Branch-and-Cut Algorithms for the Undirectedm-Peripatetic Salesman Problem
E. Duchenne, G. Laporte´ F. Semet
G–2003–02 January 2003
Revised September 2003
Les textes publi´es dans la s´erie des rapports de recherche HEC n’engagent que la responsabilit´e de leurs auteurs. La publication de ces rapports de recherche b´en´eficie d’une subvention du Fonds F.C.A.R.
m -Peripatetic Salesman Problem
Eric Duchenne ´
LAMIH-ROI, Universit´e de Valenciennes Le Mont Houy, 59313 Valenciennes
Cedex 9, France
Gilbert Laporte
GERAD and Canada Research Chair in Distribution Management HEC Montr´eal
3000 chemin de la Cˆote-Sainte-Catherine Montreal, Canada H3T 2A7
Fr´ ed´ eric Semet
LAMIH-ROI, Universit´e de Valenciennes Le Mont Houy, 59313 Valenciennes
Cedex 9, France
January, 2003 Revised: September, 2003
Les Cahiers du GERAD G–2003–02
Copyright c2003 GERAD
disjoint Hamiltonian cycles of minimum total cost on a graph. This article describes exact branch-and-cut solution procedures for the undirected m-PSP. Computational results are reported on random and Euclidean graphs.
Key Words: Peripatetic Salesman Problem, Traveling Salesman Problem, Branch- and-Cut.
R´esum´e
Dans le probl`eme dum-voyageur p´eripat´etique (m-PSP) on cherche `a d´eterminer mcycles hamiltoniens n’ayant aucune arˆete en commun et dont la somme des coˆuts des arˆetes est minimale. Cet article d´ecrit des proc´edures de branch-and-cut exactes pour lem-PSP non orient´e. On pr´esente des r´esultats num´eriques sur des graphes al´eatoires et euclidiens.
Mots Clefs: Probl`eme du voyageur p´eripat´etique, probl`eme du voyageur de com- merce, branch-and-cut.
Acknowledgments: This work was partially supported by the Canadian Natural Sciences and Engineering Research Council under grant OGP0039682. This support is gratefully acknowledged.
1 Introduction
Them-Peripatetic Salesman Problem (m-PSP) is defined on a complete graphG= (V, E), whereV ={1, . . . , n}is a vertex set andE={(i, j): i, j∈V,i < j}is an edge set. A cost matrix C = (cij) is defined on E. The problem consists of determining m edge disjoint Hamiltonian cycles of minimum total cost onG. When m= 1 the m-PSP reduces to the Traveling Salesman Problem(TSP). In the sequel we assume thatm <(n−1)/2 to avoid trivial or infeasible cases.
The m-PSP was introduced by Krarup (1975). Applications include the design of watchman tours (Wolfter Calvo and Cordone, 2003) where it is often important to assign a set of edge-disjoint rounds to the watchman in order to avoid always repeating the same tour and thus enhance security. In the same spirit, De Kort (1993) cites a network design application where, in order to protect the network from link failure, several edges-disjoint cycles must be determined. This author also mentions a scheduling application of the 2-PSP where each job must be processed twice by the same machine but technological constraints prevent the repetition of identical job sequences.
De Kort (1993) shows that the 2-PSP is NP-hard by transforming an instance of the Hamiltonian Path problem into a 2-PSP. A similar reasoning can be applied to the case wherem >2. While the undirected TSP is also an NP-hard problem, medium size instances can easily be solved in practice (see, e.g., Padberg and Gr¨otschel, 1985; Applegate, Bixby, Chv´atal and Cook, 2003). A natural question is then whether TSP algorithms can be used to provide good or optimal m-PSP solutions. A partial answer is obtained by applying the following “Krarup heuristic” (Krarup, 1975): solve a first TSP on G (exactly or by means of a heuristic) and remove from the graph all edges of the TSP solution; repeat until m Hamiltonian cycles have been obtained. Krarup shows that even if the TSPs are solved optimally, this heuristic does not always yield an optimal m-PSP solution. His counter-example consists of a graph in which costs do not satisfy the triangle inequality.
