Omid Ansary Professor of Electrical Engineering Director, School of Science, Engineering, and Technology

Transcription

1 We approve the thesis of Mufit Colpan. Date of Signature Thang N. Bui Associate Professor of Computer Science Chair, Mathematics and Computer Science Programs Thesis Advisor Sukmoon Chang Assistant Professor of Computer Science Qin Ding Assistant Professor of Computer Science Pavel Naumov Assistant Professor of Computer Science Linda M. Null Assistant Professor of Computer Science Graduate Coordinator Omid Ansary Professor of Electrical Engineering Director, School of Science, Engineering, and Technology

2 I grant The Pennsylvania State University the non-exclusive right to use this work for the University s own purposes and to make single copies of the work available to the public on a not-for-profit basis if copies are not otherwise available. Mufit Colpan

3 The Pennsylvania State University The Graduate School SOLVING GEOMETRIC TSP WITH ANTS A Thesis in Computer Science by Mufit Colpan c 2005 Mufit Colpan Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2005

4 The thesis of Mufit Colpan was reviewed and approved by the following: Thang N. Bui Associate Professor of Computer Science Chair, Mathematics and Computer Science Programs Thesis Advisor Sukmoon Chang Assistant Professor of Computer Science Qin Ding Assistant Professor of Computer Science Pavel Naumov Assistant Professor of Computer Science Linda M. Null Assistant Professor of Computer Science Graduate Coordinator Omid Ansary Professor of Electrical Engineering Director, School of Science, Engineering, and Technology Signatures are on file in the Graduate School.

5 Abstract This thesis presents an ant-based approach for solving the Traveling Salesman Problem (TSP). Novel concepts of this algorithm that distinguish it from the other heuristics are the inclusion of a preprocessing stage and the use of a modified version of an ant-based approach with local optimization in multi stages. Experimental results show that this algorithm outperforms ACS [16] and is comparable to MMAS [30] for Euclidean TSP instances. Of the 40 instances of Euclidean TSP from TSPLIB [32] that were tested, this algorithm found the optimal solution for 37 instances. For the remaining instances, this algorithm returned solutions that were within 0.3% of the optimum. iii

6 Table of Contents List of Figures List of Tables Acknowledgements vi vii viii Chapter 1 Introduction 1 Chapter 2 The TSP Problem Definition Variations and Applications Existing Algorithms Chapter 3 Ant Colony Optimization Problems Solved with ACO Using ACO To Solve TSP Variations of ACO Ant System (AS) Ant Colony System (ACS) MAX-MIN Ant System (MMAS) iv

7 Chapter 4 The ACS-TSP Algorithm The Compaction Algorithm Ant Colony Optimization Initial Tour Construction Local Optimization Chapter 5 Experimental Results 21 Chapter 6 Conclusion 26 References 27 v

8 List of Figures 4.1 The ACS-TSP Algorithm The Compaction Algorithm The macs Algorithm The Tour Conversion Algorithm lin105 TSP Benchmark Instance vi

9 List of Tables 5.1 ACS-TSP solution quality on symmetric Euclidean TSP instances ACS-TSP running time quality on symmetric Euclidean TSP instances vii

10 Acknowledgements I would like to thank Dr. Thang N. Bui for his valuable advice and help throughout this thesis. This thesis could not have been completed without his supervision and his contributive ideas. Additionally, I am thankful to the thesis committee members: Dr. Sukmoon Chang, Dr. Qin Ding, Dr. Pavel Naumov, and Dr. Linda Null for their reviews of this thesis. viii

11 Chapter 1 Introduction A well-known NP-hard optimization problem is the Traveling Salesman Problem (TSP) [11, 23, 24]. In this problem, there are n cities, and between any two cities is a direct road with a known traveling cost. A traveling salesman moving from one city to the other has to pay the traveling cost of the road between those cities. The Traveling Salesman problem is the problem of finding the tour with the cheapest total traveling cost such that the traveling salesman visits each city exactly once and returns to the starting city. Due to the complexity of the problem, many heuristic algorithms [1, 15, 16, 18, 19, 21, 22, 24, 27, 30, 31, 33] have been developed to solve it with varying degrees of efficiency. TSP arises in a variety of applications [24], such as finding the minimal route of the machine for drilling holes on a printed circuit board, minimizing the amount of time taken by a graphics plotter to draw a given figure, or minimizing the total positioning time of a diffractometer in analyzing the structures of crystals in x-ray crystallography. It is also used for placing the vanes in the best possible position to overhaul gas turbine engines in aircraft, minimizing the order-picking problems in warehouses, and designing the computer wiring. Clustering of data array, seriation in archeology, vehicle routing, scheduling, and control of robots are also common examples of this problem in practical usage. Since TSP is well-known to be NP-hard [11, 24, 23], we do not expect to find a polynomial time algorithm to solve it. Therefore, approximation algorithms

