robust model that outperforms state of the art results in CP.This paper is divided into six main parts. Section 2 provides some vocabulary and notations. Section 3 discusses the state of the art implied constraints. Next, we show in Section 4 how the reduced graph can provide useful information for pruning and improving existing models. In Section 5 we provide some improvements about the implementation of the Held and Karp method within a directed context. Section 6 shows an experimental study on the ATSP and some openings about its symmetric version (TSP). Section 7 concludes the paper with several perspectives.
2 Background
Let us consider a directed graph . A Strongly Connected Component (SCC) is a maximal subgraph of such that for each pair of nodes , a path exists from to and from to . A reduced graph of a directed graph represents the SCC of . This graph is obtained by merging the nodes of which are in the same SCC and removing any loop. Such a graph is unique and contains no circuit. We link and with two functions: and . The method can be represented by one size integer array. Also, since each node of belongs to exactly one SCC of , the method can be represented by two integer arrays: the first one represents the canonical element of each SCC while the second one links nodes of the same SCC, behaving like a linked list. Those two arrays have respectively size of and , where . The transitive closure of is a graph representing node reachability in , i.e. such that if and only if a path from to exists in .
In a CP context a Graph Variable can be used to model a graph. Such a concept has been introduced by Le Pape et al. [24] and detailed by Régin [28] and Dooms et al. [9]. We define a graph variable by two graphs: the graph of potential elements, , contains all the nodes and arcs that potentially occur in at least one solution whereas the graph of mandatory elements, , contains all the nodes and arcs that occur in every solution. It has to be noticed that . During resolution, decisions and filtering rules will remove nodes/arcs from and add nodes/arcs to until the Graph Variable is instantiated, i.e. when . It should be noticed that, regarding the TSP, , so resolution will focus on and : branching strategies and propagators will remove infeasible arcs from and add mandatory arcs of into .
3 Related Work
This section describes the state of the art of existing approaches for solving ATSP with CP. We distinguish the structural filtering, which ensures that a solution is a Hamiltonian path, from cost based pruning, which mainly focus on the solution cost. Then, we study a few representative branching heuristics.
Given, a directed weighted graph , and a function the ATSP consists in finding a partial subgraph of which forms a Hamiltonian circuit of minimum cost. A simple ATSP model in CP, involving a graph variable , can basically be stated as minimizing the sum of costs of arcs in the domain of and maintaining to be a Hamiltonian circuit with a connectivity constraint and a degree constraint (one predecessor and one successor for each node). However, it is often more interesting to convert such a model in order to find a path instead of a circuit [15, 25]. Our motivation for this transformation is that it brings graph structure that is more likely to be exploited.
In this paper, we consider the ATSP as the problem of finding a minimum cost Hamiltonian path with fixed start and end nodes in a directed weighted graph. In the following, respectively denote the start and the end of the expected path. and are supposed to be known. They can be obtained by duplicating any arbitrary node, but it makes more sense to duplicate the node representing the salesman’s home.
3.1 Structural filtering algorithms
Our formulation of the ATSP involves the search of a path instead of a circuit, the degree constraints has thus to be stated as follows: For all and for any , where (respectively ) denotes the number of successors (respectively predecessors) of . Extremal conditions, being , are ensured by the initial domain of the graph variable. An efficient filtering can be obtained with two special purpose incremental propagators. One reacts on mandatory arc detections: whenever arc is enforced, other outgoing arc of and ingoing arcs of can be pruned. The other reacts on arc removals: whenever a node has only one outgoing (or ingoing) potential arc left, this arc is mandatory and can be enforced. A higher level of consistency can be achieved by using a graphbased constraint maintaining a nodesuccessor perfect matching [27]. Deleting circuits is the second important structural aspect of the TSP. Caseau and Laburthe [7] suggested the simple and efficient constraint to remove circuits of the graph. Their fast incremental algorithm is based on the subpaths fusion process. It runs in constant time per arc enforcing event. The conjunction of this circuit elimination constraint and the above degree constraints is sufficient to guarantee that the solution is a Hamiltonian path from to .
