Revisiting the Hamiltonian p-median problem: a new formulation on directed graphs and a branch-and-cut algorithm

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1 Revisiting the Hamiltonian p-median problem: a new formulation on directed graphs and a branch-and-cut algorithm Tolga Bektaş 1, Luís Gouveia 2, Daniel Santos 2 1 Centre for Operational Research, Management Science and Information Systems (CORMSIS) Southampton Business School, University of Southampton Southampton, Highfield, SO17 1BJ, United Kingdom t.bektas@soton.ac.uk 2 Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (CMAF-CIO) DEIO, Faculdade de Ciências, Universidade de Lisboa C6 - Piso 4, , Lisbon, Portugal legouveia@fc.ul.pt, drsantos@fc.ul.pt Abstract This paper studies the Hamiltonian p-median problem defined on a directed graph, which consists of finding p mutually disjoint circuits of minimum total cost, such that each node of the graph is included in one of the circuits. Earlier formulations are based on viewing the problem as one resulting from the intersection of two subproblems. The first subproblem states that at most p circuits are required, which are usually modeled by using subtour elimination constraints known from the traveling salesman problem. The second subproblem states that at least p circuits are required, for which this paper makes an explicit connection to the so-called path elimination constraints that arise in multi-depot/location-routing problems. A new extended formulation is proposed that builds on this connection as well as on the concept of acting depot, and that allows the derivation of a stronger set of subtour elimination constraints for the first subproblem, and implies a stronger set of path elimination constraints for the second subproblem. The paper describes separation routines for the two sets of constraints that are used in a branch-and-cut algorithm. The variables in the new model also allow for an effective way of dealing with two types of symmetries inherent in this problem, those induced by the use of the concept of an acting depot and those based on the two possible orientations for a given circuit. Computational results on symmetric cost instances where two-node circuits are not allowed suggest that the algorithm is competitive with state-of-the-art methods. Keywords: Combinatorial optimization; Hamiltonian p-median; multi-cut inequalities; multi-depot routing; branch-and-cut algorithm. Corresponding author. 1

2 1 Introduction The Hamiltonian p-median problem (HpMP) is defined on a graph G with a set V = {1,..., n} of nodes, and a cost function c associated with connecting a pair of nodes. The objective of the HpMP is to find a minimum cost set of p cycles such that each node is included in one and only one of the cycles. When p = 1, the HpMP becomes the classical traveling salesman problem (TSP), which indicates that the HpMP is also NP-hard. The literature related to the HpMP is not as extensive as that of the TSP. To the best of our knowledge, the HpMP was first introduced by Branco & Coelho (1990) who present two formulations, one a set partitioning formulation and the other a directed formulation 1, similar to the one by Fisher & Jaikumar (1981) for the vehicle routing problem, as well as several heuristics, assuming symmetric cost instances. Most of the algorithmic work on this problem defines G to be undirected and that the problem does not admit solutions that include two-node cycles. The second condition was given by Gollowitzer et al. (2014) with the argument that in undirected graphs a proper cycle should have at least three nodes. This second condition is also attractive for modeling convenience, since the problem can then be easily modeled using undirected formulations with only a binary variable associated to each edge. Modeling two-node cycles on an undirected graph, although not as straightforward, is still possible using {0,1,2} variables or, equivalently, an additional set of binary variables that specifically consider this case instead, as is done in multi-depot/location-routing problems or even single-depot problems with multiple vehicles (see, e.g., Araque G. et al. 1994, Belenguer et al. 2011, Benavent & Martínez-Sykora 2013). The work by Gollowitzer et al. (2014) compares several formulations for the HpMP in terms of their corresponding linear programming (LP) relaxations. An improved version of one of the formulations presented in Gollowitzer et al. (2014), namely one with a better LP relaxation, has recently been proposed by Erdoğan et al. (2016). Recent exact algorithms are described by Erdoğan et al. (2016) and Marzouk et al. (2016) for the symmetric HpMP, the former in the form of a branch-and-cut, and the latter as a branch-and-price algorithm. In this paper, we describe a new formulation of the HpMP defined on a directed graph. Three main reasons motivate the approach of working on a directed graph. First, the asymmetric HpMP has a number of real-world applications, such as the laser multi-scanning problem (see, e.g., Glaab & Pott 2000), although this particular application requires an additional set of constraints. Second, and as we shall point out later, one set of constraints typically needed to model the problem are related with the so-called path elimination constraints that arise in multi-depot/location-routing problems. One such set was recently presented by Bektaş et al. (2017) for the directed multi-depot traveling salesman problem, for which they were shown to be effective computationally, particularly as they can be separated efficiently. These results suggest that the new constraints presented by Bektaş et al. (2017) might also be effective on similar problems, such as the HpMP. The third reason is the flexibility afforded by a directed model in easily being adapted to the symmetric HpMP, the most often studied variant, and where twonode cycles can be avoided by simply adding subtour elimination constraints written for two nodes. Furthermore, the directed formulation we will present allows for a non-trivial representation of symmetry-breaking constraints which prove to be very effective in the case of the symmetric HpMP, particularly for the variant where the twonode cycles are not permitted. Directed