Valid Inequalities and Separation for the Symmetric Sequential Ordering Problem

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1 Valid Inequalities and Separation for the Symmetric Sequential Ordering Problem Adam N. Letchford Yanjun Li Draft, April 2014 Abstract The sequential ordering problem (SOP) is the generalisation of the asymmetric TSP in which the tour must obey given precedence relations between pairs of nodes. Many solution approaches have been devised for the SOP, but the version with symmetric costs, which we call the SSOP, has received very little attention. To fill this gap, we present a new 0-1 linear programming formulation for the SSOP, and derive several new families of strong valid inequalities for it, along with efficient separation routines. Some encouraging computational results are also presented. One remarkable result is that, in the case of the SSOP, there exists an efficient separation routine for the so-called simple source-destination inequalities. No such separation routine is known for the SOP, or indeed for the symmetric or asymmetric TSP. Keywords: travelling salesman problem, sequential ordering problem, precedence constraints, cutting planes, branch-and-cut. 1 Introduction The sequential ordering problem (SOP), sometimes called the precedenceconstrained asymmetric traveling salesman problem, is defined as follows [10]. We are given a complete directed graph G = (V, A) on n nodes, with non-negative costs c ij for each arc (i, j) A. We are also given an acyclic precedence digraph H = (V, B). The task is to find a minimum cost Hamiltonian tour, starting at ending at a given depot node, that obeys the Department of Management Science, Lancaster University, Lancaster LA1 4YW, United Kingdom. A.N.Letchford@lancaster.ac.uk Krannert School of Management, Purdue University, 403 W. State Street, West Lafayette, IN , United States. li14@purdue.edu 1

2 precedences. That is, if (i, j) B, the tour must pass through node i before passing through node j. The SOP has applications in both vehicle routing and machine scheduling [1, 6, 10, 11]. It has received considerable attention in the literature, and many integer programming formulations and algorithms have been devised for it (e.g., [1, 2, 6, 7, 10, 11, 15, 16, 18, 23, 24]). On the other hand, very little work has been conducted on the symmetric sequential ordering problem (SSOP), in which, for all pairs i, j, the costs c ij and c ji are equal. Indeed, we are aware of only three works that mention the SSOP [6, 12, 13]. In this paper, we look again at the 0-1 linear programming formulations for the SSOP given in [6, 12]. We begin by proposing a modified formulation, that is defined in terms of a mixed graph, and has two copies of the depot. This formulation is more intuitive than the standard one, and easier to handle computationally. Next, we derive several new families of strong valid linear inequalities (cutting planes), that can be used to strengthen the continuous relaxation of the formulation. After that, we present exact polynomial-time separation routines for five of the families......together with heuristic separation routines for the others. Finally, we present some encouraging computational results. The paper is structured as follows. The literature review is in Section 2. The improved integer programming formulation is in Section 3. The valid inequalities are in Section 4, and the separation routines are in Section 5. The computational results are in Section 6 and some concluding remarks are in Section 7. One particularly remarkable result, given in Subsection 5.6, is that there exists an efficient exact separation routine for a huge family of valid inequalities for the SSOP, called simple mixed source-destination inequalities. No analogous separation routine is known for the SOP, or indeed for the symmetric or asymmetric TSP (see our earlier paper [17]). Throughout the paper, we assume that B contains arcs only for precedences that cannot be deduced from transitivity. We also use the following notation. For any non-depot node i, π(i) and σ(i) denote the set of predecessors and successors of i, respectively. That is, each node of π(i) must be visited before i, and each node of σ(i) must be visited after i. For any set S of non-depot nodes, we let π(s) denote the union of π(i) over all i S, and σ(s) denote the union of σ(i) over all i S. Finally, we use the convention that edges are undirected, but arcs are directed. 2