Here we present a counter-example defined on a Euclidean graph. Consider the graph defined on a plane with the following Cartesian vertex locations: vertex 1: (0,0), vertex 2: (1,0), vertex 3: (2,0), vertex 4: (2,1), vertex 5: (1,1), and vertex 6: (0,1), and where all distances are Euclidean. If m = 2, the unique TSP solution of cost 6 is given by (1,2,3,4,5,6). Removing the edges of this solution and solving the TSP once more yields a second TSP solution (1,3,5,2,6,4) of cost 5 + 2√
2 +√
5. Therefore, the total cost of the two cycles is 16.06. It can be shown that an optimal 2-PSP solution consists of the two cycles (1,2,3,4,6,5) and (1,3,5,4,2,6) of total cost 10 + 4√
2 = 15.66. Therefore, repeatedly solving a TSP to optimality only provides a heuristic for the m-PSP. We will later provide an empirical assessment of this heuristic. In addition, a simple lower bound on the optimalm-PSP value is given bymz, wherezis a lower bound on the optimal TSP solution. We will also report on the empirical accuracy of this lower bound.
In addition to Krarup’s seminal contribution, only a few scientific articles are available on them-PSP. De Kort (1991) derives a lower bound based on the solution of a capacitated transportation problem. Tests performed on instances with 42≤n≤120 andm= 2 show an average deviation of 8.9% with respect to the upper bound obtained by applying the Krarup heuristic. A branch-and-bound algorithm (De Kort, 1993) based on a 3-index
formulation was applied to instances with m = 2 and 40 ≤ n ≤ 130 for random cost matrices, and m = 2 and 40 ≤ n ≤60 for Euclidean cost matrix. Finally, De Kort and Volgenant (1994) have analyzed a generalization of the 2-PSP in which each cycle contains each vertex at most once and a penalty is incurred for vertices not included in a cycle.
Our aim is to provide exact solution procedures for the undirectedm-PSP with a general value ofm. For this we will make use of an integer linear programming formulation and of a relaxation of the problem. Section 2 describes a 3-index model while a 2-index formulation is used in Section 3. In Section 4, we report on comparative computational experiments relative to exact solution values, lower bounds and upper bounds. A conclusion follows in Section 5.
2 3-index formulation and algorithm
The undirected m-PSP can easily be modeled by means of a 3-index formulation. Let xijk(i < j) be a binary variable equal to 1 if and only if edge (i, j) appears on cycle k. The model is then
(3-index) minimize
m
k=1
i<j
cijxijk (1)
subject to
i<h
xihk+
j>h
xhjk= 2 (h∈V; k= 1, . . . , m) (2)
i,j∈S
i<j
xijk≤ |S| −1 (S⊂V, 3≤ |S| ≤ n/2; k= 1, . . . , m) (3)
m
k=1
xijk≤1 (i, j∈V; i < j) (4) xijk = 0 or 1 (i, j∈V; i < j; k= 1, . . . , m). (5) The constraints of this model are easily explained. If constraints (4) are removed, the formulation is that of a TSP for each k, where constraints (2) are degree constraints and constraints (3) are subtour elimination constraints. In other words, solving this relaxed model will yield a lower bound of valuemz∗ for them-PSP, where z∗ is the optimal TSP solution value on G. The role of constraints (4) is simply to ensure that an edge (i, j) appears on only one cycle.
This model can be strengthened through the introduction of any valid inequality for the TSP. In our implementation we have used the two most useful families of TSP valid inequalities (Gr¨otschel and Padberg, 1985): 2-matching inequalities and comb inequalities.
ForS ⊆V, let E(S) ={(i, j)∈E:i, j∈S}. For a given k, 2-matching inequalities are
(i,j)∈E(H)
xijk+
(i,j)∈E
xijk ≤ |H|+ (|E| −1)/2, (6)
where H ⊂ V, E ⊆ E, |E| ≥ 3 and odd, |{i, j} ∩ H| = 1 for all (i, j) ∈ E, and {i, j} ∩ {h, }=∅for all (i, j), (h, )∈E, (i, j)= (h, ).
The comb inequalities can be written as
(i,j)∈E(H)
xijk+
r
=1
(i,j)∈E(T)
xijk≤ |H|+
r
=1
(|T| −1)−(r+ 1)/2, (7)
where H, T1, . . . , Tr ⊂ V, r ≥ 3 and odd, H ∩T = ∅ and T\H = ∅ ( = 1, . . . , r), Th∩T =∅ (h, = 1, . . . , r, h=).