formulations for the HpMP have previously been described by Glaab & Pott (2000) and Zohrehbandian (2007) albeit without any computational experimentation. However, the type of formulations that appear in these papers have been shown to provide weak linear programming relaxations by Gollowitzer et al. (2014). As this paper will study directed formulations, we will henceforth use the term circuit to denote a directed cycle. Our approach follows the paradigm given in Gollowitzer et al. (2014) for the symmetric case in that we present models for the HpMP by partitioning the constraint set of the problem into two, namely one that guarantees that there are at most p circuits in the solution, and another to ensure that there are at least p circuits in the solution, which we will refer to as ( p) and ( p) constraint sets, respectively. At this point, it is interesting to revisit the special case of the TSP, where one wishes to obtain solutions with p = 1 circuit, for which reason one would need to eliminate solutions with more than p = 1 circuits. This leads to the observation that the ( p) constraints are generalizations of (and similar to) the subtour elimination constraints known from the TSP, and that the ( p) constraints are the odd ones that are not easy to characterize. One of the contributions of this work is to show a connection between the ( p) constraints and a different set of constraints that arise in multi-depot/location- 1 We say that a formulation is directed if it is defined on a directed graph in which a given pair of nodes defines an arc, and uses a binary variable for each arc that explicitly states the order in which the two nodes of the arc are traversed. Conversely, an undirected formulation is defined on an undirected graph where a pair of nodes induces an edge, and uses a binary variable for each edge where the direction in which the edge is traversed is irrelevant. 2

3 routing problems, namely the so-called path elimination constraints. To do so, we explore extended formulations for the HpMP which introduce the concept of acting depots into the models, which, although not a new approach, has not been thoroughly explored in previous studies. The formulation proposed in this paper can be viewed as an extended version of the acting depot formulations. In particular, we use additional (disaggregated) arc variables that describe whether an arc originates from, is destined to or is disconnected to an acting depot, and uses an adaptation of the multi-cut (path elimination) constraints introduced by Bektaş et al. (2017). The new variables also allow for an effective way of dealing with two types of symmetries inherent in this problem, one induced by the use of the concept of acting depot, and the other resulting from two possible orientations of a given circuit. The computational results for symmetric instances where two-node circuits are not allowed appear to be competitive with the state-of-the-art methods. The main contributions of this work are as follows: (i) in Section 2 we use the concept of acting depot of a circuit and review the currently known approaches to model the ( p) and ( p) constraints in this context; (ii) in Sections 3 and 4 we present the new proposed formulation, some of its properties, the way in which it is used to solve symmetric instances in which two-node circuits are not allowed, inequalities implied by the new model in the space of the variables included in the formulations using the acting depots, and comparisons of the projected inequalities with the previous known sets; (iii) in Section 5 we show how the symmetries inherent in formulations which use the concept of acting depot as well as symmetries specific to directed formulations in instances with symmetric costs can be dealt with; (iv) in Section 6 we describe a branch-and-cut algorithm based on the new formulation and present computational results for asymmetric instances with up to 171 nodes, symmetric instances with up to 100 nodes in which two-node circuits are allowed, and compare our approach to the state-of-the-art methods to solve symmetric instances with up to 159 nodes in which two-node circuits are not allowed. 2 Generic formulations for the HpMP We define the Hamiltonian p-median problem (HpMP) on a directed graph G = (V, A), with a set V = {1,..., n} of nodes, a set A = {(i, j) : i, j V, i j} of arcs, and a cost function c associated to the set of arcs. We say that a set of circuits covers the node set V, or is a cover of V, if each node is included in one and only one of the circuits. The objective of the HpMP is to find a minimum cost set of p disjoint circuits that covers V. For simplification, we will assume that G is a complete graph. Our results are applicable to incomplete graphs by simply not considering the pairs (i, j) such that (i, j) / A in all of the mathematical expressions below. The formulations use the following notation: for any general one-index variable u we write u(s) = i S u i; for any general two-index variable v we write v(s) = i,j S, i j v ij and v(s 1, S 2 ) = i S 1,j S 2 v ij, where S 1 S 2 =. For singleton node subsets, say I = {i}, we write i instead of {i}. This section presents two generic formulations for the HpMP, one defined in the space of the arc variables alone, and the other that uses an additional set of variables that differentiate the so-called acting depots from the client nodes. 2.1 A generic formulation with arc variables In many network design problems, the preferred types of formulations are the ones that use arc variables alone, as they use fewer variables as compared to other formulations that we will present below. Although such formulations usually include exponential-size sets of constraints, effective branch-and-cut algorithms can be devised if the separation of such inequalities can be done efficiently, as often is the case. The generic formulation presented below follows this line of thought and is an adaptation of the one proposed by Gollowitzer et al. (2014) for the symmetric HpMP to the asymmetric case. The model uses binary variables x ij indicating whether or not arc 3