12 1 INTRODUCTION 2 and heuristics are needed to find an optimal or at least close to optimal solution. Heuristics are algorithms that do not guarantee optimal or feasible solutions but they usually produce optimal or close to optimal solutions. In this thesis, a version of TSP called geometric TSP, in which vertices of the graph are points in the Euclidean plane and the weight of an edge is just the Euclidean distance between the two end points of the edge, is considered. An algorithm that is a combination of both Tour Construction and Tour Improvement heuristics is presented. Additionally, a preprocessing stage that allows us to deal with larger input instances efficiently was added. The results of the algorithm on a set of standard benchmarks for the geometric TSP showed that the algorithm is quite competitive against existing heuristics. In fact, it found 37 optimal solutions out of the 40 instances that were tested. The rest of the thesis is organized as follows. Chapter 2 includes a definition of the problem, applications of the problem, and previous approaches that were used to solve the problem. Chapter 3 introduces the ACO technique, problems to which it has been applied, how the ACO technique can be applied to the Traveling Salesman problem, and the variations of the ACO technique. Chapter 4 contains the description of the algorithm. In Chapter 5, the approach is examined empirically, and then compared against the benchmark results and against the results of other known algorithms. The conclusion is given in Chapter 6.

13 Chapter 2 The TSP Problem 2.1 Definition The TSP [11, 23, 24] problem is the prototypical optimization problem that is difficult to overcome but is both short and easy to state: given n cities such that there is a direct edge between any two of them, and each edge is assigned a traveling cost, the task is to find the cheapest way of visiting each city exactly once while returning to the starting city. The set of edges used in traveling the cities as stated is referred to as a tour and the total of the traveling costs of the edges in the tour is called the tour cost. As it can easily be observed, there are (n 1)!/2 possible tours if redundant tours are removed. The TSP problem asks for the smallest cost Hamiltonian tour [24] in a complete undirected graph with positive costs on the edges. A Hamiltonian tour is a cycle that visits each vertex of the graph exactly once. More formally, the TSP is defined as follows: Input: A complete undirected graph G = (V,E) with vertex set V, edge set E, and a weight function w : E R +. Output: A minimum weight Hamiltonian tour of G.

14 2 THE TSP PROBLEM Variations and Applications There are many different variations of the traveling salesman problem [24]. The Symmetric Traveling Salesman problem is to find the shortest Hamiltonian tour of a complete edge-weighted undirected graph. The Asymmetric Traveling Salesman problem is to find the shortest Hamiltonian tour of a complete edge-weighted directed graph. The Chinese postman problem is to find the shortest tour of an edge-weighted graph that visits each edge of the graph at least once. The Hamiltonian cycle problem, also called the Hamiltonian circuit problem or the Hamiltonian tour problem, is to decide if a given graph has a tour that visits each vertex of the graph exactly once. The Eulerian tour problem, also called the Eulerian circuit problem, is to decide if a given graph has a tour that uses each edge of the graph exactly once. The Traveling Salesman problem has many real world applications [24]. In the drilling problem of printed circuit boards, holes of different diameters for the pins of the integrated circuits and the electronic components need to be drilled through the board. Finding the shortest route for the machine to drill the holes of different diameters on a printed circuit board is an application of the Traveling Salesman problem. The vehicle routing problem, which asks for an optimal route of one or more vehicles through a number of pick-up points, is another application of the Traveling Salesman problem. The scheduling problem deals with the efficient allocation of tasks over resources is another application of the Traveling Salesman problem. 2.3 Existing Algorithms TSP [11, 23, 24] is a well-known problem that is extensively studied in the computer and mathematical sciences. Various papers [1, 15, 16, 18, 19, 21, 22, 24, 27, 30, 31, 33] with a variety of solutions of varying complexity and efficiency have been presented. The simplest solution, the brute force approach, generates all possible tours and takes the shortest one. This becomes impractical as the number of cities n increases, since the number of possible tours to generate is (n 1)!/2, the running time complexity is O(n!p(n)), where p(n) is a polynomial to compute the cost of

