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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“Water connects us. It connects us by flowing across imposed boundaries, by linking diverse terrains and settlements” (Federico Mayor).1
Water is essential to life. The availability of ‘good-quality’ water is not only the basis of all biological processes but also the maintenance of biodiversity and the functioning of ecosystems. Water has production functions (in terms of biological processes), maintenance functions (in terms of ecosystems) and recovery functions (through the cycles of materials).
Natural ecosystems and farming crops are the great consumers of fresh water. The competition between these consumers has been directly and indirectly intensified by urban land occupation and population growth: directly, as more land for farming means significant change to original ecosystems and, indirectly, because the growing population requires farming to become more productive and to occupy more land.
This occupation not only leads to the change in land uses in question but also to additional pollutants in the soil and associated water masses and the depletion of biological and genetic diversity. In time, the ecosystems become incapable of maintaining their functions of producing water resources with the essential characteristics to sustain human life and activities. This depletion of vital functions performed by the ecosystems endangers the sustainability of human activities like farming, aquaculture and fishing.
The river basin is the natural unit for the occurrence of water resources. River basins constitute large-scale ecosystems that combine land (forest, pasture) and aquatic components (rivers, lakes, wetlands), supplying a great diversity of vegetation and animal habitats. River basins are reference units for the hydrological cycle and the natural resources associated with it, including the filtration and regeneration of elements of the biogeochemical cycle. The main objective of water management is to provide fresh water for human consumption, for food from agriculture and fishing, for hydro-energy, and to all waterside amenities. These territorial units are also the natural link between the land and the sea that can sustain both the movement of fish species and materials between inland and marine waters (or vice versa) and the human need for navigable watercourses. This diversity of functions in the same territory is water’s specific virtue as a natural integrator. On the other hand, this virtue also contributes to the difficulties of institutional design for effective water management in several ways [1]. As Giordano and Shah ask, the basin has been always put forward as the key pillar of IWRM implementation and the natural management unit for water resources; there is no evidence that the basin approach is needed [2].
The ecological processes occur and function at various scales—from very small and effecting the proximity to very large with deferred effects in time—which can present inter-scale variations due to the interference of various, often unpredictable, factors. Accordingly, interactions between the different territorial scales condition the end results arising from the referred processes.
When attention is focused exclusively on a specific scale, it can hinder the perception of these interactions, resulting in institutional responses that are not the most suitable because they do not cover the whole process [3]. On the other hand, the issues of water security demand that different scales are considered simultaneously [4]. As referred by Uschi Eid (UN), it is evident that addressing water security problems requires integrating and taking into account different spatial and temporal scales and water governance and management have not been particularly effective in the past in doing so [5].
The basic ideas of integrated water resource management are nearing 100 years of age, being a call to consider water holistically, to manage it across sectors and to ensure wide participation in decision-making [2].
The need for territorial integration is not expressed immediately in current institutional systems and requires various adaptations and articulations between organizations and, above all, between the powers of different levels. On the other hand, the approach of the effects of human occupation on the ecological processes and of the long-term consequences of these alterations on the natural systems, and on the expectations of resource use by human activities, is a complex task for which various fields of knowledge are required. These two factors contribute to a growing need to adapt the operations of the organizations responsible for water management in a way that is not always compatible with the sectoral nature of the existing structures.
‘In earlier generations, water was seen as a technical issue – a question of organising proper water supply for human societies and agricultural production’ [6]. There is now greater recognition that while water has a strong technical component, it is more about managing people, their interaction with the natural environment and the services ecosystems provide.
Due to its cyclic, dynamic and transversal intrinsic nature, water problems have many dimensions and overlap with many scientific disciplines and fields of management, planning and policy.
Modifications of the large-scale ecosystems and natural cycles are having strong impacts on water quality and interfere with the main ‘parallel functions’ of water. Falkenmark is still very up to date in her definition of the functions for which it is absolutely crucial to create management models that are able to maintain functionality as [6]:
Health function: Safe water is crucial for protecting the survival of a healthy population—this is the perspective of the sanitary engineers.
Habitat function: Aquatic flora and fauna are critically dependent on the characteristics of the water in the water body in which they dwell—this is the perspective of ecologists.
Two carrier functions: Of dissolved material and of eroded material—this is the perspective of hydrochemists and geographers, respectively, to eliminate the pollution source since treatment of waste water is expensive.
Regional water scarcity in the future will become a limiting factor for agricultural production; socio-economic planning will, therefore, have to be adapted to actual water constraints. Inevitably, we will have to develop policy tools capable of managing the shortage of common water resources between competing actors.