Branch-and-cut algorithm
Our branch-and-cut algorithm can be summarized as follows.
Step 1 (First subproblem). Determine a first feasiblem-PSP solution of costzby means of the Krarup heuristic. Create a first subproblem defined by (1), (2) and (4) and 0≤xijk≤1 (i, j∈V, i < j; k= 1, . . . , m). Insert this subproblem in a list.
Step 2 (Termination check). If the list is empty, stop. Otherwise select a subproblem from the list according to a best first criterion (i.e. the subproblem with the smallest solution value).
Step 3 (Subproblem solution). Solve the subproblem and let z be its solution value. If z ≥z, go to Step 2. If the solution is feasible for the m-PSP, set z :=z and go to Step 2.
Step 4 (Cutting). Attempts to identify, in this order, up to 10 violated constraints of type (3), (6) and (7) by applying appropriate separation procedures. If this is suc- cessful, introduce the violated constraints to the current subproblem and go to Step 3.
Step 5 (Branching). Create two subproblems by branching on the fractional variable closest to 0.5. Ties are broken by branching on the variable with the largest cost.
In Step 1 of the algorithm the TSPs were solved by means of the Padberg and Rinaldi (1991) exact algorithm which ran in insignificant time for the instances sizes we considered.
In Step 3, we used CPLEX 6.6 for the solution of the subproblems. In Step 4, we used an exact flow separation procedure (Padberg and Gr¨otschel, 1985) for separation of subtour elimination constraints (3). The 2-matching inequalities (6) we separated by means of the heuristic proposed by Fischetti, Salazar and Toth (1997). To separate comb inequali- ties (7), we have used the heuristic proposed by Naddef and Thienel (2002). Whenever a violated constraint is identified for a givenk, it is imposed for all values ofkin the current subproblem in order to avoid equivalent solutions in which the cycle indices are merely permuted.
3 2-index formulation and algorithms
A 2-index formulation can be derived from 3-index by aggregating the decision variables, i.e., by defining binary variables xij =m
k=1xijk. Thus we obtain the model (2-index) minimize
i<j
cijxij (8)
subject to
i<h
xih+
j>h
xhj = 2m (h∈V) (9)
i,j∈S
i<j
xij ≤m(|S| −1) (S⊂V, 2m≤ |S| ≤ n/2) (10)
xij = 0 or 1 (i, j∈V; i < j). (11) The relationship between the constraints of this model and those of 3-index is imme- diate. Contrary to what is asserted in De Kort (1991), 2-index is not a valid model for the m-PSP, but only a relaxation. Any solution to 2-index is a connected regular graph of degree 2m, i.e., a graph in which the degree of each vertex is 2m. However, an m-PSP solution may not exist in such a graph and the graph is not even always Hamiltonian. To see this consider the graph depicted in Figure 1. This graph satisfies constraints (9)–(11) but is not Hamiltonian (Bauer, Broersma and Veldman, 1990).
Figure 1: Non-Hamiltonian graph feasible for 2-index withm= 2.
However, the 2-index model can still be used to compute a lower bound on them-PSP solution value and, as will be shown, it can serve as a basis for an exact algorithm. In addition, it can be reinforced through the inclusion of 2-matching and comb inequalities.
Proposition 1 The 2-matching inequalities
(i,j)∈E(H)
xij +
(i,j)∈E
xij ≤m|H|+ (|E| −1)/2, (12)
where H ⊂ V, E ⊂ E, |E| ≥ 3 and odd, |{i, j} ∩ H| = 1 for all (i, j) ∈ E, and {i, j} ∩ {h, }=∅ for all(i, j),(h, )∈E, (i, j)= (h, ), are valid for 2-index.
