Is there a solution to the traveling salesman problem?
THE TRAVELING SALESMAN PROBLEM Corinne Brucato, M.S. University of Pittsburgh, 2013 Although a global solution for the Traveling Salesman Problem does not yet exist, there are algorithms for an existing local solution. There are also necessary and su cient conditions to determine if a possible solution does exist when one is not given a ...
Is the traveling salesman problem solvable?
The traveling salesman problem is important because it is NP complete.If you can find a fast way to solve it, you have proved P=NP and changed the face of computation. The latest result shows that a special type of traveling salesman (TSP) problem is solvable in polynomial time. The TSP problem is easy to state but difficult to solve efficiently.
What is it like to be a traveling salesman?
The Algorithm of Christofides and Serdyukov
- Find a minimum spanning tree for the problem
- Create duplicates for every edge to create an Eulerian graph
- Find an Eulerian tour for this graph
- Convert to TSP: if a city is visited twice, create a shortcut from the city before this in the tour to the one after this.
What is traveling salesman problem (TSP)?
What Does Traveling Salesman Problem (TSP) Mean? The traveling salesman problem (TSP) is a popular mathematics problem that asks for the most efficient trajectory possible given a set of points and distances that must all be visited.
Has anyone solved the traveling salesman problem?
Scientists in Japan have solved a more complex traveling salesman problem than ever before. The previous standard for instant solving was 16 “cities,” and these scientists have used a new kind of processor to solve 22 cities. They say it would have taken a traditional von Neumann CPU 1,200 years to do the same task.
What is Travelling salesperson problem how is it solved?
Travelling Salesman Problem (TSP) : Given a set of cities and distances between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Note the difference between Hamiltonian Cycle and TSP.
When was the travelling salesman problem solved?
In 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.
Is the travelling salesman problem NP-complete?
Traveling Salesman Optimization(TSP-OPT) is a NP-hard problem and Traveling Salesman Search(TSP) is NP-complete. However, TSP-OPT can be reduced to TSP since if TSP can be solved in polynomial time, then so can TSP-OPT(1). I thought for A to be reduced to B, B has to be as hard if not harder than A.
Can travelling salesman problem be solved using dynamic programming?
Instead of brute-force using dynamic programming approach, the solution can be obtained in lesser time, though there is no polynomial time algorithm. Let us consider a graph G = (V, E), where V is a set of cities and E is a set of weighted edges. An edge e(u, v) represents that vertices u and v are connected.
Are there still traveling salesman?
Now, there are more customers than before with more income and more buying impulses and, as a result, there are now more traveling salesmen, an estimated one million according to the Bureau of Labor Statistics. Now, they are many and now they are not much different from the practitioners of other pursuits.
How many possible routes are possible in a TSP with 4 cities?
A four city tour has six possible routes (3 x 2 x 1)/2=3, whilst a five city tour has 12 possible distinct tours (4 x 3 x 2 x 1)/2=12.
How can we solve travel salesman problem using branch and bound?
Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible tour that visits every city exactly once and returns to the starting point. For example, consider the graph shown in figure on right side.
Which of the following is true about travelling salesman problem?
The only known way to verify that a provided solution is the shortest possible solution is to actually solve the entire TSP. Since it takes exponential time to solve NP, the solution cannot be checked in the real polynomial time. Hence, this problem is NP-hard, but not in NP.
Why is TSP so hard?
In fact, TSP belongs to the class of combinatorial optimization problems known as NP-complete. This means that TSP is classified as NP-hard because it has no “quick” solution and the complexity of calculating the best route will increase when you add more destinations to the problem.
Which algorithm is used for Travelling salesman problem?
The water flow-like algorithm (WFA) is a relatively new metaheuristic that performs well on the object grouping problem encountered in combinatorial optimization. This paper presents a WFA for solving the travelling salesman problem (TSP) as a graph-based problem.
What is TSP in AI?
The Traveling Salesman Problem (TSP) is a famous challenge in computer science and operations research. A new research competition 'AI for TSP' aims to find new solutions. The Traveling Salesman Problem (TSP) is a famous challenge in computer science and operations research.
How can we solve travel salesman problem using branch and bound?
Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible tour that visits every city exactly once and returns to the starting point. For example, consider the graph shown in figure on right side.
What is the meaning of Travelling salesman?
Definition of traveling salesman : a traveling representative of a business concern who solicits orders usually in an assigned territory.
What is travelling salesman problem in operational research?
The 'Travelling salesman problem' is very similar to the assignment problem except that in the former, there are additional restrictions that a salesman starts from his city, visits each city once and returns to his home city, so that the total distance (cost or time) is minimum.
What is a traveling salesperson called?
A travelling salesman is a travelling door-to-door seller of goods, also known as a peddler.
What is the traveling salesman problem?
The traveling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimiza tion, important in theoretical computer science and operations research .
Who is the author of Pursuit of the Traveling Salesman?
Cook, William (2012). In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press. ISBN 9780691152707.
How does TSP affect performance?
The TSP, in particular the Euclidean variant of the problem, has attracted the attention of researchers in cognitive psychology. It has been observed that humans are able to produce near-optimal solutions quickly, in a close-to-linear fashion, with performance that ranges from 1% less efficient for graphs with 10-20 nodes, and 11% less efficient for graphs with 120 nodes. The apparent ease with which humans accurately generate near-optimal solutions to the problem has led researchers to hypothesize that humans use one or more heuristics, with the two most popular theories arguably being the convex-hull hypothesis and the crossing-avoidance heuristic. However, additional evidence suggests that human performance is quite varied, and individual differences as well as graph geometry appear to affect performance in the task. Nevertheless, results suggest that computer performance on the TSP may be improved by understanding and emulating the methods used by humans for these problems, and have also led to new insights into the mechanisms of human thought. The first issue of the Journal of Problem Solving was devoted to the topic of human performance on TSP, and a 2011 review listed dozens of papers on the subject.
What is the analogous problem in geometric measure theory?
There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is , when is there a curve with finite length that visits every point in E )? This problem is known as the analyst's travelling salesman problem .
What was the new approach to solving problems in the 1960s?
In the 1960s, however, a new approach was created, that instead of seeking optimal solutions would produce a solution whose length is provably bounded by a multiple of the optimal length, and in doing so would create lower bounds for the problem; these lower bounds would then be used with branch and bound approaches .
Is bottleneck travel NP hard?
The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FP NP; see function problem ), and the decision problem version ("given the costs and a number x, decide whether there is a round-trip route cheaper than x ") is NP-complete. The bottleneck traveling salesman problem is also NP-hard. The problem remains NP-hard even for the case when the cities are in the plane with Euclidean distances, as well as in a number of other restrictive cases. Removing the condition of visiting each city "only once" does not remove the NP-hardness, since in the planar case there is an optimal tour that visits each city only once (otherwise, by the triangle inequality, a shortcut that skips a repeated visit would not increase the tour length).
Does returning to the starting city change the computational complexity of the problem?
The requirement of returning to the starting city does not change the computational complexity of the problem, see Hamiltonian path problem.
Why does the traveling salesman problem clog the work?
But the traveling salesman problem clogs the works because the number of calculations required is so huge. Adding more points on the map only increases the complexity. (Honestly, this news itself underlines how complicated the problem is: It’s major news to be able to solve just 22 cities instead of just 16!)
What are the potential applications of a more powerful salesman solver?
The potential applications of a more powerful salesman solver are myriad . The abstract problem is infamous because it’s so widely studied and difficult, but its roots are still as an abstraction of a real person’s dilemma: How do I do my job the most efficient way? Every day, taxi and Uber drivers must consider the best route to find the most passengers. Delivery drivers must arrange their addresses in an efficient way. And these applications don’t just involve minimizing distance—fresh food or the value of a fare add even more complexity.
How many cities can scientists solve in Japan?
The previous standard for instant solving was 16 “cities,” and these scientists have used a new kind of processor to solve 22 cities.
Why do commercial solvers use heuristics?
Thus, commercial solvers usually use heuristics—these are like shortcuts for our brain, eliminating a lot of math or calculations for a quick and easy solution—due to the frequency and size of real world VRPs they need to solve.
How to solve TSP?
