What is the common name of the computational problem whose task, when given a list of cities and the distances between them is to find the shortest possible tour that visits each city once?
"worse fitness than the sBest) in order to escape the local optimal. This process continues until the user specified stopping criterion is met, at which point, the best solution seen during the search process is returned (line 20). The traveling salesman problem (TSP) is sometimes used to show the functionality of tabu search. This problem poses a straightforward question — given a list of cities, what is the shortest route that visits every city? For example, if city A and city B are next to each other, while city C is farther away, the total distance traveled will be shorter"
"starting with the smallest. Whenever computing a solution requires solutions for smaller problems using the above recursive equations, look up these solutions which are already computed. To compute a minimum distance tour, use the final equation to generate the 1st node, and repeat for the other nodes. For this problem, we cannot know which subproblems we need to solve, so we solve them all. Number the cities 1, 2, . . . , ""N"" and assume we start at city 1, and the distance between city ""i"" and city ""j"" is ""d"". Consider subsets ""S"" ⊆ {2, . . ."
"tour is the one that minimizes the total amount of horizontal motion, with ties broken by Euclidean distance. The same dynamic programming algorithm that finds the optimal bitonic tour may be used to solve other variants of the traveling salesman problem that minimize lexicographic combinations of motion in a fixed number of coordinate directions. At the 5th International Olympiad in Informatics, in Mendoza, Argentina in 1993, one of the contest problems involved bitonic tours: the contestants were to devise an algorithm that took as input a set of sites and a collection of allowed edges between sites and construct a"
"for every constant r there is an instance of the traveling salesman problem such that the length of the tour computed by the nearest neighbour algorithm is greater than r times the length of the optimal tour. Moreover, for each number of cities there is an assignment of distances between the cities for which the nearest neighbor heuristic produces the unique worst possible tour. (If the algorithm is applied on every vertex as the starting vertex, the best path found will be better than at least N/2-1 other tours, where N is the number of vertexes) The nearest neighbour algorithm"
"any city, enumerate all possible permutations of cities to visit, and find the distance of each permutation and choose one of minimum distance. The total number of possible routes covering all N cities can be given as (N − 1)! and (N − 1)!/2 in aTSP and sTSP respectively. Stirling's approximation: formula_2 Faster than the exhaustive enumeration but still exponential, and the drawback of this algorithm, though, is that it also uses a lot of space: the worst-case time complexity of this algorithm is formula_3 and the space formula_4. Time: the fundamental operations employed in the computation are additions and"
", ""N""} of cities and, for ""c"" ∈ ""S"", let ""D""(""S"", ""c"") be the minimum distance, starting at city 1, visiting all cities in ""S"" and finishing at city ""c"". First phase: if ""S"" = {""c""}, then ""D""(""S"", ""c"") = ""d"". Otherwise: ""D""(""S"", ""c"") = min (""D""(""S"" − ""c"", ""x"") + ""d"" ). Second phase: the minimum distance for a complete tour of all cities is ""M"" = min (""D""({2, . . . , ""N""}, ""c"") + ""d"" ) A tour ""n"" , . . ., ""n"" is of minimum distance just when it satisfies ""M"" = ""D""({2, . ."
"Princeton University (see the Princeton external link). The total computation time was equivalent to 22.6 years on a single 500 MHz Alpha processor. In May 2004, the travelling salesman problem of visiting all 24,978 towns in Sweden was solved: a tour of length approximately 72,500 kilometres was found and it was proven that no shorter tour exists. In March 2005, the travelling salesman problem of visiting all 33,810 points in a circuit board was solved using ""Concorde TSP Solver"": a tour of length 66,048,945 units was found and it was proven that no shorter tour exists. The computation took approximately"
"problem is also a special case of the travelling salesman problem, obtained by setting the distance between two cities to one if they are adjacent and two otherwise, and verifying that the total distance travelled is equal to n (if so, the route is a Hamiltonian circuit; if there is no Hamiltonian circuit then the shortest route will be longer). There are ""n""! different sequences of vertices that ""might"" be Hamiltonian paths in a given ""n""-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow. An early"
"used to find a satisficing solution for the traveling salesman problem (that is, a solution that satisfies an adequacy criterion, although not with the high quality obtained by exploiting the graph structure). The search starts with an initial solution, which can be generated randomly or according to some sort of nearest neighbor algorithm. To create new solutions, the order that two cities are visited in a potential solution is swapped. The total traveling distance between all the cities is used to judge how ideal one solution is compared to another. To prevent cycles – i.e., repeatedly visiting a particular set"
"the number of cities the travelling salesman must visit exactly once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially. Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an inordinate amount of space. Computational complexity may be approached from many different aspects. Computational complexity can be investigated on the basis of time, memory or other resources used to solve the problem. Time and space are two of the most"