4 (i, j) A is used in any one of the p circuits. Minimize c ij x ij (i,j) A subject to: x ij = 1 i V (1) j V x ji = 1 i V (2) j V {(i, j) A : x ij = 1} forms at most p circuits (3) {(i, j) A : x ij = 1} forms at least p circuits (4) x ij {0, 1} (i, j) A. (5) Clearly, any solution of (1) (2) and (5), usually referred to as the assignment relaxation, corresponds to a set of disjoint circuits that cover V. However, such solutions may be composed of more, or of less, than p circuits. The generic constraints (3) ensure that there are at most p circuits in the solution, whereas the generic constraints (4) ensure that any solution is composed of at least p circuits. In the formulation described by Gollowitzer et al. (2014), constraints (3) are modeled as generalizations of the cut inequalities known from the TSP, whereas inequalities (4) are in the form of cycle-elimination constraints. A polyhedral study based on inequalities presented in the former work was initiated in Hupp & Liers (2013) and Gollowitzer et al. (2014). The two sets of inequalities can easily be adapted to the asymmetric HpMP. The main drawback, however, is that the separation of both sets of inequalities is NP-hard. This was proven by Gollowitzer et al. (2014) for the symmetric case. The complexity of the separation is still open for the asymmetric HpMP but there is no reason to suspect that it will be different to the symmetric case. This motivates the study of solution methods for the HpMP based on formulations using additional sets of variables such that the separation problem is polynomial, and which is one of the contributions of this paper. 2.2 A generic formulation extended with acting depot variables The HpMP can be modeled by using a connection with multi-depot/location-routing problems that considers p nodes of V to be viewed as the acting depots of the p circuits. One such formulation was reviewed by Gollowitzer et al. (2014), who compared its linear programming relaxation with those of other formulations. We now present another generic formulation for the HpMP that uses additional variables for the nodes acting as depots. The formulation uses a set of binary variables y i that equal 1 if node i V is an acting depot, and 0 otherwise. For simplicity, we will refer to a node i as a depot if y i = 1, and as a client if y i = 0, in a given solution. Minimize c ij x ij (i,j) A subject to: x ij = 1 i V (1) j V x ji = 1 i V (2) j V y i = p i V {(i, j) A : x ij = 1 and i V : y i = 1} contains no circuit with zero depots (7) {(i, j) A : x ij = 1 and i V : y i = 1} contains no circuit with two or more depots (8) x ij {0, 1} (i, j) A (5) y i {0, 1} i V. (9) Constraints (6) ensure that there are exactly p depots in any feasible solution, while constraints (7) and (8) relate the y variables with the x variables and guarantee that in each circuit one and only one node has the corresponding y variable equal to 1. To see the connection with the HpMP observe that if there exists a solution with less than p circuits then at least one of the circuits has more than one depot, and if there exists a solution (6) 4

5 with more than p circuits then at least one of the circuits will have zero depots. In this respect, constraints (7) and (8) correspond to (3) and (4), respectively. In the context of multi-depot/location-routing problems, the generic inequalities (7) are modeled as classical subtour elimination constraints and the generic inequalities (8) as path elimination constraints, which are used to prevent paths between two depots. By using the acting depot concept, we can translate these inequalities to the context of the HpMP. In fact, the generic subtour elimination constraints (7) can be used to prevent circuits composed of only client nodes (i.e., nodes such that y i = 0), whereas the generic path elimination constraints (8) prevent the formation of paths between two depots (i.e., nodes such that y i = 1). As far as we know, this is the first time that the connection between the ( p) constraints in the HpMP and the path elimination constraints in the context of multi-depot/location-routing problems is established. We shall illustrate the connection by using the directed version of two sets of inequalities taken from the literature. To model the generic inequalities (7), consider the following set of (subtour elimination) constraints that correspond to the directed version of a set of inequalities proposed by Laporte et al. (1983) (see also Gollowitzer et al. 2014), which are based on the observation that if for a given subset of nodes S we have y(s) = 0, that is, there is no acting depot in the set S, then no circuit can form that is composed only of nodes of S: x(s) S 1 + y(s) S V, S. (10) To model the generic inequalities (8), we consider the following well-known and easy to explain set of path elimination constraints y i + x(p ) + y j P + 1 i, j V, i j, (11) where P is a set of arcs defining an elementary path linking i and j. Note that if both i and j are depots, then no path between those nodes is allowed to exist since it would violate constraints (11). There are, however, three main drawbacks of the inequalities (11). The first of these are the linear programming relaxation bounds of formulations using these constraints, which are known to be weak. The second drawback is that it is not clear how the separation problem can be solved for these inequalities. The final drawback relates to a symmetry problem that is a consequence of the modeling approach that selects nodes to be the acting depots of the circuits. In particular, such an approach allows for too many alternative representations of a given integer solution. For example, a circuit (i 1, i 2,..., i m, i 1 ), with i j V, j {1,..., m}, can be represented in m different ways, depending on the m possible choices of acting depots. The situation is even worse for when solving symmetric instances, given that there exists two possible orientations of a given solution. It is not clear how the symmetry problem can be addressed with formulations such as those using constraints (11). In the next section, we present a new formulation for the HpMP that will address the three drawbacks identified above, and which yields path elimination constraints that we will show to be stronger than (11). 