Energy choices in dry climate regions are crucially influenced by water availability because water is required for almost every aspect of energy production and use: as a driving medium, as a cooling medium and as an energy transfer medium.
‘The human right to water is indispensable for leading a life in human dignity. It is a prerequisite for the realization of other rights’ [7].
The dimension introduced in the world debate by the United Nations Declaration (General Commentary no. 15 of the Committee of Economic, Social and Cultural Rights of the United Nations [7]) has recognized access to drinking water and sanitation as a human right.
Access to water must also be discussed in relation to the scarcity concept, though access to water is a human need and its satisfaction is a political objective.
The European Declaration for Water stated in 2005 that the ‘availability of quality water in unlimited quantities 24 hours a day and 365 days a year for multiple uses and at extremely reasonable rates, beyond merely satisfying the human right to a basic share of drinking water, is indeed a conquest of public health, welfare and social cohesion’ [8].
In this European level political agreement among scientists, researchers and activists for water rights, it was considered that the ‘access to this general interest equity or value must be recognized and guaranteed to all, as citizens’ social rights’. Several movements towards this attempt can be referenced in a framework that considers water as an ‘essential good’ and a ‘human right’, following the authors Morley [9], Petrella [10] and Sadler [11].
Until the 1970s, there were essentially isolated incidences of environmental problems (accumulation of pollutant residue, infected waters, polluted atmosphere, etc.) that have had more or less limited effects and generally restricted to the territory of origin. Since the 1980s, and in more particularly the late 1980s and early 1990s, environmental problems have become increasingly complex. The effects started to be cumulative and could not be explained by a single phenomenon or cause, about which there were varying degrees of knowledge, and problems started to acquire greater spatial scope. It therefore became indispensable to adopt a comprehensive perspective for analysis and intervention in order to understand and address phenomena like the greenhouse gas effect, the rarefaction of the ozone layer or climate change.
The nature of environmental problems evolved, forming combinations of symptoms and factors that required ever-increasing scrutiny. In the 1990s, there was believed to be three broad groups of problems [12]:
Generalized degradation of soil resources (loss of roughly 1.2 billion hectares of land in the previous 50 years).
Growing accumulation of gases in the atmosphere, responsible for the worsening of the greenhouse gas effect and climate change (in particular CO2, the emissions of which would have to decline about 60% at that time).
Biodiversity increasingly at risk (clearing forests, drainage of humid zones, intensification of the harvesting of living resources, degradation of ecosystems that resulted in the extinction of thousands of species).
In this context, water quality constitutes a central theme for policy action. The concept of water quality has evolved insofar as recognition is given to the growing interactions and levels of complexity in the degradation processes of water resources. These include not only the cumulative factors and effects in space and time but also the persistence of those effects caused and the difficulty in taking isolated localized actions.
In the first decade of the twenty-first century, the globalization of the problems has become more evident. Global changes like climate changes, demographic changes and economic crisis are affecting directly and indirectly the resources in water and their natural ecosystems, with growing effects at transnational and world scale, demanding international concerted action.
The last 5 years of the twentieth century were characterized by an overall tendency of continuous glacier melting. This decline will have impacts on both the sustainability of the water resources in basins, which depend on glaciers and on their ecosystems [13]. However, demographics and the increasing consumption that comes with rising per capita incomes are the most important drivers or pressure on water [14].
The world’s population is growing by about 80 million people a year, implying an increased freshwater demand of about 64 billion cubic meters a year. Most population growth will occur in developing countries, mainly in regions that are already experiencing water stress and in areas with limited access to safe drinking water and adequate sanitation facilities [14].
Human population growth and the expansion of economic activities are collectively placing huge demands on coastal and freshwater ecosystems. The increase in the number of people without access to water and sanitation in urban areas is directly related to the rapid growth of slum populations in the developing world and the inability (or unwillingness) of local and national governments to provide adequate water and sanitation facilities in these communities. The world’s slum population, which is expected to reach nearly 900 million by 2020, is also more vulnerable to the impacts of extreme weather events. It is however possible to improve performance of urban water supply systems while continuing to expand the system and addressing the needs of the poor [5].