15 2 THE TSP PROBLEM 5 a tour. Algorithmic ideas created for solving the Traveling Salesman problem can be categorized as Tour Construction, heuristic algorithms that attempt to construct feasible solutions, and Tour Improvement, heuristic algorithms that iteratively modify and try to improve a given starting solution. Examples of Tour Construction heuristics include Nearest Neighbor [25, 24], Greedy [25], Clarke-Wright [24], Christofides [24], Insertion Heuristics [24], Genetic Algorithms [29], and Ant Colony Optimization [19, 15, 16, 30]. Examples of Tour Improvement heuristics include 2-opt [25, 24], 3-opt [25, 24], Lin-Kernighan [25, 24, 27], Tabu-Search [25, 24], and Simulated Annealing [25, 24]. Varying degrees of success have been achieved by these heuristic techniques.

16 Chapter 3 Ant Colony Optimization The ants make use of pheromone, a chemical substance, to follow and mark paths as they forage much as scouts in the wilderness would mark trees to find their way back to their camp. Once the food source is located, ants begin to take bits of it back to their nest. As the shortest path to the food source is traveled over and over, the pheromone intensity on that path increases in strength and attracts other ants to take the same path. Therefore, the longer paths attract fewer ants, and the pheromone laid on the longer paths slowly fades away as the pheromone evaporates. The foraging behavior of real ants and their ability to find the shortest path between a food source and the nest have inspired Ant Colony Optimization heuristic. The Ant Colony Optimization heuristic is a population-based approach that utilizes artificial pheromones so that the artificial ants can use the same detection guides to determine the shortest path possible. The Ant Colony Optimization heuristic imitates the nature by mimicking the same patterns used by these social animals to indirectly communicate to each other so the colony as a whole has a better chance for solving the problem collectively [4, 15, 16, 18, 19, 21, 30, 31]. 3.1 Problems Solved with ACO Because ACO is a powerful heuristic, it has been applied to many combinatorial problems with great success. The Traveling salesman problem, quadratic assignment problem [17, 28], graph bisection [8], finding the maximum clique in a graph

17 3 ANT COLONY OPTIMIZATION 7 [6], graph coloring [5, 12, 10], k-cardinality tree problem [2, 7], scheduling problem [3], vehicle routing, pickup and delivery [9], the layout optimization of integrated electrical circuits [26], telecommunications network design [13], and robot controlling [14] are just a few of the problems solved with ACO. 3.2 Using ACO To Solve TSP ACO heuristics [15, 16, 18, 19, 21, 22, 30, 31, 33] initialize each edge of the problem instance with a small amount of pheromone. A number of ants are then randomly placed on different cities of the problem instance. Ants simultaneously construct tours through the problem instance by moving from one city to the other. Ants are not permitted to return to a city already visited and they randomly choose which city to move to, but this choice is biased towards the edges that are short and marked with more pheromone. Once all the ants have completed their tours by visiting all of the cities in the problem instance, the amount of pheromone assigned to the edges of the problem instance is reduced to represent pheromone evaporation and the pheromone on the edges used by each ant when constructing their tour is increased by an amount inversely proportional to the total length of those ants tours. 3.3 Variations of ACO Ant System (AS) In AS [19], each ant is placed on some randomly chosen city. An ant k currently at city r chooses to move to city s by applying the following probabilistic function: p k (r,s) = [τ (r,s) ] α [η (r,s) ] β if s J k (r) [τ (r,u) ] α [η (r,u) ] β u J k (r) 0 otherwise where J k (r) is the set of unvisited cities by ant k. (3.1)

18 3 ANT COLONY OPTIMIZATION 8 τ(r,u) is the pheromone intensity level on edge (r,u). η(r,u) is the visibility of u from r and is 1/d(r,u). d(r,u) is the distance between cities r and u. α determines the importance of pheromone intensity level, 0 α 1. β determines the importance of visibility, 0 β 1. The probabilistic function p k (r,s) is formulated to utilize the pheromone intensity and the visibility on an edge (r,s). α and β are used to tune the importance of the pheromone intensity level and the importance of visibility respectively, and they affect the behaviour of ants in choosing the next city to which ants move. Higher values of α favor choosing the edges in which there has been significant ant movements and therefore there is significant pheromone accumulation whereas higher values of β favor the shorter edges with higher probability. After all ants have completed their tours, all of the ants update the pheromone levels on the edges by applying the following function: where τ(r,s) (1 ρ) τ(r,s) + τ(r,s) (3.2) and m τ(r,s) = k τ(r,s), (3.3) k=1 Q k L τ(r,s) = k if the k th ant uses edge (r,s) (3.4) 0 otherwise and ρ is the pheromone intensity level decay parameter, 0 ρ <1, m is the number of ants, Q is a constant, and L k is the tour length of the k th ant Ant Colony System (ACS) ACS [16] differs from AS [19] in the way it chooses the next city to visit and in the way it updates pheromone levels on the edges. These changes increase the emphasis on exploitation by ensuring that most of the edges each ant follows occur arround the best known tour.

