Philipp Hungerländer Anna Jellen Stefan Jessenitschnig Lisa Knoblinger Manuel Lackenbucher Kerstin Maier September 10, 2018 Abstract In this work we analyze the multiple Traveling Salesperson Problem (mtsp) on regular grids. While the general mtsp is known to be NP-hard, the special structure of grids can be exploited to explicitly determine optimal solutions, i.e., the problem can be solved in linear time. We suggest a Mixed-Integer Linear Programming (MILP) formulation for the mtsp on regular grids where we minimize two different objective functions. The first one models the sum of the tour lengths of all salespersons and the second one considers the maximal tour length of a single salesperson. With the help of these MILPs and combinatorial counting arguments, we establish lower bounds, explicit construction schemes and hence optimal mtsp solutions for specific grid sizes, depot positions and two salespersons. Keywords: Multiple Traveling Salesperson Problem, Mixed-Integer Linear Programming, Grid. 1 Introduction The multiple Traveling Salesperson Problem (mtsp) is a generalization of the NP-hard Traveling Salesperson Problem (TSP). Given a depot d, and p points to be visited, the mtsp asks for m shortest Hamiltonian cycles, such that d is visited by all salespersons and the remaining p 1 points are visited by exactly one salesperson. In this paper we consider the mtsp on regular l n grid graphs in the Euclidean plane where the number of grid points is ln. The special structure of the grid is exploited to find lower bounds, explicit construction schemes and hence optimal mtsp solutions. Explicit construction schemes and corresponding optimal solutions are also known for another related problem, namely the TSP with Forbidden Neighborhoods (TSPFN), where consecutive points along the Hamiltonian cycle must have a minimal distance. The TSPFN was studied on regular 2D and 3D grids, see [2] and [3]. To the best of our knowledge this is the first paper suggesting lower bounds, explicit construction schemes and hence optimal mtsp solutions. Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, philipp.hungerlaender@aau.at Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, anna.jellen@aau.at Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, stefanje@edu.aau.at Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, lisakn@edu.aau.at Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, malackenbuch@edu.aau.at Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria, kerstin.maier@aau.at 1
Our research is motivated by several real-world applications, like search and rescue operations or delivering goods with swarms of unmanned aerial vehicles (UAV), see, e.g., [1] for more information about this topic. Regular grid structures can be used to divide large search areas in several equalsized squares. The size of a square is chosen as large as the sensor or camera range of a UAV. The paper is structured as follows: We suggest our Mixed-Integer Linear Programming formulation for solving the mtsp in Section 2. Our computational results are stated in Section 3. Finally, in Section 4, we propose optimal solutions for specific grid sizes, depot positions and two salespersons. 2 Mathematical Formulation In this section we propose our Mixed-Integer Linear Programming (MILP) formulation for solving the mtsp on a regular l n grid. We define [m] := {1,..., m} as the set of salespersons and [ln] := {1,..., ln} as the set of grid points. Grid points can be identified by their coordinates, for an illustration see Figure 1 a). Each salesperson starts and ends its tour at a predefined grid point, referred to as depot d. The Euclidean distance from grid point i to grid point j is given by t ij. We introduce binary variables x ijk, i, j [ln], k [m], which are set to 1, if salesperson k visits grid point j immediately after grid point i and to 0 otherwise. Motivated by applications, where m UAVs search for a missing person, we formulate two MILPs with different objective functions. MILP I (II) minimizes the average (maximal) search time for a missing person. MILP I: The first MILP minimizes the sum of the tour lengths of all salespersons, i.e., the total tour length, and can be formulated as follows: min t ij x ijk (I) s. t. k [m], i,j [ln] k [m] j [ln] k [m] i [ln] j [ln]\d, j [ln] i,j [ln]\d x ijk = 1, i [ln] \ d, (1) x ijk = 1, j [ln] \ d, (2) x djk = 1, k [m], (3) x jik = x ijk < j [ln] ln m u i u j + 1 (ln 1)(1 x ijk, k [m], i [ln], (4), k [m], (5) k [m] x ijk ), i, j [ln] \ d, i j, (6) 2