However, other implied constraints provide additional filtering that may help the resolution process. For instance, Quesada [26] suggested the general propagator which maintains the transitive closure and the dominance tree of the graph variable. However, its running time, in the worst case, makes it unlikely to be profitable in practice. A faster constraint, also based on the concept of dominance, is the constraint. It is nothing else but a simplification of the constraint [2] recently improved to a worst case time complexity [12]. Given a graph variable and a node , such a constraint ensures that is an arborescence rooted in node . More precisely, it enforces GAC over the conjunction of the following properties: has no circuit, each node is reachable from and, each node but has exactly one predecessor. Such a filtering can also be used to define the by switching with and predecessors with successors.
A dual approach consists in assigning to each node its position in the path. In such a case, the position of a node is represented by an integer variable with initial domain . Positions are different from a node to another, which can be ensured by an constraint. Since the number of nodes is equal to the number of positions, the bound consistency algorithm of constraint only requires time. Plus, a channeling has to be done between the graph variable and position variables. Such a channeling requires worst case time. In particular, lower bounds of positions are adjusted according to a single Breadth First Search (BFS) of while upper bounds of positions are shortened by a BFS of . It has to be noticed that this approach is related to disjunctive scheduling [33]: nodes are tasks of duration 1 which are executed on the same machine. The structure of the input graph creates implicit precedence constraints.
3.2 Costbased filtering algorithms
CP models often embed relaxation based constraints, to provide inference from costs. Fischetti and Toth [14] suggested a general bounding procedure for combining different relaxations of the same problem.
The most natural relaxation is obtained by considering the cheapest outgoing arc of each node: . Such a lower bound can be computed efficiently but it is in general relatively far from the optimal value.
A stronger relaxation is the weighted version of the constraint, corresponding to the Minimum Assignment Problem (MAP). It requires time [23] to compute a first minimum cost assignment but then time [6] to check consistency and filter incrementally. Some interesting evaluations are provided by [16], but are mainly related to the TSP with time windows constraints.
A widely exploited subproblem of the ATSP is the Minimum Spanning Tree (MST) problem where the degree constraint and arc direction are relaxed. We remark that a hamiltonian path is a spanning tree and that it is possible to compute a MST with a degree restriction at one node [17]. A MST can be computed in two ways. The first one is Kruskal’s algorithm, which runs in worst case time, where is the inverse Ackermann function, but requires edges to be sorted according to their weights. Sorting edges can be done once and for all in time. The second option is to use Prim’s algorithm which requires time with Fibonacci heaps [17] or time if binomial heaps are used instead. Based on Kruskal’s algorithm, Régin et al. [29, 30] made the constraint which ensures consistency, provides a complete pruning and detects mandatory arcs incrementally, within time. Dooms and Katriel [10, 11] presented a more complex constraint which maintains a graph and its spanning tree, pruning according to King’s algorithm [22].
An improvement of the MST relaxation is the approach of Held and Karp [19], adapted for CP by Benchimol et al.[4]. It is the Lagrangian MST relaxation with a policy for updating Langrangian multipliers that provides a fast convergence. The idea of this method is to iteratively compute MST that converge towards a path by adding penalties on arc costs according to degree constraints violations. It must be noticed that since arc costs change from one iteration to another, Prim’s algorithm is better than Kruskal’s which requires to sort edges. Moreover, to our knowledge neither algorithm can be applied incrementally.
A more accurate relaxation is the Minimum Spanning Arborescence (MSA) relaxation, since it does not relax the orientation of the graph. This relaxation has been studied by [14, 15] who provide a time filtering algorithm based on primal/dual linear programs. The best algorithm for computing a MSA has been provided by Gabow et al. [17]. Their algorithm runs in worst case time, but it does not provide reduced costs that are used for pruning. Thus, it could be used to create a constraint with a time consistency checking but the complete filtering algorithm remains in time. The Lagrangian MSA relaxation, with a MSA computation based on Edmonds’ algorithm, has been suggested in [7]. This method was very accurate but unfortunately unstable. Also, Benchimol et al. [4] report that the MSA based Held and Karp scheme lead to disappointing results.
3.3 Branching heuristics
Branching strategies forms a fundamental aspect of CP which can drastically reduce the search space. We study here dedicated heuristics, because the literature is not clear about which branching heuristic should be used.
Pesant et al. have introduced Sparse heuristic [25] which has the singularity of considering occurrences of successors and ignoring costs. In this way, this heuristic is based on the graph structure. It behaves as following: First, it selects the set of nodes with no successor in and the smallest set of successors in . Second, it finds the node which maximize . The heuristic then iterates on ’s successors. This process is optimized by performing a dichotomic exploration of large domains.