19 3 ANT COLONY OPTIMIZATION 9 Each ant is placed on some randomly chosen city as in AS. An ant k currently at city r chooses to move to city s specified by the following function: u such that [τ(r,u)][η(r,u)] β = max s = v J k (r) {[τ(r,v)][η(r,v)]β if q q 0 S otherwise where J k (r) is the set of cities that have not been visited by ant k. τ(r,u) is the pheromone intensity level on edge (r,u). η(r,u) is the visibility of u from r and is 1/d(r,u). d(r,u) is the distance between cities r and u. β determines the importance of visibility, 0 β 1. q is a random number, 0 q 1. (3.5) q 0 is a parameter that determines the importance of exploitation, 0 q 0 1. S is a random variable selected using the probability distribution given in the AS transition rule (Equation 3.1). The given transition function (Equation 3.5) is formulated to utilize the pheromone intensity, the visibility on an edge (r,u), and a threshold value q 0. A random number q is compared to q 0. If q is less then or equal to q 0, then the city with the highest attractivness is chosen. Otherwise, the city to move to is chosen by the probability distribution given in the AS transition rule (Equation 3.1). This change increases the rate of exploitation as compared to AS and causes many of the edges that ants follow to be the edges of the globally best tour found so far. It also directs the exploration to occur arround the globally best tour found so far. In ACS, only the edges of the globally best tour found from the beginning of the trial have pheromone deposited upon them. An edge (r,s) of the globally best tour is deposited pheromone as follows: τ(r,s) (1 α) τ(r,s) + α τ(r,s) (3.6) where α is the pheromone intensity level decay parameter 0< α <1,

20 3 ANT COLONY OPTIMIZATION 10 1 L τ(r,s) = gb if (r,s) global best tour 0 otherwise (3.7) and L gb is the length of the globally best tour found from the beginning of the trial. A local pheromone updating rule encourages exploration of unused edges and avoids a local optimum by evaporating pheromone from the edges of each ant s tour as follows. τ(r,s) (1 ρ) τ(r,s) + ρ τ(r,s), (3.8) where ρ is the pheromone intensity level decay parameter 0< ρ <1, τ(r,s) = τ 0, τ 0 is the initial pheromone intensity level on the edges and is found by the following formula: τ 0 (n L tour ) 1 (3.9) where n is the number of vertices in the given graph, and L tour is a tour length found by a nearest neighbor heuristic MAX-MIN Ant System (MMAS) MMAS [30] achieves an improved performance over AS [19] and ACS [16] by utilizing a strong exploitation of the search space while avoiding early search stagnation. Only the best ant is allowed to deposit pheromone on the edges of the tour that it has constructed. To avoid a premature convergence of the algorithm and reinforce exploration, maximum and minimum pheromone intensity levels on the edges are limited to τ max and τ min respectively. τ max and τ min are found in a problem dependent way and pheromone intensity levels on all the edges are initialized to the maximum pheromone intensity level τ max. Each ant is placed on some randomly chosen city as in AS. An ant k currently at city r chooses to move to city s by applying the same transition rule as in AS [19]. After all ants have completed their tours, only the edges of the globally best tour found from the beginning of the trial have pheromone deposited upon them and

21 3 ANT COLONY OPTIMIZATION 11 then pheromone from all the edges is evaporated. During the pheromone intensity level updates, any edge whose pheromone intensity level drops below τ min is set to τ min, and any edge whose pheromone intensity level exceeds τ max is set to τ max.