3 The PQR formulation The formulation proposed in this section is motivated by the connection previously stated between the ( p) constraints included in the HpMP and the path elimination constraints arising in the context of multi-depot/locationrouting problems. In multi-depot/location-routing problems, the node set is partitioned into depot nodes and client nodes, which we denote here by D and C, respectively. The starting point is the following set of the multi-cut (path elimination) constraints proposed by Bektaş et al. (2017) for an asymmetric multi-depot routing problem: x(c \ S, d) + x(c \ S, S) + x(d, S) 1 d D, S C. (12) The validity of these constraints and the fact that they eliminate infeasible paths which link two depots was proven by Bektaş et al. (2017). Briefly, observe that if there is a path between two depots d 1 and d 2 then the inequality (12) such that d = d 1 and the nodes in C \ S are the client nodes in the path from d 1 to d 2 is violated. Also, as pointed out by Bektaş et al. (2017), the inequalities (12) are sufficient to obtain a valid set of path elimination constraints. Since in the HpMP each node can be either a depot or a client, and in order to derive a model using a set of constraints similar to the inequalities (12), we propose three new sets of binary variables which distinguish between the cases of whether or not an arc (i, j) is used in one of the circuits where (i) i is a depot; (ii) j is a depot; and (iii) neither i nor j are depots. More precisely, for each arc (i, j), we create a binary variable p ij 5

6 which indicates whether or not arc (i, j) is used in one of the circuits where i is a depot; a binary variable q ij which indicates whether or not arc (i, j) is used in one of the circuits where j is a depot; and a binary variable r ij indicating whether or not arc (i, j) is used in one of the circuits where neither i nor j are depots. None of the three cases described above consider the situation where the two nodes i and j can be depots at the same time. Therefore, the definition of the three new sets of variables prevents solutions where two depots are directly linked. In other words, we ensure that if an (infeasible) path between two depots exists then at least one client node is included in that path. In the next two sections we show how to model the two main sets of constraints, that is, the ( p) constraints and the ( p) constraints, by using the new sets of variables. 3.1 Modeling the ( p) constraints Recall that the ( p) constraints ensure that the solution has at least p circuits which can be modeled by ensuring that there are no circuits with two or more acting depots. Following the motivation given previously, these constraints are defined as follows: q(s, i) + r(s, S) + p(i, S) y i i V, partitions (S, S) of V \ {i}. (13) The proof of the validity of the multi-cut constraints (12) in the case of the multi-depot routing problem can be applied, with a minor addition, to the case of the inequalities (13) for the HpMP. For completeness we provide the formal proof for the validity of the inequalities (13) below: Proposition 1. The inequalities (13) are valid and eliminate circuits with two acting depots. Proof. The constraints (13) are clearly valid if y i = 0. When y i = 1, let (S, S) form a partition of V \ {i} and suppose that q(s, i) = r(s, S) = p(i, S) = 0. Since i is a depot then it must be that q(s, i) = 1 = p(i, S ). Note that in the circuit of depot i there can be no more depots, hence all remaining arcs must be r arcs. But then it is not possible to complete the circuit for depot i since r(s, S) = 0. To see why these constraints cut-off solutions in which there are two depots in the same circuit suppose that i is an acting depot (y i = 1) and that there exists a node j V \ {i} such that y j = 1 and that i and j are in the same circuit. In this situation, there must exist at least one client node in the path from i to j and another client node in the path from j to i, since both nodes i and j are depots (this was the minor addition mentioned before, and it easily follows from the observation made regarding the definition of the p, q and r variables). If we consider that the set of nodes which are in the path from i to j are in S and that the remaining nodes, except i and j, are in S, we obtain p(i, S) = 0, since the arc leaving i goes to S, and q(s, i) = 0, since the arc entering i comes from S. In addition, because j is a depot, then there is no r arc incident to j, and regardless of whether j S or j S we have r(s, S) = Modeling the ( p) constraints The ( p) constraints ensure that any solution has at most p circuits which, as previously stated, can be modeled by preventing solutions containing circuits with no acting depots. From the definition of the p, q and r variables, a circuit will be composed of client nodes only, or equivalently have no acting depots, if and only if all arcs in the circuit are r arcs. Hence, in order to prevent such a situation, we can use use any set of subtour elimination constraints that is known from the literature on the TSP (possibly with minor modifications motivated by the context) rewritten with the client-only variables r. For our model, we use the well-known set due to Dantzig et al. (1954): r(s) S 1 S V. (14) However, other sets of inequalities in lieu of (14) could have been used instead. For instance, for a related problem, Bektaş (2012) uses the Miller-Tucker-Zemlin constraints (see Miller et al. 1960) which, on the one hand, have the advantage of being polynomial in number but, on the other hand, lead to substantially weaker LP relaxation bounds when compared to other formulations that use exponentially-sized sets of constraints such as (14). 6

