Tours in solution attractor for 1000-node and 10,000-node TSP instances.
\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
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Y. Tan and Jeffrey S. Amthor",authors:[{id:"160704",title:"Dr.",name:"Daniel",middleName:"K. Y.",surname:"Tan",fullName:"Daniel Tan",slug:"daniel-tan"},{id:"166320",title:"Prof.",name:"Jeffrey",middleName:null,surname:"Amthor",fullName:"Jeffrey Amthor",slug:"jeffrey-amthor"}]},{id:"45233",title:"Biosynthesis of Lipids and Hydrocarbons in Algae",slug:"biosynthesis-of-lipids-and-hydrocarbons-in-algae",signatures:"Masato Baba and Yoshihiro Shiraiwa",authors:[{id:"71335",title:"Dr.",name:"Yoshihiro",middleName:null,surname:"Shiraiwa",fullName:"Yoshihiro Shiraiwa",slug:"yoshihiro-shiraiwa"},{id:"158523",title:"Dr.",name:"Masato",middleName:null,surname:"Baba",fullName:"Masato Baba",slug:"masato-baba"}]},{id:"45158",title:"Mass Production of Microalgae at Optimal Photosynthetic Rates",slug:"mass-production-of-microalgae-at-optimal-photosynthetic-rates",signatures:"Johan U. 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What it is that makes the TSP difficulty? The difficulty of the TSP is associated with the combinatorial explosion of potential solutions in the solution space. When a TSP instance is large, the number of possible tours in the solution space is so large as to forbid an exhausted search for the optimal tour. Numerous approaches to solving the TSP have been published. Some algorithms such as enumerative search, branch-and-bound search, and linear programming are exact approaches but lack efficiency. Other approximate algorithms, based on heuristics, are quick to find a good tour but lack effectiveness and robustness. Modern approximate algorithms, with today’s fast computers, can find good solutions for extremely large TSP instances within a reasonable time, which are with a high probability just 2–3% away from the optimal tour [1, 2, 3].
\nMost approximate algorithms have been based on or derived from a general search technique known as local search. Local search algorithms iteratively explore the neighborhoods of solutions trying to improve the current solution by local changes. However, the scope of a single search trajectory is limited by the neighborhood definition. Both the TSP and local search have been hot research topics for decades, and many aspects of them have been studied. However, there is still a variety of open questions. The study of local search for the TSP continues to be a vibrant, exciting, and fruitful endeavor in combinatorial optimization, computational mathematics, and computer science.
\nA local search algorithm is essentially in the domain of dynamical systems. The goal of a dynamical system analysis is to capture the distinctive properties of certain points in the state space for a given dynamical system. The attractor theory of dynamical systems is a natural paradigm that provides the necessary and sufficient theoretical foundation to study the convergent behavior of a local search system. The TSP is believed to be NP-hard because we do not have an efficient enumerative search system for the problem. Do we need to examine all possibilities in order to solve the problem? Can we quickly narrow down the search space to a small region in which the optimal solution is located and then search that small region completely to find the optimal solution? This chapter attempts to use the solution attractor concept to answer these questions. If we can quickly identify that small region, the solution attractor, and then search that region thoroughly in reasonable time, the computational complexity of the problem can be dramatically reduced or may not exist. This chapter introduces the solution attractor concept, which not only helps us understand the behavior of a local search system for the TSP but also offers an important method to solve the problem efficiently with global optimality guarantee. This chapter presents a novel search algorithm—the attractor-based search system (ABSS)—that is a simple and quick global search system for the TSP.
\nA problem is the frame into which the solutions fall. By changing the frame, we can change the range of possible solutions and scope of the optimal solutions. The classic TSP is defined as a complete graph \n
Obviously, this definition requires a search algorithm to find any single optimal tour in the solution space for a given instance. However, many real-world optimization problems are inherently multimodal. They may contain multiple optimal solutions in their solution spaces. Finding all optimal solutions is the essential requirement for global optimization. In practice, knowledge of multiple optimal solutions is essentially helpful, providing the decision-maker with multiple best options. We assume that a TSP instance contains \n
For a given TSP instance, we do not know the number of optimal tours in the solution space until we find all of them. Obviously, this reframed TSP definition becomes even more difficult to solve. To solve this reframed TSP, we need a search algorithm that converges not just in value but also in solution. Convergence in value means that a search system can find any one of the optimal solutions in the solution space eventually. Convergence in solution means that the search system can identify the same set of optimal solutions in the solution space over and over again.