22 Chapter 4 The ACS-TSP Algorithm In this chapter we describe our ant-based algorithm for solving the geometric TSP. There are two main ideas in our algorithm: the inclusion of a preprocessing stage and the use of a modified ACS with local optimization in multi stages. The input graph is first partitioned into subgraphs of size 3, i.e. triangles. A new graph is created based on the centroids of the triangles. An ant system algorithm is used to create a tour of this graph. The tour is then converted into an initial tour for the original graph. A local optimization algorithm is then used to improve this original tour. Finally, we run an ant system algorithm on the original graph using the tour just constructed as a starting tour. The algorithm consists of two main phases. In the first phase a good starting tour is created. This tour is then used as the starting point for the second phase to create a better tour. The first phase consists of four stages. The first stage compacts the given input graph into a smaller graph. The objective is to reduce the problem size so that we can find an initial tour, quickly enabling us to deal with larger input graph. In the second stage, the compacted graph is passed into an ant colony system algorithm, called macs, to find a TSP tour. The tour found by macs is a TSP tour of the compacted graph. This tour is then converted into a tour for the original input graph in the third stage, and is locally optimized in the fourth stage. The resulting TSP tour is now a reasonably good tour of the input graph and is used as the starting point for phase two. In phase two, the macs algorithm is run using the original input graph and the TSP tour from phase one. The tour found by macs in this phase is returned as the solution to the input

23 4 THE ALGORITHM 13 instance. The main idea of using phase one is based on the expectation that a good starting tour for macs will allow macs to find a better solution and converge in much less time. The overall effect is that we can deal with larger input instances, producing good solutions in small amount of times. The ACS-TSP algorithm is given in Figure 4.1. ACS-TSP(G = (V,E,w)) Phase 1. 1 (G = (V,E,w )) Compact(G) 2 tour macs (G, ) 3 tour ConvertTour(G,G,tour) 4 tour 2-opt(tour) or 3-opt(tour) Phase 2. 5 bestt our macs (G, tour) 6 return besttour Figure 4.1. The ACS-TSP Algorithm In what follows, we describe in detail each of the four stages in the first phase. 4.1 The Compaction Algorithm In this stage we compact the given input graph into a smaller graph. Specifically, we create a new graph based on the original graph but this new graph is about three times smaller. This is accomplished by first partitioning the input graph into subgraphs of size 3, i.e., triangles, such that the vertices in each triangle are the closest to each other. Because the input graph is an instance of the geometric TSP, each vertex has coordinates in the plane. We scan the vertices based on their coordinates from each of the four directions: left, right, top and bottom. In each scan, we construct a partition of the graph into triangles as follows. For each vertex that has not been visited, we find the two nearest unvisited vertices to form a triangle. We mark all three vertices as visited and continue until all vertices have been visited. The cost of a partition is the sum of the length of the edges in all the triangles of the partition. We then mark all the vertices as unvisited and perform another scan from a different direction until we have scanned in all four directions. The resulting partition is chosen to be the one that has the smallest cost. The centroid of each triangle is computed. These centroids are then used as vertices of

24 4 THE ALGORITHM 14 the new graph. As this compacted graph is treated as an instance of TSP, we make the graph complete, and the cost of each edge between any two vertices is simply the Euclidean distance between the two vertices. The full compaction algorithm is given in Figure 4.2. Compact(G = (V,E,w)) 1 scandirection {lef t, right, top, bottom} 2 for i = 0 to length(scandirection) 3 Sort V into nonincreasing order by scandirection[i] 4 partition[i] 5 partitiont ourlength[i] 0 6 Mark all vertices of V as unvisited 7 for each v V 8 if v is unvisited then 9 Find the nearest pair of unvisited vertices (v 1,v 2 ) to v, if they exist 10 Create a new triangle p by using v,v 1, and v 2 11 Add triangle p to partition[i] 12 Find the tour length of p 13 Add the tour length of p to partitiontourlength[i] 14 Mark v,v 1,v 2 as visited 15 m min{partitiont ourlength[i] i = 1...length(scanDirection)} 16 V 17 E 18 for each triangle p in partition[m] 19 v Centroid of p 20 Add v to V 21 for every pair of vertices (u,v) V 22 Add an edge (u,v) to E 23 Assign a cost w to (u,v) 24 return (G = (V,E,w )) Figure 4.2. The Compaction Algorithm 4.2 Ant Colony Optimization In this stage we use a modified version of the ACS, as given in [16], to obtain a tour of the compacted graph obtained in Stage 1. Our modifications include flexible state transition and pheromone updating rules. We also introduce explicit mechanisms for escaping from local optima as well as increasing time efficiency. The complete algorithm, called macs, is given in Figure 4.3 and is described in detail below.