7 3.3 The complete formulation For completeness and clarity, we provide below the complete new formulation that we denote by PQR. Observe, first, that the old x and y variables and the new p, q and r variables are related by the following equalities (which are easily seen to be valid): x ij = p ij + r ij + q ij (i, j) A (15) y i = p ij = q ji i V. (16) j V \{i} j V \{i} The complete PQR formulation can be described by using only the p, q and r variables due to the above relations, however, for simplification, we will use the y variables as well. It is as follows: Minimize c ij (p ij + q ij + r ij ) subject to: (i,j) A j V \{i} j V \{i} y i = (p ij + q ij + r ij ) = 1 i V (17) (p ji + q ji + r ji ) = 1 i V (18) j V \{i} y i = p i V p ij = j V \{i} q ji i V (16) r(s) S 1 S V (14) q(s, i) + r(s, S) + p(i, S) y i i V, partitions (S, S) of V \ {i} (13) p ij {0, 1} (i, j) A (19) q ij {0, 1} (i, j) A (20) r ij {0, 1} (i, j) A (21) y i {0, 1} i V. (9) In addition, and in order to simplify the proofs in the next section, we will also consider the two following sets of equalities, (q ij + r ij ) = 1 y i i V (22) j V \{i} j V \{i} (p ji + r ji ) = 1 y i i V, (23) the validity of which can easily be proven by appropriately combining (16) with the assignment constraints (17) and (18), respectively. The formulation described above allows solutions containing circuits with only two nodes. From the point of view of minimizing the total cost of the p circuits, clearly one can obtain potentially cheaper solutions by allowing two-node circuits to exist. However, most of the literature on the symmetric HpMP is based on undirected graphs and does not allow two-node cycles, hence we show next how we can adapt the PQR formulation to this case. Possibly the most trivial way to prevent two-node circuits is to add subtour elimination constraints for two nodes, which written with the p, q and r variables are as follows: p ij + p ji + r ij + r ji + q ij + q ji 1 i, j V, i j. (24) However, the interpretation of the variables in the PQR formulation allows for other trivial ways to model this situation, as for instance by using the following set of constraints: q ki p ij i, j V, i j. (25) k j 7 (6)

8 Note that the constraints (24) state that an arc (i, j) cannot be used in both directions, whereas constraints (25) say that if an arc (i, j) is used in which i is an acting depot, then the arc that enters into i must come from a node k such that k j. It can be easily shown that, in the LP relaxation, one set does not dominate the other, however their behavior is similar in the sense that they both perform quite bad computationally. In fact, some computational experiments while attempting to solve symmetric instances in which two-node circuits are not allowed by adding (24) or (25) (or even both simultaneously) suggest that the PQR formulation with such constraints does not compete with current state-of-the-art methods. However, by using symmetry-breaking concepts that we will discuss further on, we can derive a new set of constraints that dominates both (24) and (25) and permits the resolution of unsolved benchmark symmetric instances where two-node circuits are not allowed. 4 Theoretical investigations on the PQR formulation This section presents the main theoretical findings concerning: (i) the strengthening of the subtour elimination constraints (14), (ii) a property of the multi-cut inequalities (13) for instances with a symmetric cost structure, (iii) a relationship between the multi-cut inequalities (13) and a compact formulation of the problem, and (iv) a stronger version of the path elimination constraints (11) implied by the PQR formulation. 4.1 Strengthening the subtour elimination constraints (14) In this subsection we show how we can strengthen the inequalities (14). Observe that, due to the equalities (22) (23), for any i V we have that if y i = 1 then r(i, V ) = r(v, i) = 0, hence, for any subset S V and for any i S, if y i = 1 then r(i, S) = r(s, i) = 0. Following this observation, we show next that the inequalities (14) can be strengthened in different ways depending on two cases: either S > p or S p. Proposition 2. The following liftings of (14) are valid inequalities for the PQR formulation: y(s) + r(s) S 1 S V : S > p (26) p(i, S) + y(s \ {i}) + r(s) S 1 S V, i S. (27) Proof. To prove the validity of (26) assume that y(s) = k and that K S is the subset of nodes such that y(k) = k. Then, by substitution and by using the observation made above, the inequality (26) becomes the inequality (14) written for the set S \ K, which is non-empty since S > p k = K, and thus, is valid. We now prove the validity of (27). The lifted inequality (26) for such a set S is not valid since y(s) can be equal to S. However, we can still obtain a valid but slightly weaker version of the inequality (26) which we derive in two steps. For the first step, observe that for each i S we have that y(s \ {i}) S 1 (this is trivial and follows from the fact that y i 1), and thus the reasoning used to prove the validity of the inequalities (26) can also be used to prove the validity of the following set of inequalities: y(s \ {i}) + r(s) S 1 S V, i S. (28) For the second step, notice that we can still strengthen the inequalities (28) by observing that when p(i, S) = 1, then there is a node in S \ {i} that cannot be a depot. Thus, there are at most S 2 depots in S \ {i} (e.g., if p ij = 1 with j S \ {i}, then j cannot be a depot), and the same reasoning used before leads to the validity of (27). Since the relations (16) state that y i = p(i, S) + p(i, S ), then the difference between the two lifted inequalities (26) and (27) is given by the term p(i, S ). In this second case, and as pointed out before, lifting this extra term would lead to a non valid inequality since p(i, S ) = 1 does not guarantee that there are at most S 2 depots in S \ {i}. A different version of the inequalities (27) can be derived by observing that y i = q(s, i) + q(s, i) also holds, which again follows from the relations (16): q(s, i) + y(s \ {i}) + r(s) S 1 S V, i S. (29) 8