Proof. Summing up constraints (9) over allh∈H yields
2
(i,j)∈E(H)
xij+
(i,j)∈δ(H)
xij = 2m|H|, (13)
where δ(H) ={(i, j) ∈ E :i∈ H,j ∈H or i∈ H,j ∈H}. Since E ⊂ δ(H), it follows that
2
(i,j)∈E(H)
xij +
(i,j)∈E
xij ≤2m|H|, (14)
Summing up the constraintsxij ≤1 over all (i, j)∈E yields
(i,j)∈E
xij ≤ |E|, (15)
which, combined with (14), yields
2
(i,j)∈E(H)
xij + 2
(i,j)∈E
xij ≤2m|H|+|E|
and
(i,j)∈E(H)
xij +
(i,j)∈E
xij ≤m|H|+ (|E| −1)/2, (16)
since|E|is odd and the left-hand side of (16) is integer. 2 Proposition 2 If m is odd, the comb inequalities
(i,j)∈E(H)
xij+
r
=1
(i,j)∈E(T)
xij ≤m|H|+mr
=1
(|T| −1)−(mr+ 1)/2, (17)
where H, T1, . . . , Tr ⊂V, r≥3and odd, H∩T=∅ and T\H=∅ (= 1, . . . , r), and Th∩T =∅ (h, = 1, . . . , r, h=), are valid for 2-index.
Proof. Summing up constraints (9) over all h ∈ H yields (13). Adding to (13) the non-negativity inequalities−xij ≤0 for all (i, j)∈δ(H)\(E(T1)∪ · · · ∪E(Tr)) yields
2
(i,j)∈E(H)
xij +
r
=1
(i,j)∈δ(H)∩E(T)
xij ≤2m|H|. (18)
Subtour elimination constraints (10) applied toS =T,T\H and T∩H are
(i,j)∈E(T)
xij ≤m(|T| −1) (19)
(i,j)∈E(T\H)
xij ≤m(|T\H| −1) (20)
and
(i,j)∈E(T∩H)
xij ≤m(|T∩H| −1) (21)
for= 1, . . . , r. Summing up (18) and (19)–(21) for= 1, . . . , r and dividing by 2 yields
(i,j)∈E(H)
xij +
r
=1
(i,j)∈E(T)
xij ≤m|H|+mr
=1
(|T| −1)−mr/2. (22)
Since the left-hand side of (22) is integer and mr is odd, the last term of (22) can be
replaced with (mr−1)/2 and the thesis follows. 2
Algorithms
We have developed two algorithms for the m-PSP using the 2-index formulation. In all cases, 2-index is first solved with a Branch-and-cut which can be summarized as follows.
Step 1 (First subproblem). Determine a first feasiblem-PSP solution of costzby means of the Krarup heuristic. Create a first subproblem defined by (8) and (9) and 0≤xij ≤1 (i, j∈V, i < j). Insert this subproblem in a list.
Step 2 (Termination check). If the list is empty, stop. Otherwise select a subproblem from the list according to a best first criterion (i.e. the subproblem with the smallest solution value).
Step 3 (Subproblem solution). Solve the subproblem and let z be its solution value. If z ≥z, go to Step 2. If the solution is feasible for the m-PSP, set z :=z and go to Step 2.
Step 4 (Cutting). Attempts to identify up to 10 violated constraints of type (10) by applying a procedure based on the exact flow separation procedure of Padberg and Gr¨otschel (1985). If this is successful, introduce the violated constraints to the current subproblem and go to Step 3.
Step 5 (Branching). Create two subproblems by branching on the fractional variable closest to 0.5. Ties are broken by branching on the variable with the largest cost.
In Step 3 the solution value is compared to that of a solution obtained by means of the Krarup heuristic. If the two values coincide, the Krarup heuristic solution is optimal and the process terminates. Otherwise the Krarup heuristic is applied to the solution graph of
the 2-index solution. Again, an optimal solution has been identified if its value coincides with that of 2-index. Otherwise Algorithm 1 or Algorithm 2 is applied for verifying the feasibility of the solution.
Algorithm 1 gradually constructs m cycles by selecting at each step an unvisited edge of the 2-index solution, subject to a connectivity condition. This condition states that when the sth cycle is being constructed, the graph induced by the yet unselected edges remains 2(m−s) connected. This condition is checked by solving a flow problem: an edge is added to thesth cycle only if the flow between its extremities is superior to 2(m−s). A backtracking procedure is applied whenever the connectivity condition cannot be respected.
It either identifies a feasible and optimalm-PSP solution or fails to find a feasible solution.