To solve the TSP using the Branch and Bound method, you must choose a start node and then set bound to a very large value (let’s say infinity). Select the cheapest arch between the unvisited and current node and then add the distance to the current distance. Repeat the process while the current distance is less then the bound.
How to solve TSP using brute force?
To solve the TSP using the Brute-Force approach, you must calculate the total number of routes and then draw and list all the possible routes. Calculate the distance of each route and then choose the shortest one—this is the optimal solution.
Why do academic solvers take so long to compute optimal solutions?
That’s because academic solvers strive for perfection and thus take a long time to compute the optimal solutions – hours, days, and sometimes years. If a delivery business needs to plan daily routes, they need a route solution within a matter of minutes.
What is the last mile delivery?
Last mile delivery refers to the movement of goods from a transportation hub, such as a depot or a warehouse, to the end customer’s choice of delivery. Last mile delivery is the leading cost driver in the supply chain. Companies usually shoulder some of the costs to better compete in the market. In fact, a last mile delivery costs the company an average of $10.1, but the customer only pays an average of $8.08. This is the reason why businesses strive to minimize the cost of last mile delivery.
Is TSP practical?
While academic solutions to TSP and VRP aim to provide the optimal solution to these NP-hard problems, many of them aren’t practical when solving real world problems, especially when it comes to solving last mile logistical challenges.
How many times should a traveler visit each city?
A traveler needs to visit all the cities from a list, where distances between all the cities are known and each city should be visited just once . What is the shortest possible route that he visits each city exactly once and returns to the origin city?
What is the most notorious computational problem?
Travelling salesman problem is the most notorious computational problem. We can use brute-force approach to evaluate every possible tour and select the best one. For n number of vertices in a graph, there are (n - 1)! number of possibilities.
Is a partial tour a sub-problem?
Suppose we have started at city 1 and after visiting some cities now we are in city j. Hence, this is a partial tour . We certainly need to know j, since this will determine which cities are most convenient to visit next. We also need to know all the cities visited so far, so that we don't repeat any of them. Hence, this is an appropriate sub-problem.
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What is the Traveling Salesman problem?
The Traveling Salesman Problem (TSP) is believed to be an intractable problem and have no practically efficient algorithm to solve it. The intrinsic 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 solutions in the solution space is so large as to forbid an exhaustive search for the optimal solutions. The seemingly “limitless” increase of computational power will not resolve its genuine intractability. Do we need to explore all the possibilities in the solution space to find the optimal solutions? This chapter offers a novel perspective trying to overcome the combinatorial complexity of the TSP. When we design an algorithm to solve an optimization problem, we usually ask the critical question: “How can we find all exact optimal solutions and how do we know that they are optimal in the solution space?” This chapter introduces the Attractor-Based Search System (ABSS) that is specifically designed for the TSP. This chapter explains how the ABSS answer this critical question. The computing complexity of the ABSS is also discussed.
When is a local search system able to find the set of the globally optimal tours?
It is easily verified that under certain conditons, a local search system is able to find the set of the globally optimal tours S ∗ when the number of search trajectories is unlimited , i.e.
How does the search trajectory change?
A search trajectory in a local search system changes its edge configuration during the search according to the objective function f s and its neighborhood structure. The matrix E can follow the “footprints” of search trajectories to capture the dynamics of the local search system. When all search trajectories reach their end points – the locally optimal tours, the edge configuration of the matrix E will become fixed, which is the edge configuration of the solution attractor A. This fixed edge configuration contains two groups of edges: the edges that are not hit by any of locally optimal tours (non-hit edges) and the edges that are hit by at least one of the locally optimal tours (hit edges). Figure 6 shows the edge grouping in the edge configuration of E when all search trajectories stop at their final points.
Why is the TSP intrinsically difficult?
The intrinsic difficulty of the TSP is that the solution space increases exponentially as the problem size increases, which makes the exhaustive search infeasible. When a TSP instance is large, the number of possible tours in the solution space is so large to forbid an exhaustive search for the optimal tours.
Why is the edge matrix E not included in the TSP definition?
Usually the edge matrix E is not necessary to be included in the TSP definition because the TSP is a complete graph. However, the edge matrix E is an effective data structure that can help us understand the search behavior of a local search system. General local search algorithm may not require much problem-specific knowledge in order to generate good solutions. However, it may be unreasonable to expect a search algorithm to be able to solve any problem without taking into account the data structure and properties of the problem at hand.