9 One last question concerns the validity of a lifted inequality where the two terms p(i, S) and q(s, i) are simultaneously lifted. Such an inequality is not valid since the two terms can both be equal to 1 but the number of depots in S \ {i} can still be equal to S 2 (this happens if p ij = 1 = q j i for a node j S \ {i}). If we assume that two-node circuits are not allowed, then the inequality would still not be valid since we could have r(s \ {i}) = S \ {i} 1 = S 2, that is, a path using r arcs that spans all nodes in S \ {i} and node i as an acting depot linked to the two endpoints of the path. Note that the inequalities (27) (29) are valid also if S > p, even though they are dominated by the inequalities (26) in that case. However, the inequalities (26) are not valid if S p. In addition, the inequalities (28) are also dominated by the inequalities (27) and (29) in the case where S p. Thus, by using the inequalities (26) for S > p and the inequalities (27) and (29) for S p, we obtain a valid set of ( p) constraints and, in particular, we guarantee that for any S, regardless of its cardinality, every inequality (28) is always satisfied. From here on out, we will use the inequalities (26), (27) and (29) as ( p) constraints in the PQR formulation, replacing the weaker set (14). Finally, by appropriately using the assignment constraints (17) and (18) and the relations (16), the inequalities (26) (29) can be equivalently written in cut-like form, respectively, as follows: p(v, S) + r(s, S) 1 S V : S > p (30) p(v, S) + r(s, S) 1 p(i, S ) S V, i S (31) p(v, S) + r(s, S) 1 y i S V, i S (32) p(v, S) + r(s, S) 1 q(s, i) S V, i S. (33) The cut-like form of these inequalities suggests that they can be separated by resorting to max-flow/min-cut computations on a suitable graph. Such separation algorithms will be described in Section The multi-cut inequalities (13) and instances with symmetric costs For the multi-depot routing problem studied by Bektaş et al. (2017) it was shown that the multi-cut constraints (12) have no effect on the LP bound of the assignment relaxation for instances with symmetric costs. Following a similar line of argument, one can show that the same result holds for constraints (13) in the context of the HpMP, however, for simplicity, we omit the proof. Proposition 3. Consider an instance with symmetric costs and a solution (p, q, r, y ) that satisfies the assignment constraints (17) and (18) and the relations between the p, q and y variables (16). In these conditions, there exists a solution (p, q, r, y ), with the same cost as (p, q, r, y ), that satisfies the assignment constraints (17) and (18) and the relations (16), as well as every inequality (13). The importance of this result is two-fold. Firstly, it indicates that the PQR formulation may have problems when solving instances with symmetric costs. Secondly, the proof of this result essentially relies on the fact that by using a directed formulation one is allowing any circuit to have two possible orientations. In asymmetric instances this is not an issue, however, for symmetric cost instances both orientations have the same cost and hence define alternative solutions. 4.3 A compact formulation of the ( p) constraints and connections with the multi-cut inequalities (13) The multi-cut inequalities (13) can be viewed as standard cut constraints in a corresponding 3-layered graph (see, e.g., Albareda-Sambola et al. 2005, Bektaş 2012, Bektaş et al. 2017), which allows for a better explanation of the underlying separation procedure for the inequalities (13) as well as a derivation of a compact representation of the same set of constraints. The connection between the multi-cut inequalities in the original graph and standard cut inequalities in the 3-layered graph was first pointed out by Bektaş et al. (2017) for a multi-depot routing problem. Consider a 3-layered graph L = (V L, A L ) where V L is composed of three copies of each node in V. V L is partitioned into three subsets, which are the three layers of L, each subset with a copy of each original node. The first layer and the third layer, each composed of V nodes, represent the copies of the nodes of graph G that are viewed as the acting depots. The two layers correspond to viewing them, respectively, as starting points and as ending points of one of the p circuits. The second layer represents the copies of nodes of the graph G that are 9

10 viewed as the client nodes. The arc set A L is also partitioned into three subsets. The first subset corresponds to the arcs going from the first layer to the second layer, with an arc existing for every existing p variable. The second subset corresponds to the arcs between the nodes in the second layer, which are represented by the r variables in the original graph. Finally, the second layer has arcs linking it to the third layer, with an arc existing if and only if a corresponding q variable exists. To model the HpMP in the 3-layered graph we need to guarantee the existence of p paths in L such that each path starts in a node in the first layer and ends in a node in the third layer, with the additional three sets of constraints: i) constraints that guarantee that each node is in one and only one of the paths (this is guaranteed by the assignment constraints (17) and (18)), ii) constraints guaranteeing that there are no circuits in the second layer (which is guaranteed by the ( p) constrains of the PQR formulation), and iii) constraints that guarantee that the start and end nodes of a path are copies of the same corresponding node in the original graph. Case iii) corresponds to the ( p) constraints, that can either be modeled by the multi-cut inequalities (13) of the PQR model, or alternatively by the following compact system of constraints based on the binary variables zij k, which are equal to 1 if an arc (i, j) is in the path from the copy of node k in the first layer to its copy in the third layer, and to 0 otherwise. The same variables also indicate whether or not arc (i, j) is used in the circuit of the acting depot k in the original graph. The existence of the required circuits can be modeled by the constraints below: zkj k = y k k V (34) j V,j k j V,j k j V,j i,k z k jk = y k k V (35) z k ji = j V,j i,k z k ij k V, i V : i k (36) z k kj p kj k V, (k, j) A (37) z k jk q jk k V, (j, k) A (38) z k ij r ij k V, (i, j) A (39) z k ij {0, 1} k V, (i, j) A. (40) Let (3I) be the system defined by (34) (39) and the following non-negativity constrains: z k ij 0 k V, (i, j) A (41) p ij 0 (i, j) A (42) q ij 0 (i, j) A (43) r ij 0 (i, j) A (44) y i 0 i V. (45) Using a result similar to that shown by Bektaş et al. (2017), one can prove that the projection of the polyhedron defined by (3I) onto the space of the p, q, r and y variables is given by the multi-cut inequalities (13) and the non-negativity constraints (42) (45), as the following proposition shows. The proof is similar to the one by Bektaş et al. (2017), however it is important to explicitly show it here since it will be fundamental