3 2 Literature Review We now review the relevant literature. We cover the standard formulation of the SOP in Subsection 2.1 and the symmetric SOP in Subsection 2.2. In Subsection 2.3, we recall some results on a family of cutting-planes for general integer linear programs, called {0, 1 2 }-cuts, since we will need them later on (in Subsections 4.4 and 5.6). 2.1 The standard formulation of the SOP Several different integer programming formulations of the SOP have been proposed [1, 6, 10, 11, 15, 16, 18, 23, 24], along with a dynamic programming formulation [7]. For the sake of brevity, we restrict attention to the standard 0-1 linear programming formulation, which is used, e.g., in [1, 2, 6, 11, 20]. Let the depot be node 1. For each arc (i, j) A, let x a be a binary variable taking the value 1 if and only if the arc is traversed. Given a set S V, let δ + (S) denote the set of arcs with their tail in S, δ (S) denote the set of arcs with their head in S, and A(S) denote the set of arcs with both head and tail in S. We have: min a A c ax a s.t. x(δ + (i)) = 1 (i V ) x(δ (i)) = 1 (i V ) x(δ + (S)) 1 (S V \ {1}) (1) x(j : S) + x(a(s)) + x(s : i) S x a {0, 1} ((i, j) B, S V \ {1, i, j})(2) (a A). The constraints (1) are the subtour elimination (SE) inequalities, well known in the literature on the ATSP (see, e.g., [5]). The constraints (2) are the so-called precedence-forcing (PF) inequalities. They ensure that the solution does not contain any directed path from j to i passing through the nodes in any set S, and thereby enforce the precedences. Note that the SE and PF inequalities are exponential in number. It is pointed out in [1] that the associated separation problems can solved in polynomial time by, e.g., solving a series of max-flow min-cut problems. In [1], it is pointed out that one can strengthen the above formulation by adding any of the known valid inequalities for the asymmetric travelling salesman problem (ATSP). These inequalities are surveyed in [5]. In [2, 11, 20], further families of inequalities are presented, that exploit the presence of the precedences. These include strengthened PF inequalities [11], π, σ, 3

4 (π, σ) and precedence-cycle-breaking inequalities [6], strengthened 2-matching inequalities [2] and strengthened D k inequalities [20]. It is shown in [1, 6, 10] that we can delete the arc (i, j) from A under either of the following conditions: j π(i) There exists a node k V \ {i, j} such that k σ(i) and k π(j). Some other, more complicated, rules for deleting arcs are given in [11]. 2.2 The symmetric SOP Now consider the symmetric version of the SOP, in which, instead of a directed graph G = (V, A) and costs c a for each a A, one is given an undirected graph G = (V, E) and costs c e for each e E. We call this problem the SSOP. To our knowledge, this problem first appeared in the MSc dissertation [12], which was concerned with a practical application involving helicopter scheduling. In [12], a 0-1 linear programming formulation is presented, and attributed to Balas and Pulleyblank. This formulation is an extension of the standard formulation of the symmetric travelling salesman problem (STSP) [9], which we now describe. For each edge e E, let x e be a binary variable, taking the value 1 if and only if edge e traversed. For each S V, let δ(s) denote the set of edges in E with exactly one end-node in S. Then the standard formulation of the STSP is as follows: min e E c ex e (3) s.t. x(δ(i)) = 2 (i V ) (4) x(δ(s)) 2 (S V : 2 S V /2) (5) x e {0, 1} (e E). (6) The constraints (4) are called degree equations, and the constraints (5) are the undirected version of the SE inequalities. To ensure that the precedence relations are obeyed, Balas and Pulleyblank added to the above formulation three rather complicated families of inequalities, called across-arc, generalised out-arc and generalised inarc inequalities. In [6], it is shown that those three families of inequalities are dominated, respectively, by the following three families: Symmetric precedence 2-cycle breaking inequalities of the form x(δ(s)) + x(δ(t )) 6 (7) 4