The management of scarce resources places new challenges to the institutional and organizational systems and schemes that are operating in a traditional way. This usually relates to continuing a ‘water supply focus’ of increasing the levels of services and investments in infrastructures in response to the increasing needs imposed by population growth, urbanization and intensification of irrigated areas. The need to manage the ‘demand’, increasing awareness of water scarcity, equity and sustainable uses of the available resources is still finding institutional and operational answers, though the principles are generally accepted by the agents of management and planning. This will mean a shift towards a new paradigm in water management, search for innovative and sustainable solutions, alternative sources of water (like more recycling and rainwater collection) and a new institutional framework, where the sectors may cooperate effectively and the public administration can address complex problems through new and flexible learning organization and structures.
It is still a fact that, as stated by Falkenmark almost 20 years ago, societal rules manifested in legislation and administration are quite inconsistent with natural laws [6]. Even though the same water is used for a whole set of different uses while running down the river basin, these uses are administered by different authorities as if they were not connected.
The complexity involved in water resource management and its linkages to land use adds even more difficulties to the management systems and to the decision-makers. Water management is increasingly complex and dynamic, requiring more flexible and adaptive responses.
The effects of water-depleting and water-polluting activities on human and ecosystem health remain largely unreported or difficult to measure [14], with growing need for a stronger and effective protection of ecosystems and the goods and services they produce.
Figure 1 shows project global water scarcity in 2025. The scarcity of water does not only refer to human needs. Over-use and over-allocation of water resources for human activities can lead to substantial reductions in the flow of watercourses necessary to maintain associated water-dependent ecosystems. This scarcity causes a reduction in the capacity of fresh water ecosystems to provide all the environmental services and can result in their irreversible degradation and that of the species depending on them. On the other hand, rising pressure from increasing population will endanger the maintenance of minimum environmental requirements. This situation is aggravated when growth is combined with the natural less favorable conditions and variability of resources.
Projected global water scarcity in 2025 (source: IWMI 2000).
Therefore, environment (including aquatic environment) should not be seen as complements to economic development but the foundation (and limits) on which development is built. Reflection should focus on the interdependence between the natural means and social systems. Water should be considered not as a resource which is available in varying degrees, but as a complex natural and integrating means, and our survival in this planet depends on its interdependence with other systems.
Ten years ago (2006) 13% of the world’s population did not have access to enough food to live a healthy and productive life, yet the ability, technology and resources needed to produce enough food for every man, woman and child in the world do currently exist. Lack of health, financial or natural resources such as land and water and lack of skills to link productive activities with remote markets and ensure employment are all intimately related to poverty [13]. By 2050, agriculture will need to produce 60% more food globally and 100% more in developing countries. As the current growth rates of global agricultural water demand are unsustainable, the sector will need to increase its water use efficiency by reducing water losses and increase crop productivity with respect to water [5].
By 2030, the number of urban dwellers is expected to be about 1.8 billion more than in 2005 and to constitute about 60% of the world’s population. Ninety percent of the increase in urban population is expected in developing countries, especially Africa and Asia, where the urban population is projected to double between 2000 and 2030. Coastal areas, with 18 of the world’s 27 megacities (greater than 10 million), are thought to face the greatest migration pressure. The net implication is that the world will have substantially more people in vulnerable urban and coastal areas in the next 20 years [14].
In 2000, more than 900 million urban dwellers (nearly a third of all urban dwellers worldwide) lived in slums. A slum dweller may only have 5 to 10 liters per day at his or her disposal compared to the UN suggested minimum requirement of 20–50 liters. A middle- or high-income household in the same city, however, may use some 50–150 liters per day, if not more [13].
As urban populations increase and local surface and groundwater sources are depleted or polluted, many major cities have had to draw freshwater from increasingly distant watersheds as to meet their rising demand for water [13].
Water occurrence in urban areas has the particularity of being either ‘natural’ or ‘artificially’ produced. Its presence takes the form of natural water bodies that constitute esthetical and leisure resources, the form of rainfall, flowing over built surfaces and infiltrating in subsoil; the form of treated water conducted through pipes for domestic uses and drinking; and the form of residual water in various degrees of degradation, conducted to sewage treatment systems or directly back to natural streams or the sea.
The specificity of the challenge for water management in urban areas is the need to integrate the natural hydrological systems functioning and the artificially build systems, in a balanced way and in order to minimize the negative impacts on both sides (increase of runoff due to paved surfaces, flash floods aggravated by obstruction of natural streams, overexploitation and increasing pressure on water resources sources to satisfy increasing demand and urbanization growth, contamination of groundwater’s, surface and coastal waters by the waste waters and the urban solid wastes and all kinds of particulates generated in urban areas, etc.).