25 4 THE ALGORITHM 15 We first describe the state transition, global updating, and local updating rules as described in [16], which the reader is referred to for more details. We then describe our modifications. 1. State transition rule An ant on vertex r selects a vertex s determined by u such that [τ(r,u)][η(r,u)] β = max s = v J k (r) {[τ(r,v)][η(r,v)]β if q q 0 S otherwise (4.1) where q is a random number 0 q 1, q 0 is a parameter 0 q 0 1, and S is a random variable selected using the probability distribution given in the AS [19] transition rule (Equation 3.1). 2. Global updating rule Pheromone is deposited only on the edges of the globally best tour found from the beginning of the trial. If (r,s) is an edge in the globally best tour, and if τ(r,s) denotes the amount of pheromone on the edge (r,s) then τ(r,s) (1 α) τ(r,s) + α τ(r,s) (4.2) where 1 L τ(r,s) = gb if (r,s) global best tour (4.3) 0 otherwise where α is the pheromone decay parameter, and L gb is the length of the globally best tour found from the beginning of the trial. 3. Local updating rule Whenever an ant uses an edge (r,s), it changes the pheromone level of the edge as follows: τ(r,s) (1 ρ) τ(r,s) + ρ τ(r,s) (4.4) where ρ is a parameter, 0< ρ <1 and τ(r,s) = τ 0

26 4 THE ALGORITHM 16 In the ACS of [16], the parameters q 0, α, and ρ are constants that do not change during the execution of the ACS algorithm. Intuitively, q 0 determines the relative importance of exploitation versus biased exploration, whereas α and ρ determine the desirability of the edges. In our algorithm, ants explore more at the beginning, and toward the end of the algorithm they tend to exploit more, utilizing the information that has been accumulated. We accomplish this by allowing q 0,α, and ρ to change in each cycle. Specifically, in each cycle we calculate a new value for q 0 and set the values of α and ρ as follows: q 0 = cycle ncmax 0.1 if q 0 < 0.1 q 0 = 0.9 if q 0 > 0.9 otherwise q 0 (4.5) (4.6) α ρ (1 q 0 ) (4.7) where cycle is the current cycle number and ncmax is the maximum number of cycles. As q 0 gets larger, exploitation occurs more frequently. More exploitation may direct ants to use the edges of the global best tour more frequently. To avoid a local optimum when ants find the same tour over and over again or do not improve the global best tour after a certain number of iterations, we encourage the exploration of edges not used frequently by perturbing the global best tour erasing memory from some percentage of randomly chosen edges of the global best tour. Another important parameter in ACS is τ 0, which determines the initial pheromone intensity level on the edges. If a tour is given, macs uses that tour; otherwise it finds a tour by a nearest neighbor heuristic to find τ 0 as described in [16]. Let L tour be the tour length of the given tour or the tour found by the nearest neighbor heuristic, then the value for τ 0 is defined as follows: where n is the number of vertices in the given graph. τ 0 (n L tour ) 1 (4.8)

27 4 THE ALGORITHM 17 Pheromone on the edges of the graph is initialized with τ 0. If a tour is given, an extra pheromone amount, equal to (n τ 0 ), is placed on the edges of that tour, suggesting that the ants look for better tours arround the given tour. In the original ACS, when ants choose the next vertex to move to, they consider the entire set of vertices that have not been visited yet. This can be very time consuming. To improve the efficiency of macs, we restrict the set of vertices that the ants consider to the k nearest neighbors. 4.3 Initial Tour Construction To construct an initial tour for the next phase, we convert the tour produced for the compacted graph in the previous stage into a tour for the original input graph. This is done using a greedy approach. The idea is to replace each vertex in the compacted graph, i.e., a centroid of a triangle in the original graph, by the corresponding triangle. An edge between two centroids in the compacted graph is replaced by the edge that is the shortest edge connecting two vertices one from each of the two triangles corresponding to the centroids. The remaining edges of each triangle are then selected to create a continuous tour. The complete algorithm is given in Figure Local Optimization We locally optimize the tour found in the previous stage before using it as an initial tour in the second phase. For this purpose we use k-opt-type heuristics. Among the most used and well-known tour improvement heuristics are the 2-opt [24, 25], 3- opt [24, 25], and Lin-Kernighan [24, 25, 27] heuristics. The 2-opt heuristic removes two edges from the tour and reconnects the two paths created by the edge removal step by using two new edges. If the resulting tour is better, this change is kept, otherwise the tour reverts to its original configuration. The process is repeated on another set of two edges until no further improvement can be made. The 3- opt [24, 25] heuristic works the same way by removing three edges and adding three new edges instead. In practice, 4-opt heuristic and heuristics considering a fixed number of edges are not used because their running times are larger and