\nUsually, the edge matrix E is not necessary to be included in the TSP definition because the TSP is a complete graph. However, the matrix E is a powerful data structure that can shift our point of view so that we can uncover alternative approaches. One factor contributing to algorithmic difficulty is that we lack a data structure that links the structure of the problem and the behavior of the search algorithm and that can make the complex search space traceable and tractable. It may be unreasonable to expect a search algorithm to be able to solve any problem without taking into account the structure and properties of the problem. Local search algorithms may not require much problem-specific knowledge in order to generate good solutions. However, in order to solve a problem exactly, we should design a search algorithm that is based on the structure of the problem at hand.
\nA dynamical system is a model to describing the temporal evolution of a system in its state space [4, 5, 6, 7, 8, 9]. The theory of dynamical system is an extremely broad area of study. The study of dynamical systems has discovered that many dynamical systems exhibit attracting behavior in the system trajectories. In such a system, all initial states tend to evolve toward a single final state or a set of final states. This single state or a set of states is called attractor. A heuristic local search system essentially is a discrete dynamical system and therefore natural in the domain of dynamical systems.
\nA local search system has a solution space S, a set of times T (iterations of search), and a search function \n
For the TSP, a search trajectory leads to a sequence of tours \n
Invariance, i.e., \n
Attractiveness, i.e., \n
Convexity, i.e., all locally optimal tours in A are gathered in an extremely small region of the solution space.
Centrality, i.e., the best of these locally optimal tours (the globally optimal tour) is located centrally with respect to the other locally optimal tours.
Irreducibility, i.e., the solution attractor A contains a limit number of invariant locally optimal tours.
Search trajectories and solution attractor in a local search system.
In general term, for a TSP instance with \n
\nFigure 2 presents the attractor-based search system (ABSS) for the TSP. In this algorithm, Q is a given TSP instance. K is the number of search trajectories used to generate K locally optimal tours. E is the edge matrix used to store the K locally optimal tours. \n
The ABSS algorithm for the TSP.
The critical element in the ABSS is the edge matrix E. Few search algorithms have used the edge matrix E in their search processes. An edge is the most basic element in a tour. It is a connection between two nodes and contains pieces of information about \n
It is a natural data structure that can store the edge configurations of search trajectories and thus can visually demonstrate the asymptotic behavior of the search trajectories during the search. When the search trajectories reach their final points, it records the frequency of occurrence of each of the edges in the locally optimal tours.
It is an instrument that can alter the state of what we measure for the TSP. We can change a tour-search process into an edge-search process, and thus the problem of finding the optimal tour is converted into the problem of finding a set of edges. The edge space represented by the edge matrix E is much simpler and smaller than the solution space represented by the tours.
It is a mechanism that can transform non-deterministic local search to deterministic global search. Through the matrix E, we can see that the search trajectories actually perform the process of edge inclusion and exclusion, and the temporal evolution of the edge configuration matrix E generated by different sets of K search trajectories always converges to the same small set of edges.
(a) Shows a 10-node tour and (b) shows its edge configuration in the matrix E.
A search trajectory changes its edge configuration during the search process. Let W be the total number of edges in the matrix E, \n
For a given TSP instance, W is a constant value \n
The \n\nα\n\nt\n\n\n, \n\nβ\n\nt\n\n\n, and \n\nγ\n\nt\n\n\n curves with search iterations.
This indicates that at certain point of time, the union of the edge configurations of the search trajectories will become fixed. This aggregate edge configuration will be the edge configuration of the solution attractor at limit.
\nWhen the matrix E records the edge configurations of K locally optimal tours, the edges are partitioned into two sets: the edges with hit (hit edges) and the edges without hit (non-hit edges). The hit edges include all globally superior edges, all G-edges, and some bad edges. Figure 5 shows the composition of edges in the matrix E after the edge configurations of K locally optimal tours are stored in it. The local search process can quickly make large number of edges become the non-hit edges. In our experiments, we found that the ration \n
The composition of edges in the matrix E after the edge configurations of K locally optimal tours are stored.