5 for disjoint sets S and T such that there is an arc (i, j) B with i S and j T, and also an arc (k, l) B with l S and k T. Symmetric π inequalities of the form for all S V \ {1}. Symmetric σ inequalities of the form for all S V \ {1}. x(s \ π(s) : V \ (S π(s))) 1 (8) x(s \ σ(s) : V \ (S σ(s))) 1 (9) The only other paper we are aware of concerned with the SSOP is [13]. It presents an effective constructive heuristic and a local search procedure. 2.3 {0, 1 2 }-cuts NOTE: THIS IS UNFINISHED! Another paper that will be of relevance to us is Caprara & Fischetti [8], concerned with certain cutting planes for general integer linear programs called {0, 1 2 }-cuts. A {0, 1 2 }-cut is simply a cutting plane that can be derived by summing together some given inequalities, dividing the result by two, and then rounding down. More formally, given a system of linear inequalities Ax b in non-negative integer variables x, where x Z q, A Z p q and b Z p, a {0, 1 2 }-cut is an inequality of the form λt A x λ T b for some λ {0, 1 2 }p such that λ T b / Z. In [8], it is pointed out that many of the strong inequalities known for many combinatorial optimisation problems are {0, 1 2 }-cuts with respect to a natural formulation. In particular, they mentioned that the odd closed alternating trail (odd CAT) inequalities for the ATSP, due to Balas [3], are {0, 1 2 }-cuts. They then show that the separation problem for {0, 1 2 }-cuts is N P-hard in general, but polynomially solvable in two special cases: Proposition 1 Separation... if... or its transpose is an EPT-matrix. This result can be seen as a generalisation of a result of Padberg and Rao [22], on the blossom inequalities for matching problems, due to Edmonds [?]. 5

6 (Mention also Letchford-Reinelt-Theis [19] for blossom inequalities?) For the case in which the linear system does not satisfy this condition, Caprara and Fischetti suggest to create a weakened linear system that does. This enabled them to derive a heuristic separation algorithm for the odd CAT inequalities. In Caprara et al. (2000), it is pointed out that many facet-defining inequalities for the STSP and ATSP are {0, 1 2 }-cuts. This includes the comb inequalities for the STSP, which include the 2-matching inequalities as a special case, and the SD inequalities for the ATSP, which include the comb and odd CAT inequalities as special cases. Since then, many papers have appeared that use Proposition 1 to derive exact or heuristic polynomial-time separation routines for various inequalities. Of relevance to us are the following: In [14], a heuristic separation routine is given for the cloned odd CAT inequalities for the ATSP [3]. In [17], a separation heuristic is given for the SD inequalities for the ATSP [4]. 3 A Modified Formulation of the SSOP In this section, we present a modified 0-1 linear programming formulation for the SSOP. The motivation for the modification is given in Subsection 3.1. The modification is described in Subsection 3.2, and some simple improvements are presented in Subsection Drawbacks of the original formulation Let us now consider the formulation (3) (9) in a bit more detail. Each feasible solution to the STSP constraints (4) (6) represents a tour that can be traversed in either of two directions. Such a tour is infeasible for the SSOP if and only if there exist arcs (i, j), (k, l) B such that the tour visits the nodes i, j, k and l in one of the following four orders: i, l, k, j (or the reverse), i, l, j, k (or the reverse), i, j, l, k (or the reverse), 6