However, very recently, Benchimol et al. [4] suggested a binary heuristic, based on the MST relaxation, that we call RemoveMaxRC. It consists in removing from the tree arc of maximum replacement cost, i.e. the arc which would involve the highest cost augmentation if it was removed. By tree arc, we mean the fact that it appears in the MST of the last iteration of the Held and Karp procedure. Acutally, as shown in section 6, this branching leads to poor results and should not be used.
Finally, Focacci et al. solve the TSPTW [15] by guiding the search with time windows, which means that the efficiency of CP for solving the ATSP should not rely entirely on its branching heuristic.
4 Considering the reduced graph
In this section, we consider a subproblem which is not a subset of constraints, as usual, but consists in the whole ATSP itself applied to a more restrictive scope: the reduced graph of . The structure of the reduced graph has already been considered in a similar way for path partitioning problems [3, 5]. In this section, we first study structural properties that arise from considering the reduced graph. Second, we show how to adapt such information to some state of the art implied models, including cost based reasonings.
4.1 Structural properties
We introduce a propagator, the propagator, which makes the reduced graph a (Hamiltonian) simple path and ensures by the way that each node is reachable from and can reach . It is a monotonic generalization of the algorithm depicted in [31]. Necessary conditions for this propagator have already been partially highlighted in [5]. We first modify them in order to fit with the TSP and our model. Next, we provide a linear time incremental algorithm.
Definition 1
Reduced path guarantees that any arc in that connects two SCC, is part of a simple path which go through every SCC of .
Proposition 1
Given any directed graph , its reduced graph contains at most one Hamiltonian path.
Proof
Let us consider a graph such that contains at least two Hamiltonian paths and , . Since both and are Hamiltonian then there exists at least two nodes , , such that is visited before in and after in . Thus, the graph contains a path from to and from to . This is a circuit. As , also contains a circuit, which is a contradiction. ∎
We note the reduced graph of . We remark that, as has no predecessor then its SCC is the node itself. Also, as has no successor then . To distinguish nodes of from nodes of the reduced graph, we note and . It follows that any Hamiltonian path in will be from to .
Proposition 2
If there exists a Hamiltonian path from to in then there exists a Hamiltonian path in .
Proof
Lets suppose that has no Hamiltonian path from to . Then for any path in starting at and ending at , there exist at least one node , which is not visited by . Thus, for any path in starting at and ending at , there exist at least one SCC which is not traversed by , so , then . Thus any path in starting at and ending at is not Hamiltonian. ∎
It follows that any transitive arc of must be pruned and that remaining arcs of are mandatory (otherwise the graph becomes disconnected): any SCC, but , must have exactly one outgoing arc. An example is given in figure 1: the graph contains four SCC. Its reduced graph, , has a unique Hamiltonian path . Arcs of are infeasible so and must be pruned from .
We introduce a new data structure in that we call : for each node , is the list of arcs and . We can now easily draw a complete filtering algorithm for the propagator which ensures the GAC over the property that must be a path in time:

Data structures: Compute the SCC of (with Tarjan’s algorithm [32]) and build the reduced graph .

Taking mandatory arcs into account: such that and , remove arc .

Consistency checking: Make a path if possible, fail otherwise.

for each arc such that and , ,

Pruning: if , remove arc .

Enforcing: if and is the only arc of that links and , enforce arc .

A procedure performing step 3 starts on node , finds its right successor (the one which has only one predecessor) and removes other outgoing arcs. Then, the same procedure is applied on and so on, until is reached. Such an algorithm must be performed once during the initial propagation. Then, the propagator reacts to fine events. To have an incremental behavior, the propagator must maintain the set of SCC and the reduced graph. Haeupler et al. [18] worked on maintaining SCC and a topological ordering of nodes in the reduced graph, but under the addition of arcs. We deal with arc deletions. Moreover, we may have lots of arc deletions per propagation (at least for the first ones), thus we should not use a completely online algorithm.