x ijk {0, 1}, k [m], i, j [ln], i j, (7) 2 u i ln, i [ln] \ d. (8) Equalities (1) and (2) ensure that each grid point except the depot has exactly one outgoing and one ingoing edge. Equalities (3) guarantee that each salesperson starts at the depot. Due to Equalities (4), each grid point is entered and left by the same salesperson. Inequalities (5) ensure that all salespersons visit a similar number of grid points. Inequalities (6) are the polynomial many subtour elimination constraints by Miller, Tucker, and Zemlin, where the variables u i, i [ln], represent a numbering of the grid points visited. MILP II: The second MILP minimizes the maximal tour length of a single salesperson. Therefore we replace Constraint (5) by: t ij x ijk c, k [m]. (9) i,j [ln] Inequalities (9) ensure that the tour length of each salesperson is smaller than a continuous variable c R +. Finally replacing Objective (I) by Objective (II) ensures that the maximal tour length of a salesperson is minimized: min c (II) s. t. (1) (4), (6) (9), c R +. MILP II minimizes the maximal tour length, however, it puts no restrictions on the length of the other m 1 tours. Hence, these tours are often too long for a satisfactory overall solution. To reduce the sum of the tour lengths of all salespersons we combine Objectives (I) and (II). To be more precise, we add Objective (I) and Objective (II) multiplied by the factor 5m, such that we still focus on minimizing the maximal tour length: min k [m], i,j [ln] t ij x ijk + 5mc (III) 3
3 Computational Results All experiments were conducted on a Linux 64-bit machine equipped with Intel(R) Xeon(R) CPU e5-2630 v3@2.40ghz and 128 GB RAM. We use Gurobi 8.0.1 in single thread mode as ILP-solver. We run our MILP on mtsp instances with m = 2 and the depot located in the upper left corner, i.e., in (1, 1). In Tables 1-3 we state the mtsp solutions of our MILP using Objectives (I)-(III). The MILP was able to return optimal solutions for all considered objectives and up to 25 grid points within the given time limit of six hours. Using Objective (III) instead of Objective (II) for the instance with 12 grid points results in an improved total tour length while the maximal tour length of a single salesperson stays almost the same. Illustrations of the tours for the considered instances can be downloaded from https://tinyurl.com/mtspsolutions. # grid points l n total tour length max tour length min tour length gap runtime 12 3 4 15.24 9.24 6.00 0.00 00:00:00 20 5 4 23.24 13.24 10.00 0.00 00:00:53 25 5 5 28.06 15.24 12.83 0.00 00:02:07 36 9 4 39.24 21.24 18.00 2.53 06:00:00 Table 1: Results for the mtsp with m = 2 and depot at (1, 1) obtained by our MILP using Objective (I) and Gurobi in single thread mode. The optimality gap is given in percent and the solving time [hh:mm:ss] is limited to six hours. # grid points l n total tour length max tour length min tour length gap runtime 12 3 4 16.61 8.61 8.00 0.00 00:00:02 20 5 4 24.30 12.24 12.06 0.00 00:07:29 25 5 5 28.89 14.65 14.24 0.00 00:39:08 36 9 4 40.30 20.24 20.06 4.86 06:00:00 Table 2: Results for the mtsp with m = 2 and depot at (1, 1) obtained by our MILP using Objective (II) and Gurobi in single thread mode. The optimality gap is given in percent and the solving time [hh:mm:ss] is limited to six hours. # grid points l n total tour length max tour length min tour length gap runtime 12 3 4 15.48 8.65 6.83 0.00 00:00:02 20 5 4 24.30 12.24 12.06 0.00 00:03:49 25 5 5 28.89 14.65 14.24 0.00 00:37:15 36 9 4 40.30 20.24 20.06 6.77 06:00:00 Table 3: Results for the mtsp with m = 2 and depot at (1, 1) obtained by our MILP using Objective (III) and Gurobi in single thread mode. The optimality gap is given in percent and the solving time [hh:mm:ss] is limited to six hours. 4