7 j, i, k, l (or the reverse). One can check that the symmetric precedence 2-cycle breaking inequalities (7) cut off infeasible tours of the first two kinds, the symmetric π inequalities (8) cut off infeasible tours of the third kind, and the symmetric σ inequalities (9) cut off infeasible tours of the fourth kind. This confirms that (3) (9) is a valid formulation of the SSOP. Unfortunately, however, no separation routines are given in [6] for any of the inequalities (7) (9). So, consider the separation problem for the inequalities (7). By enumerating over all possible pairs of arcs (i, j) and (k, l) in B, one can fairly easily reduce the associated separation problem to O( B 2 ) max-flow / min-cut problems. This can be conducted in polynomial time, but could be very time-consuming when B is large. The situation is even worse when it comes to the inequalities (8) and (9), since there is no obvious way to solve the associated separation problems in polynomial time. To get around this, one could work with weakened symmetric π inequalities of the form x(s : V \ (S {i, k})) 1 for all S V \ {1} such that there exist arcs (i, j), (k, l) B with i / S, j S, k / S and l S, and weakened symmetric σ inequalities of the form x(s \ {i, k} : V \ S) 1 for all S V \ {1} such that there exist arcs (i, j), (k, l) B with i S, j / S, k S and l / S. The formulation remains valid when these weakened inequalities are used in place of the original, and the separation problems for these weakened inequalities can also be reduced to O( B 2 ) max-flow / min-cut problems. Again, however, this could be very time-consuming in practice. These considerations led us to derive a slightly different formulation, for which separation becomes much easier. 3.2 A modified formulation We have found it helpful to modify the formulation by replacing the depot node with two nodes, one representing the start of the tour and the other representing the end. Then, each feasible SSOP solution can be viewed as a Hamiltonian path from the start node to the end node. To keep notation simple, let G = (V, E) now denote the expanded graph, let n denote V, 7

8 and then let us assume that the nodes have been renumbered so that the start node is node 1 and the end node is node n. We also assume that B now contains arcs (1, i) for all nodes i that have no predecessors, and arcs (i, n) for all nodes that have no successors. Accordingly, H = (V, B) now contains a dipath from 1 to n passing through any other given node. The SSOP can now be formulated as follows: min e E c ex e (10) s.t. x(δ(i)) = 1 (i {1, n}) (11) x(δ(i)) = 2 (i V \ {1, n}) (12) x(δ(s)) 2 (S V \ {1, n}) (13) x(δ(s)) 3 (1 S V \ {n} : (i, j) B : i / S, j S) (14) x e {0, 1} (e E). (15) To see this, note that the constraints (11) (13) ensure that the solution represents a Hamiltonian path from 1 to n, and the constraints (14) impose the precedences. Clearly, the formulation (10) (15) is simpler than the formulation (3) (9). More importantly, separation becomes much easier and faster. Specifically, the separation problem for (13) can be solved in the time taken to solve one max-flow / min-cut problem, and the separation problem for (14) can be reduced to B max-flow / min-cut problems. Full details will be given in Subsections 5.1 and Improving the modified formulation It turns out that the formulation given in the previous subsection can be easily improved, in three ways. First, we can eliminate any edge {i, j} from E such that there exists a node k V \ {i, j} that must be visited between nodes i and j. (That is, if either i π(k) and k π(j), or i σ(k) and k σ(j).) The x variables associated with these edges can be eliminated (i.e., fixed at zero). Second, we can convert some edges into arcs a priori. Indeed, if the arc (i, j) is in B, the edge {i, j} E can be traversed only in the direction from i to j. Accordingly, each such edge can be replaced with the arc (i, j). In particular, each edge {1, i} can be replaced with the arc (1, i), and each edge {i, n} can be replaced with the arc (i, n). These replacements do not make the formulation stronger, but they give us additional information, which will prove to be very useful for deriving valid inequalities later on. 8