Algorithm 2 attempts to determine a feasible solution by solving 3-index on the re- stricted set of edges that are part of the 2-index solution. However, whenm= 2, we apply the following modification to the branching rule in Step 5 of the Branch-and-Cut proposed for the 3-index model. Consider a vertexi for which one variable xijk is fractional. The standard branching rule is to create the dichotomy xijk = 0 or xijk = 1. Ifm = 2, there are four variablesxij1k,xij2k,xij3k andxij4kassociated withiandk, two of which must be equal to 1 in a feasible solution. The following three-partite branching is therefore valid:
xij1k= 1,xij1k= 0 andxij2k= 1,xij1k =xij2k= 0 andxij3k=xij4k= 1. Given a variable xij1kselected for branching, the choice ofj2, j3 andj4is made arbitrarily. This rule applies whenever none of the variables xij1k, xij2k, xij3k, xij4k has previously been fixed by the branching process. Otherwise, the number of branching options is more restricted.
4 Computational results
The algorithms just described were coded in C++and run on a Compaq AlphaServer DS20 biprocessor EV6/500. All integer linear programs were solved with CPLEX 6.6. Euclidean TSPs were solved using the Padberg and Rinaldi (1991) code. Random TSPs were solved by means of our own algorithm for 3-index, with m= 1.
Tests were performed on two types of cost matrices. As in De Kort (1993), we solved random cost instances on which thecijs were random distributed on [1,1000] according to a discrete uniform distribution and Euclidean instances with vertices randomly distributed in [0,100]2 according to a discrete uniform distribution, and cijs computed as Euclidean distances. Five instances of each type were generated forn= 10, . . . ,60 andm= 1, . . . ,5.
We also solved TSPLIB instances form= 1, . . . ,5 to facilitate future comparisons between differents algorithms.
Computational results are summarized in Tables 1 to 8. The first three tables corre- spond to random instances while the next three are for Euclidean instances and the last two for TSPLIB instances. The table headings are as follows:
n: number of vertices inG;
m: number of edge-disjoint cycles in the solution;
LB: the simple lower bound equal to mz, where z is a lower bound on a TSP defined on G (see Section 1); in our implementation z was taken as the optimal TSP solution value;
UB: solution value obtained by applying the Krarup heuristic; again the TSPs were solved optimally;
OPT: optimalm-PSP solution value;
SUCC: success rate of the algorithm: number of optimal solutions found within the time limit divided by the number of attempted instances.
SROOT: success rate at the root of the search tree before branching;
LB0: lower bound value obtained after solving the subproblem at the root of the search tree but before the introduction of valid inequalities;
LB1: lower bound value obtained at the root of the search tree immediately before branching;
SEC: number of subtour elimination constraints generated in the search tree;
2MI: number of 2-matching inequalities generated in the search tree;
COM: number of comb inequalities generated in the search tree;
NODES: number of nodes in the search tree;
SECONDS: CPU time in seconds.
All statistics are averages over the number of successful instances, i.e., instances that could be solved optimally within 3600 CPU seconds.
Our results indicate that random instances of size n = 30 and m = 2 can be solved optimally by the 3-index model and algorithm; much larger instances of size n= 60 and m = 5 can be solved by the 2-index model. For Euclidean instances, the corresponding figures aren= 50 andm= 2 for the 3-index model andn= 50 andm= 2 for the 2-index model. Difficulty increases with m. Tables 2 and 5 show that the 2-index relaxation is relatively easy to solve since rather large instances are solved to optimality. Tables 3 and 6 clearly show that the 2-index model followed by Algorithm 1 or 2 is superior to the 3-index model both in terms of success rate and of computation time. If the 2-index model is used, it is preferable (apart from a few exceptions) to use Algorithm 2 than Algorithm 1.
Tables 1 and 4 show that the simple lower bound LB is of very poor quality, especially in the case of random instances. On the other hand, the Krarup upper bound is generally quite sharp. An examination of Tables 1, 2, 4 and 5 reveals that the LB0/OPT and LB1/OPT ratios are very close to 1 both for 3-index and for 2-index but these ratios are slightly higher for 3-index. Very often, introducing violated valid inequalities at the root node will have little impact on the lower bound value (the LB0/OPT and LB1/OPT ratios are often equal), but this is not always true. Thus, if the 2-index model is applied to Euclidean instances (Table 5), valid inequalities will often increase the lower bound significantly. In all cases, these inequalities will help the linear relaxation move toward less fractional solutions. As observed for the TSP (Padberg and Gr¨otschel, 1985), subtour elimination constraints are very frequently generated. In fact only these inequalities were used in the case of 2-index.