4 Optimal Solutions for Specific Grid Sizes In this section we provide optimal tour lengths for specific grid sizes, depot positions, two salespersons and Objective (I). Theorem 1. Let the depot for all salespersons be located in one of the four corners of the l n grid. Then the value of the optimal mtsp solution considering Objective (I) with m = 2 and n = 4 is (ln 1) + 2 + 5. Proof. We use the grid numbering depicted in Figure 1 a) and consider the regular grid as a chessboard with black and white squares. Due to Constraint (5), the maximal number of squares a salesperson is allowed to visit is 4l 2 = 2l. W.l.o.g. we assume that the depot is located at (1, 1) and colored black. There are only two possibilities for a move of length 1 out of the depot. Hence, there must be two moves of length > 1, such that both salespersons are able to leave and return to the depot. The shortest such moves have length 2 and 2 and thus a trivial lower bound for the value of a mtsp solution on a l 4 grid considering Objective (I) is (4l 1) + 2 + 2. The steps of length 1 and 2 are unique, i.e., they end at (1, 2), (2, 1), and (2, 2) respectively. There are two possibilities for the step of length 2, i.e., (1, 3) or (3, 1). No matter which step of length 2 is chosen, either (1, 2) or (2, 1) can not be connected to any square by a step of length 1. Hence, we derive the improved lower bound of (4l 1) + 2 + 5. For obtaining this lower bound the steps to (1, 2), (2, 1), and (2, 2) are unique and we have two possibilities to choose the step of length 5, i.e., (3, 2) or (2, 3), we start with (3, 2). If we do not consider the four steps connecting the two tours to the depot, it remains to draw two paths that are connected to (1, 2), (2, 1), (2, 2), and (3, 2). These paths consist only of steps of length 1 and have length 2l 1 and 2l 2 respectively due to Constraint (5). One path has to start and end at (2, 1) and (3, 2), and the other one at (1, 2) and (2, 2), otherwise the paths would intersect, i.e., one path would contain at least one additional step of length 2, which is a contradiction. If the paths start and end at their respective squares, it is possible to produce paths of lengths 2l 4 and 2l + 1 or 2l and 2l 3, but in each case Constraint (5) is contradicted. If (2, 3) is chosen instead, the argument is analog, and we derive paths of lengths 2l + 4 and 2l 7 or 2l and 2l 3, which again contradicts Inequalities (5). Now let us consider further possible lower bounds, where similar arguments as above can be applied (see Figure 2 for illustrations): (4l 1) + 2 2: The steps of length 1 and 2 are unique, they end at (1, 2), (1, 3), (2, 1), and (3, 1). Either from (1, 2) or (1, 3) a step of length > 1 is necessary and this is a contradiction. (4l 2) + 2 2 + 2: Steps to (1, 2), (2, 2), and (2, 1) are unique. For the step of length 2 we have two possibilities: First, let us choose (3, 1). Then a step from (2, 1) to (3, 2) is implied. As above it is not possible to find two paths that fulfill Constraint (5). Second, let us choose (1, 3). Then a step from (1, 2) to (2, 3) is implied. Again there exist no paths that fulfill Inequalities (5). (4l 2) + 2 2 + 5: Steps to (1, 2), (2, 2), and (2, 1) are unique. We have two possibilities for the step of length 5, either to (3, 2) or (2, 3). In both cases one path uses a step of length 2, Constraint (5) is fulfilled, but we additionally need one step of length 2 on the other path. 5
We finally derive a lower bound of (4l 1) + 2 + 5 and the construction scheme in Figure 1 b) shows that a mtsp solution with this value always exists. Q.E.D. a) Numbering b) Construction Scheme (1, 1) (1, 2) (1, 3) (1, 4) (2, 1) (2, 2) (2, 3) (2, 4) (l 1, 1) (l 1, 2) (l 1, 3) (l 1, 4) (l, 1) (l, 2) (l, 3) (l, 4) Figure 1: The picture on the left shows the grid numbering, and the drawing on the right depicts the construction scheme of an optimal mtsp solution with m = 2 and the depot located at (1, 1) on the l 4 grid with value (4l 1) + 2 + 5. (4l 1) + 2 + 2 (4l 1) + 2 + 5 (4l 2) + 2 2 + 2 (4l 1) + 2 2 + 5 Figure 2: The four drawings indicate why mtsp solutions with m = 2 and the depot located at (1, 1) of the given values cannot exist on the l 4 grid. Theorem 2. Let the depot for all salespersons be located in one of the four corners of the l n grid. Then the value of the optimal mtsp solution considering Objective (I) with m = 2 and l, n 5 odd is (ln 2) + 2 2 + 5. The proof of the above theorem is omitted due to space restrictions and will be stated in a forthcoming paper. For future work it remains to investigate lower bounds, explicit construction schemes and hence optimal mtsp solutions for further grid sizes, different locations of the depot and more than two salespersons. Additionally, it would be interesting to determine corresponding results for Objectives (II) and (III). 6
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