"Canadian traveller problem In computer science and graph theory, the Canadian traveller problem (CTP) is a generalization of the shortest path problem to graphs that are ""partially observable"". In other words, the graph is revealed while it is being explored, and explorative edges are charged even if they do not contribute to the final path. This optimization problem was introduced by Christos Papadimitriou and Mihalis Yannakakis in 1989 and a number of variants of the problem have been studied since. The name supposedly originates from conversations of the authors who learned of the difficulty Canadian drivers had - traveling a"
"Route inspection problem In graph theory, a branch of mathematics and computer science, the Chinese postman problem (CPP), postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of an (connected) undirected graph. When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. It"
"Nearest neighbour algorithm The nearest neighbour algorithm was one of the first algorithms used to determine a solution to the travelling salesman problem. In it, the salesman starts at a random city and repeatedly visits the nearest city until all have been visited. It quickly yields a short tour, but usually not the optimal one. Below is the application of nearest neighbour algorithm to the travelling salesman problem . These are the steps of the algorithm: The sequence of the visited vertices is the output of the algorithm. The nearest neighbour algorithm is easy to implement and executes quickly, but"
"applications, such as the traveling salesman problem and many questions in linear programming. There are standard techniques for transforming function and optimization problems into decision problems. For example, in the traveling salesman problem, the optimization problem is to produce a tour with minimal weight. The associated decision problem is: for each ""N"", to decide whether the graph has any tour with weight less than ""N"". By repeatedly answering the decision problem, it is possible to find the minimal weight of a tour. Because the theory of decision problems is very well developed, research in complexity theory has typically focused on"
"is a comprehensive review of many issues related to simulation and optimization. The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes"
"city. The last constraints enforce that there is only a single tour covering all cities, and not two or more disjointed tours that only collectively cover all cities. To prove this, it is shown below (1) that every feasible solution contains only one closed sequence of cities, and (2) that for every single tour covering all cities, there are values for the dummy variables formula_2 that satisfy the constraints. To prove that every feasible solution contains only one closed sequence of cities, it suffices to show that every subtour in a feasible solution passes through city 1 (noting that the"
"Held–Karp algorithm The Held–Karp algorithm, also called Bellman–Held–Karp algorithm, is a dynamic programming algorithm proposed in 1962 independently by Bellman and by Held and Karp to solve the Traveling Salesman Problem (TSP). TSP is an extension of the Hamiltonian circuit problem. The problem can be described as: find a tour of N cities in a country (assuming all cities to be visited are reachable), the tour should (a) visit every city just once, (b) return to the starting point and (c) be of minimum distance. Broadly, the TSP is classified as symmetric travelling salesman problem (sTSP), asymmetric travelling salesman problem"
"the book ""Century/Acorn User Book of Computer Puzzles"". Knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is ""closed"", otherwise it is ""open"". The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students."
"market only and each market sells only one item. Since TSP is NP-hard, TPP is NP-hard. Approaches for solving the traveling purchaser problem include dynamic programming and tabu search algorithms. Traveling purchaser problem The traveling purchaser problem (TPP) is an NP-hard problem studied in theoretical computer science. Given a list of marketplaces, the cost of travelling between different marketplaces, and a list of available goods together with the price of each such good at each marketplace, the task is to find, for a given list of articles, the route with the minimum combined cost of purchases and traveling. The traveling"
"example is finding minimum T-joins (i.e., edge sets whose only endpoints of odd degree are those in a specified vertex set T). An important special case is the seemingly difficult problem of finding a shortest tour in a graph that traverses every edge at least once, known as the Postman problem. The stark contrast between the polynomial solvability of this problem and the intractability of the traveling salesman problem in which the tour is supposed to traverse vertices rather than edges, helped focus attention on the phenomenon so typical of combinatorial structures: two seemingly very similar problems turn out in"
"bitonic tour using those edges that included as many sites as possible. As with the optimal bitonic tour, this problem may be solved by dynamic programming. Bitonic tour In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the chain at most twice. The optimal bitonic tour is a bitonic tour of minimum total length. It is a standard exercise in dynamic programming to devise a polynomial time algorithm that constructs the optimal bitonic"
"Bitonic tour In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the chain at most twice. The optimal bitonic tour is a bitonic tour of minimum total length. It is a standard exercise in dynamic programming to devise a polynomial time algorithm that constructs the optimal bitonic tour. The problem of constructing optimal bitonic tours is often credited to Jon L. Bentley, who published in 1990 an experimental comparison of many heuristics for"