SCC maintenance: Let us consider an arc such that and . If and if is removed from then it must be removed from also. If then the removal of may split the SCC , so computation is required. As many arcs can be removed from one propagation to another, we suggest a partially incremental algorithm which computes exclusively SCC that contains at least one removed arc, each one exactly once. We introduce a bit set to mark nodes of . Initially, each node is unmarked, then when a removed arc, inside an unmarked SCC , is considered, we apply Tarjan’s algorithm on and mark . Tarjan’s algorithm will return either if is still strongly connected, or the new set of SCC induced by all arc removals from . In both cases, we can ignore other arcs that have been removed from . Since the SCC of a graph are nodedisjoint, the overall processing time of a propagation dealing with arc deletions involving some SCC is .
maintenance and filtering: Algorithm 1 shows how to get an incremental propagator that reacts to SCC splits. When a SCC is split into new SCC, the reduced graph gets new nodes and must be turned into a path while some filtering may be performed on . The good thing is that there is no need to consider the entire graph. We note the set of nodes induced by the breaking of SCC . Since was a path at the end of the last propagation, we call the predecessor of in and its successor. Then, we only need to consider nodes of in . To compute new arcs in it is necessary to examine arcs of , but only and arcs that have the tail in a SCC of need to be considered. Note that we filter during the maintenance process.
Once we get those data structures, then it is worth exploiting them the most we can, to make such a computation profitable. Especially, a few trivial ad hoc rules come when considering SCC. We call an indoor a node with a predecessor outside its own SCC, an outdoor, a node with a successor outside its SCC and a door a node which is an indoor or/and an outdoor.
Proposition 3
If a SCC has only one indoor , then any arc is infeasible.
Proof
First we remark that cannot be since has no predecessors. Let us then suppose that such arc is enforced. As the TSP requires nodes of to have exactly one predecessors, all other predecessors of will be pruned. As was the only indoor of , then is not reachable anymore from , which is by Proposition 2 a contradiction. ∎
By symmetry, if a SCC has only one outdoor , then any arc is infeasible. Moreover, if a SCC of more than two nodes has only two doors , then arcs and are infeasible.
4.2 Strengthening other models
In general, the reduced graph provides three kinds of information: Precedences between nodes of distinct SCC; Reachability between nodes of the graph; Cardinality sets . Such information can be directly used to generate lazy clauses [13]. It can also improve the quality of the channeling between the graph variable and position variables by considering precedences: When adjusting bounds of position variables (or time windows), the BFS must be tuned accordingly, processing SCC one after the other.
Some propagators such as [26], require the transitive closure of the graph. Its computation requires worst case time in general, but since the reduced graph is now a path, we can sketch a trivial and optimal algorithm: For any node , we call the set of nodes reachable from in and the set of nodes reachable from in , including . More formally, and , where . Then, for any node , . As is a path, iterating on requires operations. Also, since SCC are nodedisjoints, computing takes because and . As , the computation of the transitive closure takes which is bounded by . It can be performed incrementally by considering SCC splits only.
Finally we show how the MST relaxation of the TSP can be improved by considering the reduced graph. We call a Bounding Spanning Tree (BST) of a spanning tree of obtained by finding a minimum spanning tree in every SCC of independently and then linking them together using the cheapest arcs:
. 
The resulting spanning tree provides a tighter bound than a MST. Indeed, since BST and MST both are spanning trees, , otherwise MST is not minimal.
We will now see how to improve the (WST) constraint, leading to the (BST) propagator. We assume that the reader is already familiar with this constraint, otherwise papers [29, 30] should be considered as references. The BST can replace the MST of the WST constraint: the pruning rules of WST constraint will provide more inference since the bound it tighter. Actually, we can do even better by slightly modifying the pruning rule of the WST constraint for arcs that are between two SCC: an arc linking two SCC can only replace (or be replaced by) another arc linking those two same SCC. Consider a BST of cost , the upper bound of the objective variable , an arc and a tree arc , we can rephrase the pruning rule by: Any arc is infeasible if . The reader should notice that no Lowest Common Ancestor (LCA) query is performed here. This do not only accelerate the algorithm, it also enables more pruning, because a LCA query could have returned an arc that does not link SCC and . Such an arc cannot replace since exactly one arc of is mandatory.
We now briefly describe a simple and efficient way to compute the BST. We assume that the propagator has been applied and that is thus a path. Initially the BST is empty. First, we add to the BST mandatory arcs of , then for each we add . Finally, we run Kruskal’s algorithm as described in [29, 30] until the BST has arcs. A faster way to compute a BST is to perform Prim’s algorithm on successive SCC, but this method does not enable to use the efficient filtering algorithm of Régin [29].