The use of unexploited resources like storage of rainwater in buildings and open places (gardens, parking areas) is one very relevant action to take seriously inside urban areas and must be considered as a top priority in any management plan, not only in dry or semidry regions.
Emerging first at Mar del Plata and evolving through subsequent water fora was recognition of the need to integrate all aspects and dimensions of the water cycle, to achieve integrated water resource management (IWRM). More recently, there is increasing recognition that operating only within the water domain is not sufficient as the greatest factors influencing water management are beyond water policy in stricto sensus, in areas such as national economic policy.
Following the call for a more holistic approach to water resource management, water utilities and land-use planners need also to coordinate to overcome these water resource challenges in the urban areas [15]. Planners and utilities need to work together to implement more green infrastructure to better manage storm water, more onsite reuse to reduce potable demand and more green buildings to reduce potable demand and better manage water within buildings [15].
Agreeing with Engle that IWRM is fundamentally about governance arrangements, we also note that IWRM stresses the interconnectedness of catchments and users; there is no universal model for the way in which management institutions are structured and linked [16].
The need to improve assessment of the effectiveness of water governance and its impact on the implementation of IWM in an integrated way with territorial planning in urban areas relates to the responsibility of people and institutions involved in water management and urban planning, in any country, to determine and prioritize actions necessary to improve water and territorial governance. The central question therefore revolves around the premise that water should not be managed as a mere resource but as a complex natural entity, whose frontiers extend to environmental, institutional and social (including the economic aspects) spheres and whose reference is always territorial. This premise results not only from the specific characteristics of the water cycle, closely associated with the physical territory (including the soil and subsoil, vegetation cover, ecosystems, atmosphere and climatic factors). It also results from the fact that the human communities have occupied and transformed this territory, from the beginnings of sedentary societies and the time when man learnt to store water, interfering with this cycle and being also conditioned by it.
Consequently, a ‘territorial’ approach shall consider all the means in which water is manifested and in which it remains or contributes to the various biological, geological and chemical cycles, as well as all the occurrence, draining and natural infiltration processes on one hand and all the reserve, abstraction, consumption, change of water courses and/or physical-chemical and biological conditions on the other. This approach does not exclude social relations of cooperation or conflict towards access to water, in particular regarding the occupation and changing processes of the use of land and territory.
Falkenmark has deeply developed the analysis of the water and land use interaction. Her studies on the articulation of the cyclical support functions of the materials by the hydrological cycle, through the physical territories of the river basins, provide greater understanding of the interactions between the water functions and the physical territory, as illustrated in Figure 2.
Water-soil interactions (source: Falkenmark [6]).
Falkenmark discusses the different administrative divisions and the need to consider flows of water imports from upstream to downstream through these jurisdictions. The buffering capacity of ecosystem services (flood regulation, drought mitigation, wetland water storage, preparing room for the river, clearing invasive trees, etc.) is essential, and these dimensions need to be considered in a holistic management approach.
Figure 3 presents the division of a basin in three regions and the resulting precipitation—evapotranspiration and its circulation to the surface or through the subsoil of regions from upstream to downstream.
Water flow through the river basin territory (source: Falkenmark [6]).
The conclusions of the different fora and international committees on worldwide water management give ever-greater priority to the recognition that water is not a ‘mere resource’ but a complex and multidimensional entity. They highlight the need for concerted action between the different actors (government and society, stakeholders, etc.), in order to respond to the need for more ‘effective’ water governance [1].
Areas of further research for better operational results from integrated water management practices and for improving water governance can be identified [1].
Coherent articulation of spatial plans aiming at the integration of land uses, water and sustainability.
The role of state agencies aiming at integration and participation.
Enhancing opportunities for local solutions
Capacity building among planners aiming at adequate response to new challenges, adequate knowledge concerning contemporary planning problems (participation, governance, climatic change, etc.).
A river basin or drainage basin is a territory that contains a watercourse and its tributaries. The river basin should be the starting point for the analysis of the water resources, associated ecosystems and habitats that depend on them, as well as the analysis of all the interdependencies resulting from the scale(s) of occurrence from natural processes or the dysfunction’s triggered by human occupation and activities.
However, to effectively integrate water resource management with other natural resources, we need to think beyond the basin, and planning at any level always involves the public administration and the elected representatives of the state. Good government requires adequate networking and the integration of all sectors in planning activities. As quoted before, the UN recognizes that drivers and policies outside the water sector often have more impact on water management than those within [17].