Different sets of K search trajectories will generate a little different edge configuration in the matrix E. However, the underlying edge configuration of the solution attractor in the matrix E is structurally stable because small differences in the final edge configurations generated by different sets of K search trajectories do not mean the qualitative difference in the dynamical behavior of search trajectories. The core structure of the edge configuration of the solution attractor keeps unchanged. In our experiments, we observed that in the aggregated edge configurations of the different sets of K locally optimal tours, the set of globally superior edges and the G-edges is always the same. This empirical fact indicates that a local search system actually is a deterministic system. Although a single search trajectory appears stochastic, there is an important aspect of order hidden in the local search system that makes all different sets of K search trajectories converge to the same set of core edges.
\nIn order to make sure that the ABSS is an effective and efficient search system, we should answer the following fundamental questions:
“How can we construct the edge configuration of the solution attractor without large number of search trajectories?” that is, “What is a proper size of K?”
What is the relationship between the size of the constructed solution attractor and the size of the TSP instance?
How does the ABSS meet the requirements of a global optimization system?
Is the best tour in the solution attractor the best tour in the solution space?
It is easy to verify that the edge configuration of a true solution attractor can be obtained if all search trajectories are performed and all search trajectories reach their real locally optimal points. In other words, the probability of finding all globally optimal points is one if all possible search trajectories are performed. However, the required search effort may be very huge—equivalent to enumerating all possibilities in the solution space. In fact, we can construct the edge configuration of the solution attractor with a limited number of K locally optimal tours. In a heuristic local search system, K search trajectories start a sample of initial points from a uniform distribution over the solution space S and generate a sample of locally optimal points uniformly distributed over the solution attractor A. The fundamental theory behind using K search trajectories is the information theory. According to the information theory [13], each solution point in the solution space contains some information about its neighboring points that can be modeled as mapping \n
Another related question is “how many moves a local search trajectory has to make before it reaches a real locally optimal tour?” So far we do not have an answer to this question. We even do not know any nontrivial upper bounds on the number of moves that may be needed to reach local optimality [14, 15, 16, 17]. In practice, we are rarely able to find a true locally optimal point because we simply do not allow the local search process run enough long time. We usually let a search trajectory run a predefined number of iterations, accept whatever solution it generates, and treat it as a locally optimal solution. Therefore, the size of the constructed solution attractor depends not only on the problem structure and the neighborhood function used in the local search process but also on the amount of search time invested in the local search process. If we spend more time in the local search process (\n
The size of a constructed solution attractor is also determined by the time spent in the local search process.
Let \n
Therefore, at any search time t before the K search trajectories reach their true end points, the edge configuration of the true solution attractor \n
What is the relationship between the size of the constructed solution attractor and the size of the given problem? So far there is no theoretical or analytical tool available in the literature that can be used to answer this question. We have to depend on empirical results to lend some insights. If the size of the constructed attractor increases exponentially with the size of the problem increases, the ABSS still does not fundamentally reduce the computational complexity of the problem. The ABSS consists of two search phases: the local search phase that construct the solution attractor (from line 5 to line 10 in the ABSS algorithm) and the exhausted search phase that find the best tour in the solution attractor (line 11). For the TSP, the solution space can be represented by a search tree. The local search phase actually performs the task of pruning off the edges that cannot possibly be included in the globally optimal tours. When the first edge is discarded by all K search trajectories, \n
The relationship between the size of the constructed solution attractor and the size of the problem.