Figure 2 illustrates this relaxation : the input directed graph, on figure 2(a), is composed of four SCC and . For simplicity purpose, costs are symmetric. Its minimum hamiltonian path, figure 2(b), costs 28 and we will suppose that such a value is the current upper bound of the objective variable. The MST of the graph, figure 2(c), only costs 19, which is unfortunately too low to filter any arc. Instead, the BST, figure 2(d), is much more accurate. It actually consists of the MST of each SCC, with respective costs , and the cheapest arcs that connect SCC each others: with respective costs . Thus, the entire BST costs 27. It is worth noticing that it enables to filter arcs and . Indeed, can only replace in the relaxation, so its marginal cost is which is strictly greater than the upper bound of the objective. The same reasoning enables to prune .
5 The Held and Karp method
The Lagrangian relaxation of Held and Karp has initially been defined for solving symmetric TSP. Instead of converting asymmetric instances into symmetric ones, through the transformation of Jonker and Volgenant [4], we can directly adapt it to the asymmetric case: at each iteration , we define two penalties per node , and , respectively equal to and . We note the indegree of in the MST of iteration whereas is its outdegree and is a constant whose calculation is discussed in [19, 20]. As a path is expected, we post . Arc costs are then changed according to: . It has to be noticed that, since it relies on the computation of successive MST, such a model is equivalent to what would give the usual Held and Karp scheme used on a transformed instance. However, this framework is more general and can also handle the computation of Minimum Spanning Arborescence.
Such a method should be implemented within a specific propagator to be easily plugged, or unplugged, into a constraint. We noticed that keeping track of Lagrangian multipliers from one propagation to another, even upon backtracking, saves lots of computations and provides better results on average. Our approach is based on a few runs. A run is composed of iterations in which we compute a MST according to Prim’s algorithm and update and the cost matrix. Then, we run a Kruskal’s based MST to apply the complete filtering of [4, 29]. We first chose but this led to disappointing results when scaling up to a hundred nodes. We thus decided to fix to a constant. The value appeared as a good compromise between speed and accuracy. Remark that, as we perform a fix point, the method may be called several times per search node, and since it is relatively slow, we always schedule this propagator at the end of the propagation queue.
This procedure has the inconvenient of not being monotonic^{1}^{1}1A propagator , involving a graph variable and a filtering function is said to be monotonic [31] iff for any , where . (it is not even idempotent): filtering, related to other propagators, can slow down the convergence of the method. The intuition is that to go from a MST to an optimal tour, it may be easier to use some infeasible arcs during the convergence process. One can see an analogy with local search techniques that explore infeasible solutions in order to reach the best (feasible) ones more quickly [8]. This fact, which occured during some of our experiments involving static branching heuristics, breaks the usual saying the more filtering, the better. Moreover, it follows that we cannot measure precisely the improvement stemming from additional structural filtering. We mention that the BST relaxation can be used within the Held and Karp scheme, however, this may also affect the convergence of the method and thus sometimes yield to poorer results. For that reason, we recommend to use a Lagrangian BST relaxation in addition to, rather than in replacement of, the usual Held and Karp procedure.
6 Experimental study
This section presents some experiments we made in order to measure the impact of the graph structure. We will show that branching according to graph structure only outperforms current state of the art results while using implied filtering based on graph structure avoids pathological behaviors on hard instances at a negligible time consumption. Our implementations have been done within the solver which is an open source Java library. Tests have been performed on a Macbook pro under OS X 10.7.2 and with a 2.7 GHz Intel core i7 and 8Go of DDR3. We set a limit of 3 Go to be allocated to the JVM. We tested TSP and ATSP instance of the TSPLIB. For each one, we refer to the number of search nodes by nodes and report time measurements in seconds. As in [4], we study optimality proof and thus provide the optimal value as an upper bound of the problem. We computed equivalent state of the art results (SOTA) (referred as 1tree with filtering in [4]), to position our model in general. Their implementation is in C++ and has no memory restriction.