9 We denote the mixed graph that remains after making these two improvements G = (V, E B). We let L = E B denote the set of links. (Note that, between any two nodes in G, there is at most one link.) For any S, we let δ E (S) denote the set of edges in E with exactly one end-node in S, δ + (S) denote the set of arcs in B with their tail in S, δ (S) denote the set of arcs in B with their head in S, and δ(s) denote δ E (S) δ + (S) δ (S). Using this notation, the formulation resulting from the first two improvements can be written as: min l L c lx l (16) s.t. x(δ + (1)) = 1 (17) x(δ (n)) = 1 (18) x(δ(i)) = 2 (i V \ {1, n}) (19) x(δ(s)) 2 (S V \ {1, n}) (20) x(δ(s)) 3 (1 S V \ {n} : (i, j) B : i / S, j S) (21) x l {0, 1} (l L). (22) We call the constraints (20) and (21) mixed subtour elimination (mixed SE) and mixed precedence-forcing (mixed PF) inequalities, respectively. The third improvement is to add the following inequalities: x(δ + (i)) 1 (i V \ {1, n}) (23) x(δ (i)) 1 (i V \ {1, n}). (24) We call these out-bound and in-bound inequalities. If B is large, they can improve the lower bound substantially. 4 Valid Inequalities In this section, we introduce several new (and exponentially-large) families of valid inequalities for the SSOP. 4.1 Cutset inequalities Suppose that D is a dipath from 1 to n in H. For a given S, let D(S) be the set of arcs in D that have one end-node in S. Then the following cutset inequalities are valid: x(δ(s)) D(S) (S V ). (25) These dominate the mixed SE and mixed PF inequalities... 9

10 4.2 Balancing inequalities According to Ford and Fulkerson, a mixed graph is Eulerian if and only if it is connected and balanced... This leads to the following inequalities: x(δ E (S) δ + (S)) x(δ (S)) (S V \ {1, n} : S 2). (26) We call these balancing inequalities. (A similar family of inequalities was derived by Nobert & Picard [21] for a completely different vehicle routing problem, known as the mixed Chinese postman problem.) 4.3 Mixed blossom and comb inequalities We now define some valid inequalities for the SSOP that are analogous to the blossom and comb inequalities for the STSP. Let H V \ {1, n} be the handle, let T δ E (H) be the set of teeth, and let T be odd. Then we have the mixed blossom inequality: x(e(h) B(H)) + T x e H +. 2 e T (There is no point making an arc a tooth... since one would get a stronger cut by replacing the bound x a 1 with an out-bound or in-bound inequality...) (What about mixed combs? teeth?) What condition should we impose on the 4.4 Mixed SD inequalities We now define some valid inequalities for the SSOP that are analogous to the SD inequalities for the ATSP. They turn out to generalise the mixed comb inequalities... We start first with the primitive version of the these inequalities. Let H V \ {1} be the handle, let T δ E (H) be the set of teeth, S H be the source nodes and D H be the destination nodes. Assume that the 10

11 teeth, S and D have no nodes in common, and let T + S + D be odd. Then we have: x(e(h) B(H))+ T + S + D x e +x(b(s : V \H))+x(B(V \H : D)) H +. 2 e T We call these primitive mixed SD inequalities. Lemma 1 Primitive mixed SD inequalities are valid for the SSOP. Proof. Sum together the degree equations (19) over all nodes in H, the upper bounds x e 1 for all e T, the out-bound inequalities (23) over all nodes in S, and the in-bound inequalities (24) over all nodes in D, to obtain: x(e(h) B(H))+ e T x e +x(b(s))+x(b(d))+x(b(s : V \S))+x(B(V \D : D)) 2 H + T + S + D Divide this by two and round down all coefficients to the nearest integer. Note that, when S = D =, the primitive mixed SD inequalities reduce to blossom inequalities. The general class of mixed SD inequalities is obtained by applying cloning to any or all nodes in S or D, or any end-nodes of the teeth. These take the following form... When... these are analogous to the comb inequalities for the STSP. When... they are analogous to the odd CAT inequalities for the ATSP. We say that a mixed SD inequality is simple if it is obtained from a primitive SD inequality by cloning some nodes in S and D at most once. The simple mixed SD inequalities include all mixed blossom and mixed odd CAT inequalities. 5 Separation Algorithms Surprisingly, for five of the classes of inequalities that we have defined, there are polynomial-time separation algorithms! 11