Finally, Tables 7 and 8 present results on some of the TSPLIB instances. These results are similar to those obtained for the Euclidean instances.
Table1:Computationalresultsforthe3-indexmodel:randominstances. Simplebounds3-indexmodel nmLB/OPTUB/OPTLB0/OPTLB1/OPTSROOTSEC2MICOMNODESSUCCSECONDS 10111111000010 1020.6691.023110.426.800.81.211 1030.4861.0160.9990.99904744.829.466.6151 1040.3881.010110.4205.60.856.219 20111111000010 2020.7091.0040.9990.9990.2189.603.416.2190 2030.5191.005110.2479.31.51118.30.8349 2040.4071.0050.9980.9990.22358238.589.20.81932 30111111000010 3020.6821.0101101540.72.38.30.6364
Table 2: Computational results for the 2-index model: random instances.
2-index model
n m LB0/OPT LB1/OPT SROOT SEC NODES SUCC SECONDS
10 1 1 1 1 0.2 0 1 0
10 2 1 1 1 0 0 1 0
10 3 0.999 0.999 0.8 0 0.4 1 0
10 4 1 1 1 0 0 1 0
20 1 0.987 0.997 0.4 3.2 4.6 1 1
20 2 1 1 0.8 0 0.2 1 0
20 3 1 1 0.8 0 0.2 1 0
20 4 1 1 1 0 0 1 0
20 5 1 1 1 0 0 1 0
30 1 0.986 0.998 0.4 2.6 1.2 1 2
30 2 0.999 0.999 0.6 0 0.6 1 2
30 3 1 1 0.8 0 0.2 1 2
30 4 1 1 1 0 0 1 3
30 5 1 1 0.6 0 0 1 4
40 1 0.995 0.996 0.4 3.4 6.6 1 28
40 2 1 1 0.8 0 0.6 1 8
40 3 1 1 0.6 0 0.6 1 12
40 4 1 1 1 0 0 1 10
40 5 1 1 0.6 0 2.4 1 35
50 1 0.994 0.998 1 3.2 2.8 1 37
50 2 1 1 0.6 0 1 1 26
50 3 1 1 1 0 0 1 21
50 4 1 1 0.8 0 0.4 1 37
50 5 1 1 0.8 0 0.2 1 42
60 1 0.997 0.999 0.8 5.6 2.8 1 105
60 2 1 1 1 0 0 1 31
60 3 1 1 0.8 0 0.4 1 60
60 4 1 1 0.8 0 0.6 1 84
60 5 1 1 0.8 0 0.2 1 87
Table 3: Success rate and computation time for the various algorithms: random instances.
2-index + 2-index +
3-index Algorithm 1 Algorithm 2
n m SUCC SECONDS SUCC SECONDS SUCC SECONDS
10 1 1 0 1 0 1 4
10 2 1 1 1 0 1 5
10 3 1 51 1 0 1 3
10 4 1 9 1 0 1 1
20 1 1 0 1 1 1 1
20 2 1 90 1 1 1 27
20 3 0.8 349 1 29 1 179
20 4 1 1932 1 41 1 1663
20 5 0 – 1 59 0.4 3251
30 1 1 0 1 2 1 2
30 2 0.6 364 1 162 1 257
30 3 0 – 0.8 173 0.6 2
30 4 0 – 0.6 487 0.4 42
30 5 0 – 0.4 510 0.8 5
40 1 1 0 1 29 1 29
40 2 0 – 0.6 1160 0.8 1544
40 3 0 – 0.2 643 0.2 76
40 4 0 – 0 - 0.2 10
40 5 0 – 0 – 0.2 13
50 1 1 0 1 37 1 37
50 2 0 – 0 – 0.2 1628
60 1 1 0 1 105 1 105
60 2 0 – 0 – 0.2 32
Table4:Computationalresultsforthe3-indexmodel:Euclideaninstances. Simplebounds3-index nmLB/OPTUB/OPTLB0/OPTLB1/OPTSROOTSEC2MICOMNODESSUCCSECONDS 10111111000010 1020.8101.0060.9880.9910.850.501.50.510 1030.6511.002110.4186.704.73.3313 1040.5741110.85928202318 20111111000010 2020.8011.0050.9980.9990.2173.23.25131110 2030.6631.003110.26002.38.514.81259 2040.5811.00311022161.662.86012500 2050.4731110.200000.20 30111111000010 3020.8221.0060.99210.22131.50.54.3192 3030.6961110.2267040.50.4470 40111111000010 4020.8061.0080.9930.9980268503.50.4475 50111111000010 5020.8391.0180.9890.99107724270.22263
Table 5: Computational results for the 2-index model: Euclidean instances.