Our implementation (referred as BASIC) involves one graph variable, one integer variable (the objective) and one single constraint that is composed of several propagators. Subtour elimination is performed by a special purpose incremental propagator, inspired from the constraint [7]. The degree constraint is ensured by special purpose incremental propagators described in section 3.1. The objective is adjusted by the natural relaxation and an implied propagator, based on the Held and Karp method. We mention that we solved rbg instances (that are highly asymmetric) by replacing the tree based relaxation by a Minimum Assignment Problem relaxation (also in SOTA). For that, we have implemented a simple Hungarian algorithm. Indeed, it always provided the optimal value as a lower bound at the root node. When a relaxation finds an optimal solution, this one can be directly enforced [4]. However, it could be in contradiction with side constraints. Thus, we unplugged such a greedy mode. The solver works under a trailing environment.
6.1 Dedicated heuristics
We experimentally compare the branching heuristics RemoveMaxRC and Sparse of section 3.3.
We also introduce three variants of these methods:
 EnforceMaxRC, consists in enforcing the tree arc of maximum replacement cost. It is the opposite of RemoveMaxRC.
 RemoveMaxMC
, consists in removing the non tree arc of maximum marginal cost, i.e. the arc which would involve the highest cost augmentation if it was enforced. This heuristic may require an important number of decisions to solve the problem. There are low probabilities to make wrong decisions, but if a mistake has been performed early in the search tree, it might be disastrous for the resolution.
 EnforceSparse, which first selects the set of nodes with no successor in and the smallest set of successors in . Second, it finds the node which maximize . Then it fixes the successor of by enforcing the arc such that is maximal.
All branching heuristics are performed in a binary tree search. RemoveMaxRC, RemoveMaxMC and Sparse can be said to be reduction heuristics. They respectively involve a worst case depth for the search tree of , and . In contrast, EnforceMaxRC and EnforceSparse perform assignments, leading to a depth in the worst case. Assignment heuristics perform strong decisions that bring more structure in left branches of the search tree while it is the opposite for reduction branchings that restrict more right branches. An exception is Sparse which has a balanced impact on both branches.
6.2 Structural filtering
We then study the benefit we could get from adding some implied filtering algorithms to the BASIC model
with RemoveMaxMC and Sparse heuristics:
 ARB: and propagators used together.
 POS: The model based on nodes position, with an constraint performing bound consistency.
 AD: propagator, adapted to graph variables, with GAC.
 BST: propagator with a Lagrangian relaxation based on a BST, in addition to the usual HeldKarp scheme.
 ALL: combine all above propagators.
Cost based heuristics  Graph structure based heuristics  
SOTA[4]  RemoveMaxRC  EnforceMaxRC  RemoveMaxMC  Sparse  EnforceSparse  
instance  nodes  time  nodes  time  nodes  time  nodes  time  nodes  time  nodes  time 
br17  223,603  26.64  513  1.17  14  0.02  120  0.55  12  0.02  11  0.13 
ft53  1  0.07  7  0.09  7  0.12  7  0.13  3  0.08  3  0.10 
ft70  138  0.81  31  0.19  33  0.15  52  0.22  16  0.12  16  0.13 
ry48p  364  1.01  143  0.38  53  0.19  1,135  2.25  71  0.28  50  0.20 
ftv33  3  0.02  2  0.01  2  0.01  2  0.01  2  0.01  2  0.01 
ftv35  41  0.08  12  0.03  29  0.05  120  0.19  25  0.08  27  0.05 
ftv38  87  0.15  42  0.08  26  0.06  201  0.32  18  0.05  20  0.04 
ftv44  227  0.47  101  0.22  62  0.13  584  1.22  35  0.10  34  0.09 
ftv47  471  0.89  144  0.34  247  0.43  648  1.55  81  0.24  105  0.32 
ftv55  2,155  4.22  614  1.24  596  1.32  2,580  5.00  54  0.24  90  0.33 
ftv64  2,111  7.22  1,724  4.34  695  1.90  12,665  25.02  104  0.57  115  0.64 
ftv70  5,992  25.43  6,294  19.08  2,936  8.79  68,673  144.63  88  0.59  151  0.88 
kro124p  5,670  53.09  1,742  7.52  1,671  8.32  8,845  37.50  158  1.31  184  1.43 
ftv170  85,244  342,691  391,842  411,955  23,457  275.33  14,331  155.36  
p43  990,440  2,681,727  2,569,776  2,546,104  1,383,073  1,628.26  33,251  53.48  