How does heuristic local search work?
In a heuristic local search algorithm, there is a great variety of ways to construct initial tour, choose candidate moves, and define criteria for accepting candidate moves . Most heuristic local search algorithms are based on randomization. In this sense, a heuristic local search algoorithm is a randomized system. There are no two search trajectories that are exactly alike in such a search system. Different search trajectories explore different regions of the solution space and stop at different final points. Therefore, local optimality depends on the initial points, the neighborhood function, randomness in the search process, and time spent on search process. On the other hand, however, a local search algorithm essentially is deterministic and not random in nature. If we observe the motion of all search trajectories, we will see that the search trajectories go towards the same direction, move closer to each other, and eventually converge into a small region in the solution space.
Should search systems be deterministic?
The search system should be deterministic and have a rigorous guarantee for finding all globally optimal solutions without excessive computational burden.
What is the Travelling Salesman Problem (TSP)?
The Travelling Salesman Problem (TSP) is a combinatorial problem that deals with finding the shortest and most efficient route to follow for reaching a list of specific destinations.
Optimization Challenge Posed By The Travelling Salesman Problem
The Traveling Salesman Problem is an optimization problem studied in graph theory and the field of operations research. In this optimization problem, the nodes or cities on the graph are all connected using direct edges or routes. The weight of each edge indicates the distance covered on the route between two cities.
Which is the Most Optimal Solution to Travelling Salesman Problem?
These are some of the near-optimal solutions to find the shortest route to a combinatorial optimization problem.
Some Other Optimal Solutions to Traveling Salesman Problem
Multi-Agent System: Involves distributing cities into m groups. Then assign a single agent to discover the shortest path, covering all the cities in the assigned group.
A New Method Gave Birth to a New Algorithm
There hasn’t been a method which can find the shortest trip. But the method has made it possible to find solutions that are almost as good. This was done by Christofides algorithm.
Real-life Travelling Salesman Problem Uses Route Optimization
Academic solutions do try really hard to offer possible solutions to these NP-hard problems. But most of them are not very practical when it comes to solving real-world TSP problems. Solving last-mile logistical challenges aims at getting instant results, even if they are not entirely accurate.
Overview
Computing a solution
The traditional lines of attack for the NP-hard problems are the following:
• Devising exact algorithms, which work reasonably fast only for small problem sizes.
• Devising "suboptimal" or heuristic algorithms, i.e., algorithms that deliver approximated solutions in a reasonable time.
History
The origins of the travelling salesman problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment.
The travelling salesman problem was mathematically formulated in the 19th century by the Irish mathematician William Rowan Hamilton and by the British …
Description
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 a complete graph (i.e., each pair of vertices is connected by an edge). If no path exists bet…
Integer linear programming formulations
The TSP can be formulated as an integer linear program. Several formulations are known. Two notable formulations are the Miller–Tucker–Zemlin (MTZ) formulation and the Dantzig–Fulkerson–Johnson (DFJ) formulation. The DFJ formulation is stronger, though the MTZ formulation is still useful in certain settings.
Common to both these formulations is that one labels the cities with the numbers and takes to b…
Special cases
In the metric TSP, also known as delta-TSP or Δ-TSP, the intercity distances satisfy the triangle inequality.
A very natural restriction of the TSP is to require that the distances between cities form a metric to satisfy the triangle inequality; that is the direct connection from A to B is never farther than the route via intermediate C:
Computational complexity
The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FP ; see function problem), and the decision problem version ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-complete. The bottleneck travelling salesman problem is also NP-hard. The problem remains NP-hard even for the case when the cities are in the plane with Euclidean distances, as well as in a number of other restricti…
Human and animal performance
The TSP, in particular the Euclidean variant of the problem, has attracted the attention of researchers in cognitive psychology. It has been observed that humans are able to produce near-optimal solutions quickly, in a close-to-linear fashion, with performance that ranges from 1% less efficient for graphs with 10-20 nodes, and 11% less efficient for graphs with 120 nodes. The apparent ease with which humans accurately generate near-optimal solutions to the problem ha…