2-index
n m LB0/OPT LB1/OPT SROOT SEC NODES SUCC SECONDS
10 1 0.938 1 1 1.3 0 1 0
10 2 0.988 1 1 0.6 0 1 0
10 3 1 1 1 0 0 1 0
10 4 1 1 1 0 0 1 0
20 1 0.866 1 0.8 10.2 0.6 1 0
20 2 0.998 1 0.8 0.6 0.4 1 0
20 3 1 1 0.8 0 0.2 1 0
20 4 1 1 0.8 0 0.6 1 0
20 5 1 1 1 0 0 1 0
30 1 0.949 1 0.4 9.8 2.2 1 3
30 2 0.992 1 0.8 2 0.2 1 1
30 3 0.999 1 0.8 0.6 1.2 1 4
30 4 1 1 1 0 0 1 2
30 5 1 1 0.8 0 0.2 1 3
40 1 0.934 0.995 0.4 23.2 19.6 1 61
40 2 0.995 1 0.4 2.6 1.2 1 11
40 3 1 1 0.4 0.2 3 1 24
40 4 1 1 0.6 0.4 0.8 1 16
40 5 1 1 0.6 0 2.2 1 31
50 1 0.923 0.994 0 44 55.4 1 404
50 2 0.998 1 0.4 5.4 23.2 1 261
50 3 1 1 0.6 0.4 9.4 1 161
50 4 1 1 0.8 0 1.4 1 58
50 5 1 1 0.8 0 0.2 1 37
Table 6: Success rate and computation time for the various algorithms: Euclidean instances.
2-index + 2-index +
3-index Algorithm 1 Algorithm 2
n m SUCC SECONDS SUCC SECONDS SUCC SECONDS
10 1 1 0 1 0 1 0
10 2 1 0 1 0 1 0
10 3 1 3 1 0 1 13
10 4 1 7 1 0 1 0
20 1 1 0 1 0 1 0
20 2 1 110 1 20 1 10
20 3 1 259 1 26 1 133
20 4 1 2500 1 117 0.6 117
20 5 0.2 0 1 87 0.4 87
30 1 1 0 1 3 1 3
30 2 1 92 0.6 1336 1 72
30 3 0.4 470 0 – 0 –
30 4 0 – 0 – 0.4 2
30 5 0 – 0 – 1 4
40 1 1 0 1 61 1 61
40 2 0.4 475 0 – 0.8 120
50 1 1 0 1 404 1 404
50 2 0.2 2263 0 – 0.4 182
Table7:Computationalresultsforthe3-indexmodel:TSPLIBinstances. Simplebounds3-index ProblemnmLBUBLB0LB1SEC2MICOMNODESOPTSECONDS burma141413323332333233323000033230 burma1414266467577738735375000075371 burma1414399691326412759129023150281290224 burma141441329221034210342103410244183021034196 burma141451661530265297222972221150515729722600 gr171712085208520852085000020850 gr1717241704971486249156000049153 gr17173625590509005900525201510900591 gr17174834013900136681366811040354813668662 gr212112707270727072707000027070 gr21212541469176847.56900178001690011 gr212138121124931248612486660091612486295 gr242411272127212721272000012720 gr2424225443160314731472668673147130 gr242433816561456145614000056140 gr242445088843584358435000084350 fri2626193793793793700009370 fri262621874223522022218174404221839 bayg292911610161016101610000016100 bayg29292322037613718.5373780000373726 bays292912020202020202020000020200 bays292924040473146944694100000469412
Table 8: Success rate and computation time for the various algorithms: TSPLIB instances.