rbg323  3,134,515  3,343  26.90  262  3.09  563  10.55  339  13.76  
rbg358  2,636,522  268  2.72  643  7.95  284  4.47  
rbg403  34,132  316  10.34  1,000  46.10  313  18.90  
rbg443  5,596  339  13.59  1,121  62.18  342  26.83 
BASIC  ARB  POS  AD  BST  ALL  
instance  nodes  time  nodes  time  nodes  time  nodes  time  nodes  time  nodes  time 
br17  11  0.13  11  0.05  11  0.06  11  0.04  11  0.10  11  0.08 
ft53  3  0.10  2  0.13  3  0.12  3  0.11  3  0.13  1  0.15 
ft70  16  0.13  13  0.01  16  0.15  10  0.12  10  0.26  13  0.35 
ry48p  50  0.20  56  0.20  50  0.19  75  0.26  49  0.26  41  0.23 
ftv33  2  0.01  2  0.01  2  0.01  2  0.01  2  0.01  2  0.01 
ftv35  27  0.05  22  0.06  27  0.05  8  0.03  24  0.06  8  0.04 
ftv38  20  0.04  20  0.05  20  0.05  14  0.04  20  0.06  14  0.06 
ftv44  34  0.09  34  0.08  34  0.09  37  0.09  32  0.10  27  0.15 
ftv47  105  0.32  127  0.39  105  0.32  67  0.19  93  0.37  60  0.24 
ftv55  90  0.33  99  0.35  90  0.35  54  0.24  74  0.38  67  0.36 
ftv64  115  0.64  113  0.56  115  0.60  124  0.65  96  0.66  123  0.76 
ftv70  151  0.88  132  0.85  137  0.87  105  0.64  132  1.01  104  0.80 
kro124p  184  1.43  147  1.37  184  1.43  223  1.68  219  1.94  150  1.50 
ftv170  14,331  155.36  3,900  40.18  8,972  99.36  2,561  28.63  5,368  71.15  2,565  31.94 
p43  33,251  53.48  5,255  13.53  4,597  12.24  15,251  30.47  1,003  6.39  1,742  10.37 
rbg323  339  13.76  339  13.53  339  14.55  267  3.38  339  14.05  267  3.59 
rbg358  284  4.47  284  5.44  284  6.68  283  5.83  284  5.45  283  6.94 
rbg403  313  18.90  313  24.83  313  21.69  308  22.72  313  20.69  308  23.56 
rbg443  342  26.83  342  26.62  342  28.20  339  27.63  342  26.69  339  28.82 
6.3 Results and analysis
Table 2 provides our experimental results over the impact of branching strategies. RemoveMaxRC can be seen as our implementation of the SOTA model. The main differences between these two are the fact that we perform a fixpoint and implementation details of the Held and Karp scheme. As it can be seen, our Java implementation is faster and more robust (br17, kro124p and rbg323). Results clearly show that the most recently used heuristic [4] is actually not very appropriate and that the more natural EnforceMaxRC is much more efficient. EnforceMaxRC is in general better than RemoveMaxRC, showing the limits of the first fail principle. The worst heuristic is clearly RemoveMaxMC while the best ones are EnforceMaxRC, Sparse and EnforceSparse. More precisely, graph based heuristics are the best choice for the ftv set of instances whereas EnforceMaxRC behaves better on rbg instances. This efficiency (not a single failure for instances rbg) is explained by the fact that EnforceMaxRC is driven by the MAP relaxation, which is extremely accurate here. In contrast, Sparse heuristic does too many decisions (because of the dichotomic branching), even if the number of wrong ones is negligible. Results show that assignment based branchings work better than reduction heuristics. The most robust branching strategy is EnforceSparse. Thus, graph structure can play a significant role during the search. However, side constraints of real world applications might require to use dedicated heuristics, so results should not entirely rely on the branching strategy.
We now study the impact of implied filtering algorithms on robustness of the model. For that, we consider the best heuristic, EnforceSparse, and solve TSPLIB’s instances within different model configurations. Results can be seen on Table 2. It can be seen that implied algorithms do not significantly increase performances on all instances, but it seriously improves the resolution of hard ones (ftv170 and p43 are solved 5 times faster by ALL). Moreover, those extra algorithms are not significantly time expensive. The eventual loss is more due to model instability rather than filtering algorithms’ computing time. Indeed, a stronger filtering sometimes yields to more failures because it affects both the branching heuristic and the Langrangian relaxation’s convergence. In general, no implied propagator outperforms others, it depends on instances and branching heuristics. The combination of them (ALL) is not always the best model but it provides robustness at a good trade off between filtering quality and resolution time.