After the local search phase, majority of unnecessary branches have been cut off from the search tree. Usually, when using tree search enumerative algorithm, the effective branching factor is used to measure the computing complexity of the algorithm. An effective branching factor\n\n
where N is total number of nodes generated from the origin node and n is the size of the TSP instance, representing the depth of the tree. We conducted several experiments on different TSP instances. The tree search process always starts from node 1 (the first row of the matrix E). N is the total number of nodes that are processed to construct all valid and invalid tours in the matrix E from the node 1. N does not count the node 1 (the origin node), but includes node 1 as the end node of a valid tour. Figure 8 shows the result of one experiment, using the same instances and setting reported in Figure 7. The effective branching factors in all our experiments are very small, all less than 2. This result indicates that the edge configuration of the solution attractor presents a tree with extremely sparse branches, and the degree of sparseness does not change as the problem size increases if we properly increase local search time for a larger instance. It also indicates that the exhausted search phase is polynomial time if we polynomially increase local search time for larger instances. Therefore, the tree represented by the edge configuration of the constructed solution attractor has a manageable size that can be searched completely in \n
The \n\n\nb\n∗\n\n\n values for different problem size n.
The ABSS is a global optimization system. The goal of a global optimization system is to find all absolute best solutions in the solution space. There are two major tasks in a global optimization system: (1) finding all globally optimal points in the solution space and (2) making sure that they are globally optimal. To complete these tasks, the global optimization system should meet the following requirements: (1) its search behavior should be globally convergent, (2) it should be deterministic and has a rigorous guarantee for finding all globally optimal solutions without excessive computing burden, and (3) it should have a self-evident optimality criterion.
\nIn the ABSS, two different search phases have different search objectives. The objective of the local search phase is “searching for most promising tours in the solution space.” It tries to provide an answer to the question “In which small region of the solution space is the best tour located?” The objective pursued by the exhausted search phase is “finding the best tour among the most promising tours.” It tries to provide an answer to the question “In this small region, which tour is the best one?” Putting these two objectives together, the ABSS tries to provide an answer to the question “Which tour is the best tour in the solution space?”
\nThe ABSS combines beautifully two crucial aspects in search: exploration and exploitation. In the local search phase, K search trajectories explore the solution space independently and individually to collect the edges for constructing the solution attractor. The K search trajectories create and maintain diversity from beginning to the end. Randomization in the local search process makes the local search process become a randomized process. A search trajectory changes its edge configuration according to the objective function and its neighborhood structure. The local search phase actually uses the Monte Carlo simulation to sample locally optimal tours. Monte Carlo simulation is defined as simulations used to model the probability of different outcomes in a process that cannot easily be predicted due to intervention of random variables. The essential idea of Monte Carlo method is to use randomness to solve problems that might be deterministic in principle. In the ABSS, K search trajectories start a sample of initial tours from uniform distribution over the solution space and, through a randomized local search process, generate a sample of locally optimal tours that are uniformly distributed in the constructed solution attractor. Therefore the edge configuration of the solution attractor is constructed through this Monte Carlo sampling process. The distribution of the hit edges in the matrix E converges to a small set of edges, and the set of the edges is statistically fixed. This fixed edge configuration is not sensitive to the selection of K search trajectories. Convergence and stability are two desirable properties of the solution attractor: all search trajectories will converge to the solution attractor and remain there forever. The ability of K search trajectories to explore the entire solution space and thus collect all globally superior edges and G-edges can help the ABSS achieve its required function—finding all globally optimal tours.
\nThe global convergence and deterministic property of the search trajectories make the ABSS converge in solution, that is, the ABSS always find the same set of the best tours. This argument was empirically confirmed in our experiments. For a given TSP instance, we repeated the same search process on the same instance many times, each time using a different set of K search trajectories, and the search system always generates the same set of the best tours in all trials. Table 1 shows the result of one experiment. This experiment generated two TSP instances \n
Trial # | \nNumber of tours in A\n | \nRange of tour cost | \nNumber of best tours in A\n | \n
---|---|---|---|
1000 nodes (Q\n1) | \n(6000 initial tours) | \n\n | \n |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | \n5,703,833 5,703,785 5,703,479 5,703,829 5,703,868 5,703,499 5,703,253 5,703,791 5,703,742 5,703,990 5,703,637 5,703,457 5,703,642 5,703,626 5,703,727 | \n[3926, 4437] [3926, 4521] [3926, 4509] [3926, 4495] [3926, 4540] [3926, 4500] [3926, 4556] [3926, 4488] [3926, 4498] [3926, 4551] [3926, 4526] [3926, 4536] [3926, 4534] [3926, 4546] [3926, 4522] | \n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | \n
10,000 nodes (Q\n2) | \n(60,000 initial tours) | \n\n | \n |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | \n9,428,645 9,428,571 9,428,032 9,429,004 9,428,625 9,428,819 9,428,815 9,429,021 9,428,950 9,428,847 9,428,749 9,428,978 9,428,767 9,428,933 9,428,799 | \n[81,967, 85,287] [81,967, 84,979] [81,967, 85,286] [81,967, 85,365] [81,967, 85,348] [81,967, 85,345] [81,967, 85,232] [81,967, 85,254] [81,967, 85,320] [81,967, 85,286] [81,967, 85,036] [81,967, 85,248] [81,967, 85,076] [81,967, 85,223] [81,967, 85,337] | \n3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 | \n
Tours in solution attractor for 1000-node and 10,000-node TSP instances.