12 Let E denote... let B denote... and let G denote Mixed SE inequalities These are easy to separate exactly, as usual... one can use the Nagamochi et al. (1994) algorithm. 5.2 Mixed PF inequalities These are polynomially-separable too. Solve one max-flow / min-cut problem for each arc in B. 5.3 Cutset inequalities The complexity of separation is unclear... we can always just take a violated or near-violated mixed SE or mixed PF inequality, and then solve a maximum-weight dipath algorithm to find the best possible value of the right-hand side Balancing inequalities The balancing inequalities (26) can be separated in polynomial-time by reduction to a single max-flow / min-cut problem. See [21] Mixed blossom inequalities These can be separated as in Padberg-Rao, or as in Letchford-Reinelt-Theis. In the former case, one needs to solve O( E ) max-flow / min-cut problems on a graph with... nodes and... edges. In the latter case, one needs to solve O( V ) max-flow / min-cut problems on G itself. 12

13 5.6 Simple mixed SD inequalities Observe that every feasible solution to our SSOP formulation satisfies the following linear inequalities: x(δ(i)) = 2 (i V ) (27) x(δ + (i)) 1 (i V \ {1}) (28) x(δ (i)) 1 (i V \ {1}) (29) x e 1 (e E) (30) x e 0 (e E) (31) x a 0 (a B). (32) The remarkable thing is that the linear system (27) (32) satisfies one of the two conditions in Proposition 1. Theorem 1 The matrix given by the left-hand side coefficients of the inequalities (27) (29) is an EPT-matrix. From the above theorem it follows that one can separate in polynomial time over all of the simple mixed SD inequalities mentioned in Subsection 4.4. Perhaps say something about the details of the algorithm. One must take the tree mentioned in the proof of the theorem, label some edges and nodes odd, and then add other edges representing the x variables with positive value at x. Finally, one must compute a minimum-weight odd cut in the resulting graph. Next, we need to analyse the running time. Let L be the set of links with positive value at x. The odd cut must be computed in a graph with O( L ) nodes and edges, where the number of odd edges and odd nodes seems to be O( L ). Then, using a standard max-flow algorithm, the time taken by the separation algorithm seems to be O(( L 3 log L ). 5.7 Shrinking We could probably apply shrinking to paths of 1-edges, and to 1-arcs, before running the mixed blossom and simple mixed SD separation algorithm. Then, the exact separation algorithm for mixed blossom inequalities becomes a heuristic for comb inequalities, and the exact separation algorithm for simple mixed SD inequalities becomes a heuristic for general mixed SD inequalities. But is such shrinking valid? 13

14 6 Computational Results It would be nice to compare some different bounds. Our original formulation, formulation after edge elimination, then after out-bound and in-bound inequalities, then with various combinations of cutset, balancing, mixed blossom and mixed SD inequalities... 7 Conclusion We could probably derive additional valid inequalities from conflicts between pairs of edges / arcs. The same applies to the standard SOP! References [1] N. Ascheuer, L.F. Escudero, M. Grötschel & M. Stoer (1993) A cutting plane approach to the sequential ordering problem (with applications to job scheduling in manufacturing). SIAM J. Optim., 3, [2] N. Ascheuer, M. Jünger & G. Reinelt (2000) A branch-and-cut algorithm for the asymmetric traveling salesman problem with precedence constraints. Comput. Optim. Appl., 17, [3] E. Balas (1989) The asymmetric assignment problem and some new facets of the traveling salesman polytope. SIAM J. Discr. Math., 2, [4] E. Balas & M. Fischetti (1993) A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets. Math. Program., 58, [5] E. Balas & M. Fischetti (2007) Polyhedral theory for the asymmetric traveling salesman problem. In G. Gutin & A.P. Punnen (eds.) The Traveling Salesman Problem and Its Variations, pp Springer. [6] E. Balas, M. Fischetti & W. Pulleyblank (1995) The precedenceconstrained asymmetric traveling salesman polytope. Math. Program., 68, [7] L. Bianco, A. Mingozzi & S. Ricciardelli (1994) Exact and heuristic procedures for the travelling salesman problem with precedence constraints, based on dynamic programming. INFOR, 32,

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