6.4 Consequences on symmetric instances
In this section, we show the repercussion of our study on the (symmetric) Traveling Salesman Problem (TSP), which can be seen as the undirected variant of the ATSP. For that, we use an undirected model, as in [4]: Each node has now two neighbors. Previously mentioned implied structural filtering algorithms are defined for directed graphs, and thus cannot be used for solving the TSP. However, our study about search heuristics can be extended to the symmetric case. It is worth noticing that the Sparse heuristic cannot be used here because the dichotomic exploration is not defined for set variables that must take two values. We suggest to measure the impact of EnforceSparse heuristic that we have introduced and which appeared as the best choice for solving the ATSP. As can be seen in Table 3, the EnforceSparse heuristic can dramatically enhance performances on TSP instances. While the SOTA model fails to solve most instances within the time limit of 30 minutes, our approach solves all instances that have up to 150 nodes in less than one minute (kroB150 appart) and close half of the others that have up to 300 nodes. This seems to be the new limit of CP. Scaling further on simple instances would be easy: the number of iterations in the Lagrangian relaxation should be decreased to get reasonable computing times, but solving bigger and still hard instances would require serious improvements.
SOTA [4]  EnforceSparse  
instance  nodes  time  nodes  time 
eil76  335  0.40  8  0.06 
eil101  1,337  2.54  55  0.42 
pr124  123,938  443.87  1,795  12.57 
pr144  3,237  20.82  128  1.58 
pr152  33,602  224.43  1,168  13.96 
pr226  5,980  56.30  440  6.84 
pr264  994  21.37  291  8.42 
pr299  99,612  91,584  
gr96  317,005  714.23  899  3.87 
gr120  19,231  65.20  239  1.76 
gr137  427,120  4,622  24.16  
gr202  225,054  2,079  20.68  
gr229  166,297  151,159  
bier127  465,699  460  4.13  
ch130  595,541  7,662  40.66  
ch150  445,976  593  4.76  
u159  188,943  988.25  817  5.86 
SOTA [4]  EnforceSparse  
instance  nodes  time  nodes  time 
kroA100  947,809  12,973  52.35  
kroA150  403,254  7,068  54.65  
kroA200  240,552  171,805  
kroB100  883,399  3,223  13.94  
kroB150  384,255  95,638  668.73  
kroB200  270,47  163,894  1772.19  
kroC100  42,690  82.95  1,472  6.55 
kroD100  3,382  8.71  71  0.34 
kroE100  702,011  1,343.25  7,854  30.73 
si175  285,444  423,760  
rat99  742  1.44  72  0.25 
rat195  286,495  8,535  79.48  
d198  235,513  3,079  32.48  
ts225  195,217  113,827  
tsp225  207,868  170,906  
gil262  175,408  145,000  
a280  148,522  34,575  305.96 
7 Conclusion
We have provided a short survey over solving the ATSP in CP and shown how general graph properties, standing from the consideration of the reduced graph, could improve existing models, such as the Minimum Spanning Tree relaxation. As future work, this could be extended to scheduling oriented TSP (TSPTW for instance) since finds some sets of precedences in linear time.
We also provided some implementation guidelines to have efficient algorithms, including the Held and Karp procedure. We have shown that our model outperforms the current state of the art CP model for solving both TSP and ATSP, pushing further the limit of CP. Our experiments enable us to state that graph structure has a serious impact on resolution: not only cost matters. More precisely, the EnforceSparse heuristic provides impressive results while implied structural filtering improves robustness for a negligible time consumption.
We also pointed out the fact that non monotonicity of the Lagrangian relaxation could make implied filtering decrease performances. An interesting future work would be to introduce some afterglow into the Held and Karp method: when the tree based relaxation is applied, it first performs a few iterations allowing, but penalizing, the use of arcs that have been removed since the last call of the constraint. This smoothing could make the convergence easier and thus lead to better results.
Acknowledgement
The authors thank Pascal Benchimol and LouisMartin Rousseau for interesting talks and having provided their C++ implementation as well as Charles Prud’Homme for useful implementation advise. They are also grateful to the regional council of Pays de la Loire for its financial support.
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