28 4 THE ALGORITHM 18 macs(g = (V,E,w),tour) 1 Set macs parameters 2 if tour = then 3 Find τ 0 using a nearest neighbor heuristic 4 else 5 Find τ 0 using the given tour 6 Initialize pheromone trails 7 if tour then 8 Lay extra pheromone to the edges of tour 9 globalbestt our 10 globalbestt ourlength 11 perturbationcounter 0 12 repeat 13 Place the ants randomly on the vertices of G 14 Find new values for q 0, α, and ρ 15 for i=1 to n //n is the number of vertices 16 for j=1 to m //m is the number of ants 17 Find the next unvisited vertex v for j th ant 18 from j th ant s current vertex 19 by using state transition rule and 20 candidate lists 21 Append v into j th ant s tabu list 22 if currentcycle 1/3 rd of the total Cycles then 23 Locally optimize the tour of each ant 24 Locally update the edges used by the ants 25 tourlength length of the best tour found in this Cycle 26 if tourlength < globalbestt ourlength then 27 globalbestt ourlength tourlength 28 globalbesttour best tour found in this Cycle 29 perturbationcounter 0 30 if perturbationcounter = 100 then 31 Randomly select 1/3 rd of the edges from globalbesttour 32 Evaporate (1 q 0 ) of the pheromone 33 from the selected edges 34 perturbationcounter 0 35 else 36 Perform global pheromone update 37 perturbationcounter perturbationcounter until(terminationcondition) 39 return globalbestt our Figure 4.3. The macs Algorithm they do not yield significant improvements over 2-opt or 3-opt. The Lin-Kernighan algorithm is a variable k-opt algorithm. Usually the quality of the tour improved by the given heuristics is determined by the Held-Karp lower bound [25]. 2-opt, 3-opt, and Lin-Kernighan heuristics

29 4 THE ALGORITHM 19 ConvertTour(G = (V,E,w),G = (V,E,w ),tour) 1 //We consider tour as an array of vertices, i.e., centroids 2 convertedt our 3 previoust riangle null 4 for i = 0 to length(tour) 1 5 Let p 1 be the triangle whose centroid is tour[i] 6 Let p 2 be the triangle whose centroid is tour[i + 1] 7 Find the shortest edge (u,v) between p 1 and p 2 8 where u p 1 and v p 2 9 if previoust riangle null then 10 if u convertedtour then 11 remove u from convertedtour 12 u 2 last vertex convertedtour 13 if w(u 2,u) w(u,v) then 14 Append u to convertedtour 15 u 2 u 16 Find the shortest edge (u,v) between p 1 and p 2 17 such that u u 2 18 else 19 Find the vertex v 2 p 1 incident to 20 the shortest edge from u 2 such that v 2 u 21 Append v 2 to convertedtour 22 Append v 2 to convertedtour such that 23 v 2 p 1 and 24 v 2 convertedtour and 25 v 2 u 26 Append u to convertedtour 27 Append v to convertedtour 28 previoustriangle p 2 29 return convertedt our Figure 4.4. The Tour Conversion Algorithm yield tours that are within 5%, 3%, and 2% respectively of the Held-Karp lower bound. In our implementation, we use 2-opt and 3-opt tour improvement heuristics. A naive implementation of 2-opt and 3-opt would take O(n 2 ) and O(n 3 ) time, respectively. This is impractical when the problem size is large. To speed up 2- opt or 3-opt, we use candidate lists [1, 25, 31] when considering which edges to be added. This significantly reduces the running time. Our implementation uses candidate lists of size k = 20. In fact, the running times of our implementation of 2-opt and 3-opt drop to O(kn) and O(k 2 n), respectively. For most instances that we tested, 2-opt was sufficient. For some instances, however, 3-opt was needed in

30 4 THE ALGORITHM 20 the algorithm to obtain competitive results.

31 Chapter 5 Experimental Results In this chapter, we report the results obtained by running ACS-TSP on a collection of benchmark problems from TSPLIB [32], a database of benchmark instances for the TSP problems created and maintained at TSPLIB, and compare them against the known optimal solutions and best known results from two other algorithms: ACS [16] and MMAS [30]. Because the best solutions for some algorithms were not available and no running times for any of the problem instances were provided, we could not compare the best solutions and the running times for those problem instances obtained by our algorithm against the others. Our algorithm was implemented in C++ and run on a Pentium IV 3.2GHz processor PC with 1GB of RAM. We tested our algorithm on symmetric Euclidean TSP instances. All tests have been carried out for 2500 cycles with 20 trials per instance. The value of the parameter β was 2 and the number of ants was 10. We have purposely chosen the values for β and the number of ants as specified in order to obtain an equal basis for comparison with ACS. The results shown in Table 5.1 list the best tour length(best), the optimality ratio(ratio), the average tour length of our algorithm, and the standard deviation(σ) of the tour lengths found for 20 trials. The values shown in the RATIO column are calculated using the following formula: BEST - OPTIMUM RATIO = BEST (5.1) Table 5.1 also includes optimal solution(optimum) and the best tour lengths found by ACS and MMAS. As can be seen from Table 5.1, the optimality ratios are zero for most of the