2-index + 2-index +
3-index Algorithm 1 Algorithm 2
Problem n m SUCC SECONDS SUCC SECONDS SUCC SECONDS
burma14 14 1 1 0 1 0 1 0
burma14 14 2 1 1 1 0 1 1
burma14 14 3 1 24 1 1 1 11
burma14 14 4 1 196 1 1 1 1
burma14 14 5 1 600 1 5 1 528
gr17 17 1 1 0 1 0 1 0
gr17 17 2 1 3 1 3 1 0
gr17 17 3 1 91 1 2 1 72
gr17 17 4 1 662 1 4 1 1
gr17 17 5 0 - 1 71 0 -
gr21 21 1 1 0 1 0 1 0
gr21 21 2 1 11 1 19 1 20
gr21 21 3 1 295 1 6 1 328
gr21 21 4 0 - 1 634 0 -
gr21 21 5 0 - 1 2971 0 -
gr24 24 1 1 0 1 0 1 1
gr24 24 2 1 130 1 197 1 126
gr24 24 3 1 0 1 3242 1 1
gr24 24 4 1 0 1 56 1 1
gr24 24 5 0 - 1 860 1 8
fri26 26 1 1 0 1 1 1 1
fri26 26 2 1 39 1 255 1 1
fri26 26 3 0 - 1 3589 1 2
fri26 26 5 0 - 0 - 1 3
bayg29 29 1 1 0 1 4 1 4
bayg29 29 2 1 26 1 2632 1 2
bayg29 29 3 0 - 1 24 0 -
bays29 29 1 1 0 1 3 1 3
bays29 29 2 1 12 1 142 1 2
bays29 29 3 0 - 1 2211 1 2
bays29 29 4 0 - 1 3457 0 -
bays29 29 5 0 - 0 - 1 4
eil51 51 1 1 0 1 353 1 352
eil51 51 3 0 - 0 - 1 82
5 Conclusion
We have presented new models, valid inequalities and algorithms for the undirected m- Peripatetic Traveling Salesman Problem, a variant of the classicalTraveling Salesman Prob- lem. The largest size solved optimally attain n= 60 and m = 5. To our knowledge, this is the first time exact results are presented for m ≥ 3. We have presented two models:
3-index is an exact description of the problem whereas 2-index is only a relaxation. This model is superior when followed by a procedure aimed at restoring feasibility (Algorithm 1 or Algorithm 2).
References
Applegate, D., Bixby, R., Chv´atal, V., and Cook, W., 2003, “Implementing the Dantzig- Fulkerson-Johnson Algorithm for Large Traveling Salesman Problems”, Mathematical Programming, Series B 97, pp 91–153.
Bauer, D., Broersma, H.J., and Veldman, H.J., 1990, “On Smallest Non-Hamiltonian Regular Tough Graphs”, Congressus Numerantium 70, pp 95–98.
De Kort, J.B.J.M., 1991, “Lower Bounds for Symmetric K-Peripatetic Salesman Prob- lems”,Optimization 22, pp 113–122.
De Kort, J.B.J.M., 1993, “A Branch and Bound Algorithm for Symmetric 2-Peripatetic Salesman Problems”, European Journal of Operational Research 70, pp 229–243.
De Kort J.B.J.M., and Volgenant, A., 1994, “On the Generalized 2-Peripatetic Salesman Problem”, European Journal of Operational Research 73, pp 175–180.
Fischetti, M., Salazar, J.J., and Toth, P., 1997, “A Branch-and-Cut Algorithms for the Symmetric Generalized Traveling Salesman Problem”, Operations Research 45, pp 378- 394.
Gr¨otschel, M., and Padberg, M.W., 1985, “Polyhedral Theory”, inThe Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, eds, Wiley, Chichester, pp 251–305.
Krarup, J., 1975, “The Peripatetic Salesman and Some Related Unsolved Problems”,Com- binatorial Programming, Methods and Applications, B. Roy, ed, Reidel, Dordrecht, pp 173–178.
Naddef, D., and Thienel, S., 2002, “Efficient Separation Routines for the Symmetric Trav- eling Salesman Problem I: General Tools and Comb Separation”, Mathematical Pro- gramming 92, pp 237–255.
Padberg, M.W., and Gr¨otschel, M., 1985, “Polyhedral Computations”, in The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, E.L. Lawler, J.K.
Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, eds, Wiley, Chichester, pp 307–360.
Padberg, M.W., and Rinaldi, G., 1991, “A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Travelling Salesman Problems”,SIAM Review 33, pp 60-100.
Wolfter Calvo, R., and Cordone, R., 2003, “A Heuristic Approach to the Overnight Security Service Problem”, Computers & Operations Research 30, pp 1269–1287.