"instance of the shortest path problem: given fares and for trips from departure A to some intermediate destination B and from B to final destination C, determine the least expensive combined trip from A to C. X + Y sorting In computer science, X + Y sorting is the problem of sorting pairs of numbers by their sum. Given two finite sets and , the problem is to order all pairs in the Cartesian product by the key . The problem is attributed to Elwyn Berlekamp. This problem can be solved using a straightforward comparison sort on the Cartesian product,"
"main goal is to reduce the run time by reducing the search space. Dijkstra thought about the shortest path problem when working at the Mathematical Center in Amsterdam in 1956 as a programmer to demonstrate capabilities of a new computer called ARMAC. His objective was to choose both a problem as well as an answer (that would be produced by computer) that non-computing people could understand. He designed the shortest path algorithm in about 20 minutes without aid of paper and pen and later implemented it for ARMAC for a slightly simplified transportation map of 64 cities in the Netherlands"
"Knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is ""closed"", otherwise it is ""open"". The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students. Variations of the knight's tour problem involve chessboards"
"(in the worst case). Closest pair of points problem The closest pair of points problem or closest pair problem is a problem of computational geometry: given ""n"" points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane was among the first geometric problems that were treated at the origins of the systematic study of the computational complexity of geometric algorithms. A naive algorithm of finding distances between all pairs of points in a space of dimension ""d"" and selecting the minimum requires time. It turns"
"cities.. Multi-fragment algorithm The multi-fragment (MF) algorithm is an heuristic or approximation algorithm for the travelling salesman problem (TSP) (and related problems). This algorithm is also sometimes called the ""greedy algorithm"" for the TSP. The algorithm builds a tour for the traveling salesman one edge at a time and thus maintains multiple tour fragments, each of which is a simple path in the complete graph of cities. At each stage, the algorithm selects the edge of minimal cost that either creates a new fragment, extends one of the existing paths or creates a cycle of length equal to the number"
"Delbert Ray Fulkerson and Selmer M. Johnson from the RAND Corporation, who expressed the problem as an integer linear program and developed the cutting plane method for its solution. They wrote what is considered the seminal paper on the subject in which with these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. Dantzig, Fulkerson and Johnson, however, speculated that given a near optimal solution we may be able to find optimality or prove optimality by adding a small amount of extra inequalities (cuts). They"
"trip through all sites in Milan whose total length is at most 10 km. For this reason, complexity theory addresses computational problems and not particular problem instances. When considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet (i.e., the set {0,1}), and thus the strings are bitstrings. As in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in"
"equalities ensure there can only be one such tour). For if we sum all the inequalities corresponding to formula_6 for any subtour of ""k"" steps not passing through city 1, we obtain: which is a contradiction. It now must be shown that for every single tour covering all cities, there are values for the dummy variables formula_2 that satisfy the constraints. Without loss of generality, define the tour as originating (and ending) at city 1. Choose formula_9 if city ""i"" is visited in step ""t"" (""i"", ""t"" = 1, 2, ..., n). Then since formula_2 can be no greater than"
"potential applications, many natural questions still remain open. Is there a constant-factor approximation or is the problem APX-hard? Is i-SSPPR #P-complete? An even more fundamental question has been left unanswered: is there a polynomial-size ""description"" of an optimal policy, setting aside for a moment the time necessary to compute the description? Canadian traveller problem In computer science and graph theory, the Canadian traveller problem (CTP) is a generalization of the shortest path problem to graphs that are ""partially observable"". In other words, the graph is revealed while it is being explored, and explorative edges are charged even if they do"
"2006, Cook and others computed an optimal tour through an 85,900-city instance given by a microchip layout problem, currently the largest solved TSPLIB instance. For many other instances with millions of cities, solutions can be found that are guaranteed to be within 2-3% of an optimal tour. TSP can be modelled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's weight. It is a minimization problem starting and finishing at a specified vertex after having visited each other vertex exactly once. Often, the model is"
"Analyst's traveling salesman theorem The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks under what conditions may a set ""E"" in two-dimensional Euclidean space formula_1 be contained inside a rectifiable curve of finite length. So while in the original traveling salesman problem, one asks for the shortest way to visit every vertex in a graph with a discrete path, this analytical version requires the curve to visit perhaps infinitely many points. A posteriori, for ""E"" to be contained in a rectifiable curve Γ, since Γ"