32 5 EXPERIMENTAL RESULTS 22 instances with the exception of the fl1577 and u1060 instances, and the standard deviations are very small, except for instance u1060. Figure 5.1 shows the commonly used TSP benchmark instance lin105 from TSPLIB [32] and the stages of the algorithm in finding the optimal tour. To make the diagram more legible, only edges in the tour or in the compaction triangles have been drawn. (One must assume there is an edge between any two vertices even if it is not drawn.) Figure 5.1(a) shows the problem instance with its vertices. Figure 5.1(b) shows the graph after the Compaction stage, depicted in Figure 4.2. Points in the partitioned graphs denote the centroids of the partitions and the set of these points make up the compacted graph. Figure 5.1(c) shows a tour of the compacted graph after macs stage, depicted in Figure 4.3. Figure 5.1(d) shows the initial tour for the original graph after ConvertTour stage, depicted in Figure 4.4. Figure 5.1(e) shows the optimized initial tour. Finally, Figure 5.1(f) shows the tour found after macs stage, depicted in Figure 4.3, by using the optimized tour as a starting tour. In fact, this final tour is the optimal tour for the lin105 TSP benchmark instance.

33 5 EXPERIMENTAL RESULTS 23 Table 5.1. ACS-TSP solution quality on symmetric Euclidean TSP instances ACS-TSP INSTANCE OPTIMUM BEST RATIO AVERAGE σ ACS MMAS att48 10, , , att532 27, , , , , a280 2, , , berlin52 7, , , bier , , , ch130 6, , , ch150 6, , , d198 15, , , , , d , , , , eil eil eil fl , , , fl , , , , , kroa100 21, , , , , kroa200 29, , , krob150 26, , , krob200 29, , , kroc100 20, , , krod100 21, , , kroe100 22, , , lin105 14, , , lin318 42, , , , oliver pcb442 50, , , , pcb , , , , pr76 108, , , pr107 44, , , pr124 59, , , pr136 96, , , pr144 58, , , pr152 73, , , pr226 80, , , pr , , , rat575 6, , , rat783 8, , , , tsp225 3, , , u159 42, , , u , , , , , vm , , ,

34 5 EXPERIMENTAL RESULTS 24 Table 5.2. ACS-TSP running time quality on symmetric Euclidean TSP instances Running Time in seconds INSTANCE OPTIMUM BEST Average σ att48 10, , att532 27, , , a280 2, , berlin52 7, , bier , , ch130 6, , ch150 6, , d198 15, , , d , , , eil eil eil fl , , , fl , , , kroa100 21, , kroa200 29, , krob150 26, , krob200 29, , kroc100 20, , krod100 21, , kroe100 22, , lin105 14, , lin318 42, , oliver pcb442 50, , , pcb , , , pr76 108, , pr107 44, , pr124 59, , pr136 96, , pr144 58, , pr152 73, , pr226 80, , pr , , rat575 6, , , , rat783 8, , , , tsp225 3, , u159 42, , u , , , , vm , , , ,148.20

35 5 EXPERIMENTAL RESULTS 25 (a) Original Graph (b) Partitioned Graph after Compact(G) Algorithm (c) Tour of Compacted Graph after macs(g, ) Algorithm (d) Initial Tour after ConvertT our(g, G, tour) (e) Optimized Initial Tour after Local Optimization Figure 5.1. lin105 TSP Benchmark Instance (f) Final Tour after macs(g,tour) Algorithm

36 Chapter 6 Conclusion In this thesis, we have presented a hybrid ant system algorithm called ACS-TSP. The given algorithm includes compacting the graph, running an ant system to find a tour of the compacted graph, finding an initial tour for the original graph, optimizing the initial tour, and finally running an ant system by using the optimized tour. We have shown that ACS-TSP is very good in finding close to optimal, mostly optimal solutions. Experimental results show that our algorithm outperforms ACS [16] in solution quality and is comparable to MMAS [30] for symmetric Euclidean TSP instances. The running time of our algorithm can be improved further by using more sophisticated data structure such as those suggested by [1, 20].

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