to understand another result further on. Proposition 4. The projection of the polyhedron defined by (3I) onto the space of the p, q, r and y variables is given by the multi-cut inequalities (13) and the non-negativity constraints (42) (45). Proof. The result follows from the max-flow/min-cut theorem and the interpretation of the (3I) equation system in the 3-layered graph L. The max-flow/min-cut theorem states that for each node k, y k units of flow are sent from its copy in the first layer, say k 1, to its copy in the third layer, say k 3, without passing through its copy in the second layer (due to the flow-conservation constraints (36) which are not defined for the copy of node k in the second layer) with arc capacities given by the values of the variables p, q and r if and only if every cut separating k 1 from k 3 has capacity with value at least y k. This last requirement corresponds precisely to the constraints q(s, k) + r(s, S) + p(k, S) y k for all partitions (S, S) of V \ {k}, which are the desired multi-cut inequalities (13). 10

11 The compact formulation can be strengthened by replacing (39) in the (3I) system with the stronger set of constraints, zij k r ij (i, j) A, (46) k V \{i,j} and define a equation system (3I + ) that is preferable as it includes fewer but a stronger sets of constraints as compared to (3I). It is also possible to generalize the multi-cut inequalities (13), in the same way as was done by Bektaş et al. (2017) for the multi-depot routing problem where the computational results did not indicate significant gains by the use of the stronger system. This suggests that a similar result may hold for the HpMP, for which reason we do not pursue such generalizations in this paper. 4.4 Inequalities in the (x, y)-space implied by the PQR model Due to the relations (15) and (16) we can view the PQR formulation as an extended version of an (x, y)-space formulation. Therefore, it would be interesting to see which ( p) and ( p) inequalities defined on the (x, y)-space can be obtained by projecting the feasible set of the linear programming relaxation of the PQR formulation onto the (x, y)-space. Obtaining the complete description of the required projected polyhedron does not appear to be an easy task. It is, however, possible to establish two results concerning the way in which the inequalities (10) and (11) are related to the projection. Proposition 5. The LP relaxation of the PQR model implies inequalities (10). Proof. We consider a generic set S V such that S p. First, by adding y(s) + p(i, S ), with i S, to each side of (27) (a similar proof holds if we start with inequalities (29)) and using the relation (16) between the y and the p variables for node i, we obtain y(s) + y(s) + r(s) S 1 + y(s) + p(i, S ). Now by using the relations (16) between the y and p variables for the first y(s) term on the left-hand side, and the relations (16) between the y and q variables for the second y(s) term on the left-hand side, we obtain p(s) + p(s, S ) + q(s) + q(s, S) + r(s) S 1 + y(s) + p(i, S ). From the relation (15) between the x and the p, r and q variables, we know that x(s) = p(s) + r(s) + q(s). Hence, we can write the previous equation as follows: x(s) S 1 + y(s) + p(i, S ) p(s, S ) q(s, S). Finally, since the expression p(i, S ) p(s, S ) q(s, S) is non-positive, we obtain an inequality that implies (10) for the same set S. In the case where S is such that S > p, we can use the same reasoning (in this case it is even simpler) but starting with the inequalities (26). We now return to our claim in Section 2.2 stating that the path elimination constraints (11) are weak in terms of the linear programming relaxation bounds. Below, we present a new set of inequalities that are stronger than inequalities (11), which were obtained by using an analogy with the multi-cut inequalities (12) proposed by Bektaş et al. (2017) for the multi-depot routing problem: x(s, i) + x(s, S) + x(i, S) y i + y j 1 i, j V, i j, partitions (S, S) of V \ {i, j}. (47) An inequality (47) is only of interest when the two nodes i and j are chosen to act as depots (i.e., y i = y j = 1). By using the assignment constraints (1) (2) we can rewrite the inequalities (47) in the following packing form, y i + x(i, S) + x(s) + x(s, j) + y j + x ij S + 2 i, j V, i j, S V \ {i, j}, (48) which immediately shows that they imply constraints (11). In addition, the packing representation (48) allows for a straightforward interpretation of the inequalities. In particular, if there is a path from a node i to a node j that spans a set S of nodes, then the path will have S + 1 arcs given that x(i, S) = x(s, j) = 1 and x(s) = S 1. 11

12 In this case, inequality (48) is violated if the two nodes i and j are both depots (that is, y j = y i = 1). The fact that the inequalities assume any spanning path in S is a clear improvement over the original path elimination inequalities (11) that are restricted to a specific path in the set S. The inequalities also prohibit direct travel between two depots, which is easily seen through the additional term x ij in their packing representation (48). In this case, given that S =, the constraints will read y i +y j +x ij 2 and consequently ensure that no arcs can be used between nodes chosen as depots. The PQR formulation implicitly models this case through the definition of the variables. The above observations show that the inequalities (47) provide a valid set of ( p) constraints which can be separated in polynomial time and can, therefore, be used effectively in a formulation completely defined on the (x, y)-space. The following proposition provides a formal proof of the validity of the inequalities (47) and show that the PQR formulation implies these constraints (47) in the (x, y)-space. Proposition 6. The LP relaxation of the PQR model implies the inequalities (47). Proof. Consider a node i V, let (S, S) form a partition of V \ {i} and consider the inequality (13) written for i, S and S q(s, i) + r(s, S) + p(i, S) y i. Consider any other node j V \ {i} and assume, without loss of generality, that j S (if j S, the proof is very similar). For the sake of simplicity, let S = S \ {j}, thus (S, S ) defines a partition of V \ {i, j}. Then the above inequality can