Territorial integration of different sectoral strategies is the only way to match objectives at each scale of analysis. This integration demands an approach to water problems relating to all sectors, taking into account the other scales of analysis, beyond the river basin.
Decisions on the appropriate jurisdictional level for water resource allocation and management need to take the impact of all sectors at all scales into account. Furthermore, the level for jurisdiction must also recognize the specific role of the state in defining the priorities for the use of the available water. The promotion of good ‘governance’ in addition to good government depends primarily on the political will of the government and public administration bodies. Participation of stakeholders and sectors, as well as end users of water infrastructures and water amenities, is a central issue in governance. The need for integrated representation of all relevant human settlements, which are not always wholly within one basin, is also a key issue for good water governance [6, 18, 19, 1].
Water policy is a broad field where new perspectives for water management and planning can be framed. Le Meur defines this distinction between water management and ‘water politics’, based on the more comprehensive perspective and the inclusion of the social dimension, as the most relevant difference. Water politics should replace water management in order to involve diverse social actors and groups and follow an approach that goes apart from the managerial perspective that the ‘right solution’ (technical and institutional) exists and can be worked out with expertise and participation [20].
Allocation of available resources, according to priorities defined by the state and the control of these resources, is a prerogative of the state action, through its policies. This responsibility of how to deploy a resource to the national advantage is the key to water governance at the beginning of the twenty-first century, accordingly, and it is ‘how, through politics, the State can achieve this fairly and equitably, without reducing incentives for efficient use of the resource’ [20].
Fragmentation at administration levels is a traditional problem in environment and water institutions, with the different authorities handling different issues so that it becomes more difficult to allow interdependencies.
Even though reality is multidimensional and complex, all our institutional organizations are oriented to unidirectional and unidimensional interventions, leading to the dilemma stressed by Falkenmark, and discussed previously. It also happens that individual institutions have specific and often opposing objectives [6, 18]. The legislative framework is built to frame action in compartmentalized ‘worlds’, often hindering any interaction and cooperation that would be crucial to address these complex nature problems.
The ‘land/water dichotomy characterized by a mental image of land as opposed to water’ that was present at the time (1999) is still a reality, with policy structure based on the separation between land and water as entities to be managed in different ways and by different management systems. What usually happens in Europe, and many other regions in the world, is that land and water are administratively linked mainly through the use of environmental impact assessments in the case where land use changes produce side effects on water, not going beyond that legal guiding framework.
Consideration of the territorial character of water represents a complex challenge for water management but unavoidable in the urban areas. While cities are impacted by global changes, some urban structures become more and more static in comparison to the unexpected and faster changing contexts [21]. Accepting also that urban settlements incorporate an increasing complexity of interlinks and interdependencies within their economic and social tissue, effective solutions will not arise exclusively through ‘traditional, institutional or formal’ approaches. Therefore, out-of-the-box, experimental and cross scale responses may be necessary [21].
There is a crucial need for better understanding of the extended urban water cycle within the river basin processes at regional level. River basins are the territorial units for hydrological reference and for understanding the natural cycle in all its dimensions, including the links with other natural resources. Integrated water management calls for a territorial integration of these dimensions and the processes.
The diversity of functions in the same territory is one of water’s specific virtues as a natural integrator. On the other hand, this virtue also contributes to the difficulties of institutional design for effective water management in several ways. The need for territorial integration is not automatically reflected in the current institutional frameworks, and existing processes require several levels of adaptation, as well as articulation between organizations. On the other hand, the approach of the effects of human occupation on the ecological processes and of the long-term consequences of these alterations on the natural systems, and on the expectations of resource use by human activities, is a complex task for which various fields of knowledge are required. These two factors contribute to a growing need to adapt the operations of the organizations responsible for water management in a way that is not always compatible with the sectoral nature of the existing structures [1, 22].
There is a clear call for better integrated land and water management in the urban sphere by developing analytical frameworks and operational tools to assess the climate change impacts and the role of territorial planning in the urban water cycle. A new conceptual framework based on ‘territorial integration’, ‘sustainable urban development’ and ‘integrated water management’ was proposed to address the new and more complex challenges: an ‘integrated urban water management policy’ [22]. This approach is based on previous conceptual discussion [6, 19, 1, 23, 24, 25].
This contemporary approach aims to provide the policy making sphere with more operational and comprehensive tools to address the multidimensional challenges imposed by urban water management in the context of the current global changes and challenges.
This chapter is based on my PhD thesis presented at the University of Lisbon in 2010 [1] and also reflects more recent work [22].
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
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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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