One factor that makes the TSP difficult to solve is that we have not found a simple optimality criterion to decide whether or not a locally optimal tour is also a globally optimal tour. Selecting the best tour among a set of tours and knowing it is the best one are the full challenges of the TSP. A brute-force algorithm that sorts through all tours in the solution space can be certain that it meets the challenge. However, it lacks practical efficiency. For a TSP instance, there are an unknown number of globally and locally optimal tours. The ABSS uses a simple and practical optimality criterion: the best tours in the set of all locally optimal tours are the globally optimal tour. In fact, this criterion is the necessary and sufficient condition for a locally optimal tour to be a globally optimal tour. In the ABSS, the local search phase identifies the solution attractor, and no tour outside the solution attractor can be better than any tour inside. Then the exhausted search phase examines all tours in the solution attractor and finds the best tours. In fact, this optimality criterion describes how the ABSS models and solves the TSP.
\nFor a tour \n
\n\n
\n\n
\n\n
\n\n
\n\n
For the TSP, the computational complexity is associated with the combinatorial explosion of potential solutions in the solution space. If we accept the argument that the number of tours in the solution space indicates the difficulty of the TSP, then the fact that the solution space can be significantly reduced to a small solution attractor means that the difficulty of the TSP can be dramatically reduced. The novel perspective of solution attractor in a local search system for the TSP gives us an opportunity to overcome combinatorial complexity. The solution attractor shows us where the best tour can be found in the solution space. If we concentrate the exhausted search effort in this much smaller region, the number of possibilities in search space is no longer prohibitive. Our experiments showed that the ABSS can significantly reduce the computational complexity for the TSP and thus can solve the TSP much efficiently with global optimality guarantee. The ABSS is an obvious finite algorithm in computing complexity of \n
The edge matrix E is the data structure that is defined by the TSP naturally and is used in the ABSS to separate the solution attractor from the entire solution space. In the ABSS, the combination of an efficient local search process, a powerful data structure (the matrix E), and an exhausted search process provides a highly effective and efficient search system. If some other NP-hard problems have the same nice data structure that can be used to reduce the search space, these problems can also be solved in polynomial time.
\nThis chapter focuses on the solution attractor of the local search system for the TSP. Does it appear to be technical archetypes for other combinatorial optimization problems? Each optimization problem has its own specifics and data structure. In order to fully understand the search process for a particular problem, we must put our attention to the data structure that is defined by the problem. The combination of a proper data structure and simple search strategy can make the highly complex solution space become tractable and lead to more knowledge about the problem and provide opportunities for new algorithmic designs.
\nThe TSP is the most prominent problem in NP-hard problems. It is hoped that this chapter will serve as a pioneer in this field and bring more and better works from other researchers and practitioners. The ultimate goal of this chapter is to encourage readers to take up their own pursuit of interesting problem-by-problem methods for attacking diverse optimization problems.
\nThe solution attractor theory provides some important insights into the power of efficient computations and a line of reasoning that may lead to a proof in the near future about P vs. NP problem. The P vs. NP problem is an important computational issue in nearly every scientific discipline [18]. It is about how efficient we can search through a huge number of possibilities. Computational complexity theory suggests that there are limits of the power of general-purpose optimization techniques. Majority of people are in favor of \n
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