be written as: q(s, i) + r(s, S ) + p(i, S ) y i r(s, j) p ij. Now if we add p(s, i)+r(s, i)+p(s, S )+q(s, S )+q(i, S )+r(i, S ) to both sides of the previous inequality, and use the relations (15) between the x and the p, q and r variables, we obtain: x(s, i) + x(s, S ) + x(i, S ) y i r(s, j) p ij + p(s, i) + r(s, i) + p(s, S ) + q(s, S ) + q(i, S ) + r(i, S ). Notice that from the equalities (23) we can derive that r(s, j) + p ij 1 y j, or equivalently, r(s, j) p ij y j 1, thus we can obtain a weaker inequality than the previous one as follows: x(s, i) + x(s, S ) + x(i, S ) y i + y j 1 + p(s, i) + r(s, i) + p(s, S ) + q(s, S ) + q(i, S ) + r(i, S ). Finally, since p(s, i) + r(s, i) + p(s, S ) + q(s, S ) + q(i, S ) + r(i, S ) 0, we can easily see that the above inequality implies the inequality which completes the proof. x(s, i) + x(s, S ) + x(i, S ) y i + y j 1, The arguments used for the proof of Proposition 6 suggest that it is possible to obtain a generalization of the inequalities (47), the investigation of which we leave for future research. However, we can state that this generalization would certainly be related with the generalization of the multi-cut inequalities (13) of the PQR formulation suggested in the previous section. 5 Breaking symmetries in the PQR formulation This section discusses two types of symmetries observed in the solutions obtained by the PQR formulation and describes how they could be resolved. 5.1 Symmetry of type I To address the symmetry problem induced by the m different ways a circuit can be represented (see Section 2.2) by selecting each node of the circuit as an active depot, we use constraints that are motivated by the well-known idea (see, e.g., Campêlo et al. 2004) that the acting depot in any circuit (i 1, i 2,..., i m, i 1 ) should be the node with the lowest index to reduce the m possible representations into one (e.g., choosing node 2 as an acting depot for 12

13 the circuit (3, 2, 7, 6, 3)). This idea can be easily implemented in the compact model presented in Section 4.3 as follows, z k kj z k jk = 0 k V, (k, j) A : k > j (49) = 0 k V, (j, k) A : k < j (50) z k ij = 0 k V, (i, j) A : i < k or j < k, (51) which collectively impose that the depot of each circuit should be the node with the lowest index by disallowing the use of arcs which do not satisfy this condition, and which can be partially adapted to the PQR formulation by considering the following set of symmetry-breaking constraints: p ij = 0 (i, j) A : i > j (52) q ij = 0 (i, j) A : i < j. (53) Constraints (52) (53) prevent some alternative representations of the same circuit, but do not guarantee that the depot of a circuit will be the node with the lowest index. For example, whereas they eliminate the solution (3, 2, 7, 6, 3) since p 32 = 0, they would not eliminate solutions such as the circuit (3, 7, 2, 6, 3). Constraints to deal with such symmetries have already been proposed by Gollowitzer et al. (2014) by using binary variables that indicate whether or not a node is in the circuit of a given acting depot. It is not straightforward to adapt a similar idea in the context of the p, q and r variables. However, we can indirectly provide such a set of constraints by using (49) (51) together with the projection result in Proposition 4. In particular, since there exists a path of value y i from the copy of node i in the first layer to the copy of the same node in the third layer if the capacity of any cut separating the node copies of node i is at least y i, restricting the arcs allowed to be in the path by removing arcs in the second layer according to the inequalities (49) (51) also restricts the arcs allowed in the cut sets. This leads to the following restricted multi-cut inequalities that can be used for symmetry-breaking purposes, q(s, i) + r(s, S) + p(i, S) y i i V, partitions (S, S) of V \ {1,..., i}, (54) which use all the information provided by the inequalities (49) (51) and therefore ensure that the acting depot in any circuit will be the node with the lowest index. As the computational results will show, the computational times substantially decrease when the restricted set of constraints (54) is used instead of (13). In addition, although equalities (52) (53) are not needed in the presence of the restricted multi-cut inequalities, we will still use them as they help to eliminate nearly half of the p and q variables from the PQR formulation. In many situations, symmetry-breaking constraints do not improve the value of the LP relaxation. However this is not the case for the restricted multi-cut constraints, as shall be seen in the computational results, the use of which leads to non-negligible improvements on the LP bounds. Proposition 3 still holds when we use the restricted multi-cut constraints (54) since any solution of the assignment relaxation of the PQR formulation can be rewritten in the form described above, that is, in any solution the acting depots should be the nodes with the lowest index. Hence, this increase in the LP bound is only observed in asymmetric instances. 5.2 Symmetry of type II In the symmetric HpMP, one other symmetry arises for a circuit and its reverse, both of the same cost, and which therefore represent equivalent solutions even if they are structurally different. This can pose a significant problem since any solution with p circuits can be represented in 2 p different ways by combining both possible orientations for the p circuits. We discuss below how such symmetries can be broken using the PQR formulation. We first observe that in a given circuit, one of the two nodes adjacent to the acting depot has a smaller index than the other. In order to break the symmetry, we enforce that the first node visited after the acting depot is given a lower index than the node visited just before the acting depot. In the case of the circuits c 1 = (2, 5, 7, 4, 2) and c 2 = (2, 4, 7, 5, 2), for example, the circuit c 1 would be considered infeasible, leaving circuit c 2 to be one of the p possible circuits. This idea has already been discussed for solving symmetric instances of the TSP using directed formulations, and it can be introduced to the PQR formulation by the use of the following constraints: q ki p ij, i, j V, i j. (55) k j 13